# The Art of Scientific Discovery β€” full corpus Each page below begins with its canonical URL followed by its original Markdown, OKF frontmatter included. Pages are grouped by section in navigation order; the linked outline is at https://tyson-swetnam.github.io/aosd/llms.txt. Relative links have been rewritten to absolute URLs that point at each linked page's Markdown twin (its URL plus `index.md`), so you can traverse the bundle without leaving Markdown. ---8<--- https://tyson-swetnam.github.io/aosd/section1/ --- title: "Section 1: Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes" description: "The opening six sessions of Winfree's course: two contrasting challenges, facts before explanations of facts, the N-ray affair and pathological science, ways to check for errors and hidden assumptions, with every problem and reading the schedule assigns." type: Lesson tags: [course, student-facing, section-1, error-checking, false-assumptions, detecting-nonsense] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Section 1: Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *The introductory meeting and five sessions on catching your own errors before they catch you* ## Overview The syllabus calls its exercises "practice scrimmages" in, among other things, "spotting and taking advantage of your own mistakes, and especially of learning a positive attitude toward mistakes, because they are often the most available doors to discovery." Section 1 builds that attitude. It moves in three steps. Sessions 1 and 2 set up "two contrasting challenges": a small puzzle you can test with your hands, and a large question that may have no checkable answer. Sessions 3 to 5 practise separating what you were given, or saw, from what you supplied or imagined. Session 6 names the methods: "Some ways to check for errors", "hidden assumptions", and what is "understand"? The syllabus says the exercises "depend as little as possible on knowledge of any particular subject area". It adds: "The purpose of the puzzles (many of them silly) is to slow you down for a few minutes so you can examine the working of your own mind." Getting one wrong is expected, and worth writing down. ## The sessions ### Session 01: Demonstrating the need **Readings due:** introduction and handouts; Adams, preface and Chapter 1. The [Triangle Problem](https://tyson-swetnam.github.io/aosd/problems/triangle-problem/index.md) comes first, under the label "Demonstrating the need". Which triangle question Winfree asked is not recorded; its page offers two candidates, each a familiar question whose confident answer fails. Then comes the "First of two contrasting challenges", [13 Nails](https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/index.md), or "alternatively, the celts problem": balance thirteen nails on the head of one, or explain a top that reverses its own spin. ### Session 02: The second contrasting challenge **Readings due:** Adams, preface and Chapter 1; Platt, *The Art of Creative Thinking*; Feynman, *Cargo Cult Science*. The "Second of two contrasting challenges" is [Conscious Machines](https://tyson-swetnam.github.io/aosd/problems/conscious-machines/index.md): could a machine be conscious, and how would you tell? A wrong idea about the nails falls on the table; here nothing falls. Platt's essay is the source of the daily [GamesWorth](https://tyson-swetnam.github.io/aosd/gamesworth/index.md) habit. The syllabus explains the name: "the allusion is to how much thought it takes to play one game of serious chess". The syllabus gives only this challenge's title, so its page reconstructs the wording. ### Session 03: A solo problem **Readings due:** Judson, Chapter 1: *The Rage to Know*. Discuss [Square Windows](https://tyson-swetnam.github.io/aosd/problems/square-windows/index.md), which the syllabus marks as a "solo problem". Read as the Carroll and Dudeney puzzle its page identifies, its only obstacle is an assumption you were never given. ### Session 04: Facts before explanations of facts **Readings due:** none listed. [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md): "facts before explanations of facts", Fontenelle's story of books explaining a boy's gold tooth before anyone checked it. [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md), a parable about two retired scientists studying a car factory from a hill. [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md): "distinguishing things we know vs only imagine". ### Session 05: Pathological science **Readings due:** [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md); Langmuir, *Pathological Science*. Trained physicists confirmed a radiation that was not there until R. W. Wood's tests exposed it. Discuss the [Phone Cord Problem](https://tyson-swetnam.github.io/aosd/problems/phone-cord-problem/index.md), a coiled cord that tangles though neither plug ever turns, and [Stockholm Restrooms](https://tyson-swetnam.github.io/aosd/problems/stockholm-restrooms/index.md), a travel oddity to describe before you explain it. The syllabus gives only their names, so both pages reconstruct the problems. ### Session 06: Ways to check for errors **Readings due:** packet of readings on valuing mistakes; Ehrlich, Chapter 1 (Introduction). The syllabus lists "Some ways to check for errors", "hidden assumptions" and what is "understand"?, then [Evaporated Gold](https://tyson-swetnam.github.io/aosd/problems/evaporated-gold/index.md). It also starts the first group effort in the schedule, [Collective Reproduction](https://tyson-swetnam.github.io/aosd/problems/collective-reproduction/index.md). No statement of either problem survives, so both pages offer reconstructions. The schedule does not list Collective Reproduction again, so how long it ran is not recorded. ## Key ideas ### Mistakes are data The syllabus asks you to write down "your approaches, your lucky insights, how you got into and out of blind alleys". A recorded mistake can be studied later, in a morning-after on the notebook's left-hand pages. ### Facts before explanations of facts The mind jumps to the cause and skips the fact; the golden tooth, the N-rays and the phone cord show that habit on three scales. Before explaining anything, ask: has this been observed, by whom, and could a simple check settle it? Then check the answer by a second route that shares no arithmetic with the first, as the reconstructed [Evaporated Gold](https://tyson-swetnam.github.io/aosd/problems/evaporated-gold/index.md) asks. ### Know versus imagine Many puzzles are hard only because you added something to them: a condition nobody stated, or a picture of the situation you imagined instead of checking. When you are stuck, list the conditions you were given. Then list the ones you supplied yourself. ### Symptoms of pathological science Langmuir described honest scientists fooled by their own expectations. His warning signs include effects that stay near the limit of detection, claims of great accuracy, fantastic theories contrary to experience, and criticisms met by ad hoc excuses. Wood did not argue with the theory of N-rays: he changed something the observers could not see and watched whether their reports changed. ## GamesWorth focus for this section - Before each problem, list your assumptions, marked as something you *know*, *assume*, or could *look up*. - On 13 Nails, log each failed arrangement and the belief it disproved. - On Conscious Machines, write down what observation would change your mind. If none would, say so. - On Golden Tooth, Bookworm's Journey and the Phone Cord Problem, keep two columns: seen and inferred. - On Evaporated Gold, reach the answer by two independent routes and record whether they agreed. - Do a morning-after on the Triangle Problem: what made the confident answer feel safe? ## Readings for this section - **James L. Adams**, *Conceptual Blockbusting*, preface and Chapter 1 (sessions 1 and 2). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#adams); [publisher](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674059/?lens=basic-books){target=_blank} πŸ”’; 1974 first edition at the [Internet Archive](https://archive.org/details/conceptualblock00adam){target=_blank} πŸ”“ *(borrow)* - **John R. Platt**, "The Art of Creative Thinking", in *The Excitement of Science* (1962) (session 2). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#platt-creative-thinking); [Internet Archive](https://archive.org/details/excitementofscie0000john){target=_blank} πŸ”“ *(borrow)* - **Richard P. Feynman**, *Cargo Cult Science* (1974) (session 2). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#feynman-cargo-cult-science); [Caltech](https://resolver.caltech.edu/CaltechES:37.7.CargoCult){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions*, Chapter 1: *The Rage to Know* (session 3). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#judson); [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* - **N-Rays** (session 5). Which text Winfree handed out is not recorded. [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#n-rays); the [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md) page quotes Wood's 1904 letter; [overview](https://en.wikipedia.org/wiki/N-ray){target=_blank} πŸ”“ - **Irving Langmuir**, *Pathological Science* (1953 colloquium) (session 5). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#langmuir); [transcript](https://www.cs.princeton.edu/~ken/Langmuir/langmuir.htm){target=_blank} πŸ”“ - **Packet of readings on valuing mistakes** (session 6). Its contents are not recorded. [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#valuing-mistakes) - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True*, Chapter 1, Introduction (session 6). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ The session 4 problem [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md) is also a text to read: William T. Sullivan's parable, at the [Sullivan Lab, UC Santa Cruz](https://sullivanlab.sites.ucsc.edu/salvation-of-doug/){target=_blank} πŸ”“. ## Problems in this section | Session | Problem | Kind | What it trains | | :-- | :-- | :-- | :-- | | 01 | [Triangle Problem](https://tyson-swetnam.github.io/aosd/problems/triangle-problem/index.md) | puzzle | Doubting a confident, unanimous answer (editors' reconstruction) | | 01 | [13 Nails](https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/index.md) | puzzle | Exposing unspoken assumptions with your hands | | 02 | [Conscious Machines](https://tyson-swetnam.github.io/aosd/problems/conscious-machines/index.md) | thought experiment | Noticing when a question cannot be checked | | 03 | [Square Windows](https://tyson-swetnam.github.io/aosd/problems/square-windows/index.md) | puzzle | Telling given conditions from imported ones | | 04 | [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md) | case study | Facts before explanations of facts | | 04 | [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md) | case study | Separating observation from inference | | 04 | [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md) | puzzle | Distinguishing things we know vs only imagine | | 05 | [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md) | case study | Detecting nonsense by blind tests | | 05 | [Phone Cord Problem](https://tyson-swetnam.github.io/aosd/problems/phone-cord-problem/index.md) | puzzle | Looking closely before explaining | | 05 | [Stockholm Restrooms](https://tyson-swetnam.github.io/aosd/problems/stockholm-restrooms/index.md) | discussion | Recording what you saw before why (editors' reconstruction) | | 06 | [Evaporated Gold](https://tyson-swetnam.github.io/aosd/problems/evaporated-gold/index.md) | puzzle | Cross-checking an estimate by independent routes (editors' reconstruction) | | 06 | [Collective Reproduction](https://tyson-swetnam.github.io/aosd/problems/collective-reproduction/index.md) | puzzle | Checking, as a group, the hidden assumptions in a plausible story (editors' reconstruction) | See the [problem index](https://tyson-swetnam.github.io/aosd/problems/index.md) for every problem in the course. --- *Remember: the goal is not to avoid mistakes, but to learn how to make them productively and recover from them creatively.* ---8<--- https://tyson-swetnam.github.io/aosd/section2/ --- title: "Section 2: Creative Blocks" description: "Sessions 7 to 12 of Winfree's course: Adams's perceptual, emotional, cultural and intellectual blocks and his blockbusters, practised on trap puzzles, a cross-checking exercise and the start of an outdoor lab, with the readings each session assigns." type: Lesson tags: [course, student-facing, section-2, creative-blocks, conceptual-blockbusting] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Section 2: Creative Blocks Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *Six sessions on the blocks that stop ideas before they start* ## Overview Section 2, which the [syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md) calls simply "Creative Blocks", asks why good ideas fail to arrive at all. Its backbone is James Adams's *Conceptual Blockbusting*, one of the course's two required books. Sessions 7 to 10 take his chapters on perceptual, emotional, cultural and intellectual blocks in turn; session 11 takes his chapter on blockbusters. Some people suppose problem-solving cannot be taught: you are born with the ability or you are not. Winfree disagreed: "I think we are all born with it and mostly lose it during and because of schooling and the general intimidation that comes with any competitive society." A block is one form that loss takes. Several puzzles here are traps whose obstacle is a rule you added yourself. Getting stuck is part of the design: the course aims for a feeling of "disorientation and hopeless lost-ness", Winfree wrote, so that you "learn not to despair in paralysis but instead focus on method, generate several alternative guesses, and test them for workability." The section ends with an exercise in cross-checking and the start of an outdoor lab that leads into Section 3. ## The sessions ### Session 07: Perceptual blocks **Readings due:** Adams Chapter 2: Perceptual blocks. Discuss [Weird Organism](https://tyson-swetnam.github.io/aosd/problems/weird-organism/index.md), reconstructed as a familiar organism described so plainly that a stereotype hides it. Discuss [Rearranged Triangle](https://tyson-swetnam.github.io/aosd/problems/rearranged-triangle/index.md): the same four pieces fill a triangle, then leave one square over. Start the [Dominoes (group lab)](https://tyson-swetnam.github.io/aosd/problems/dominoes-lab/index.md), reconstructed as the cut-chessboard puzzle done with plain blocks. ### Session 08: Emotional blocks **Readings due:** [The Barometer Story](https://tyson-swetnam.github.io/aosd/problems/barometer-story/index.md) (Calandra) and Adams Chapter 3: Emotional blocks. Discuss [Telltale Number](https://tyson-swetnam.github.io/aosd/problems/telltale-number/index.md), a ten-digit number that describes itself. Then "About assumptions and [Tying Knots](https://tyson-swetnam.github.io/aosd/problems/tying-knots/index.md)". The puzzle as reconstructed: tie a knot in a rope while holding one end in each hand, without letting go. ### Session 09: Cultural blocks **Readings due:** Adams Chapter 4: Cultural blocks; Platt, *Diversity*. Discuss [Mercury's Mysterious Hidden Hemisphere](https://tyson-swetnam.github.io/aosd/problems/mercurys-hidden-hemisphere/index.md): a "fact" repeated in textbooks for 75 years until radar tested it in 1965. ### Session 10: Intellectual blocks **Readings due:** Adams Chapter 5: Intellectual blocks. [Taboo Questions](https://tyson-swetnam.github.io/aosd/problems/taboo-questions/index.md): the questions a scientist is not supposed to ask. [Paths Through Mazes](https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/index.md): count the ways to read ABRACADABRA down a diamond of letters, then check the count another way. ### Session 11: Blockbusters **Readings due:** Adams Chapter 7: Blockbusters. [Walking Through Walls](https://tyson-swetnam.github.io/aosd/problems/walking-through-walls/index.md), the only problem listed that day: a drill in restating the problem before solving it. ### Session 12: Cross-checks and unfashionable pursuits **Readings due:** Dyson, *Unfashionable Pursuits*; a one-page biography of Mario Capecchi; Narlikar on venture funding. [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md): "like a jig-saw puzzle of cross-checks". Find the formula for 1 + 2 + ... + *n* by as many independent routes as you can. Then the [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md) (lab), reconstructed as a group log of a real street crossing. It begins here and continues in session 13, the first of [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), where the class discusses its "observations outdoors"; it is listed with that section's problems. ## Key ideas ### Adams's four kinds of block - **Perceptual blocks** stop you seeing the problem, or the information that would solve it: seeing what you expect (stereotyping), drawing the problem's boundaries too tightly, never changing viewpoint. Rearranged Triangle and Weird Organism are built to be misperceived. - **Emotional blocks** include fear of looking foolish, dislike of ambiguity, and judging ideas before you have finished producing them. The syllabus asks for the opposite in class: "braving social opprobrium by blurting out nutty ideas, and risking devastating counter-attack by publicly objecting to nonsense blurted by others." - **Cultural blocks** come from the society around you: taboos, the belief that play wastes time, the belief that tradition beats change. A claim every textbook repeats, like Mercury's frozen hemisphere, is hard to question. - **Intellectual blocks** are about your tools: thinking in words when the problem needs a sketch or a count, reusing the same few strategies, working with missing or wrong information. Paths Through Mazes rewards whoever stops listing paths one by one. ### Hidden rules and restated problems Several of these puzzles cannot be solved until you notice a rule you invented. Tying Knots seems impossible because of how you pick up the rope. Telltale Number collapses once you notice a fact the puzzle never states. In The Barometer Story, an instructor wanted to give zero to a correct answer that was not the one he had in mind. Each time, list the rules you are obeying and check which were actually given. Walking Through Walls, set for the blockbusters session and built on Adams's door example, attacks the same trap from the other side: ask for "a better door" and you get a better slab on hinges; ask for a better way through a wall and the answers multiply. ### Cross-checks and unusual routes One derivation is only a claim; several independent routes that agree, "like a jig-saw puzzle of cross-checks", are evidence. The final exam asks for exactly this: "exhibit as many distinct approaches as you can, and as many cross-checking distinct solutions as you can." The session 12 readings push from another direction: Dyson on unfashionable pursuits and Narlikar on venture funding both concern the value of a route few others are taking. ## GamesWorth focus for this section - Before you try Tying Knots, Telltale Number or Rearranged Triangle, write down every rule you think you must obey. Afterwards, mark which rules were really stated. - When stuck, name the block: perceptual, emotional, cultural or intellectual. Tally which one catches you most often. - Solve Paths Through Mazes and Sums of Integers at least two independent ways, and record where the routes agree or expose a mistake. - For Walking Through Walls, write three ever-broader restatements of the problem, and list every idea for each before judging any of them. - In a morning-after entry on a left-hand page, ask which block slowed you down on a Section 1 problem you struggled with. - At the crosswalk, write down what happens before you try to explain it. Winfree's later *Adventures in Discovery* columns ([introduction, archived](https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html){target=_blank} πŸ”“) collect puzzles he worked through as his own daily GamesWorths. See also the [GamesWorth page](https://tyson-swetnam.github.io/aosd/gamesworth/index.md). ## Readings for this section - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, Chapter 2: Perceptual blocks (session 7), Chapter 3: Emotional blocks (session 8), Chapter 4: Cultural blocks (session 9), Chapter 5: Intellectual blocks (session 10) and Chapter 7: Blockbusters (session 11). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#adams); [publisher](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674059/?lens=basic-books){target=_blank} πŸ”’; 1980 second edition at the [Internet Archive](https://archive.org/details/conceptualblockb0000adam){target=_blank} πŸ”“ *(borrow)*. Chapter numbers may differ slightly between editions. - **Alexander Calandra**, *The Barometer Story* ("Angels on the Head of a Pin"), *Saturday Review*, 21 December 1968 (session 8). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#barometer-story); [text](https://www.stephenhicks.org/2022/07/10/angels-on-the-head-of-a-pin-by-alexander-calandra/){target=_blank} πŸ”“ - **John R. Platt**, *Diversity*, *Science* 154, 1132-1139 (1966) (session 9). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#platt-diversity); [Science](https://doi.org/10.1126/science.154.3753.1132){target=_blank} πŸ”’ - **Freeman Dyson**, *Unfashionable Pursuits*, *The Mathematical Intelligencer* 5(3), 47-54 (1983) (session 12). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#dyson); [journal](https://doi.org/10.1007/BF03026573){target=_blank} πŸ”’ - **Capecchi one-page biography** (session 12). The syllabus does not say which one. [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#capecchi); Capecchi's later [Nobel Prize autobiography](https://www.nobelprize.org/prizes/medicine/2007/capecchi/biographical/){target=_blank} πŸ”“ (2007) is a free account of his life. - **Jayant V. Narlikar**, *Venture funding for new ideas*, *Nature* 404, 707 (2000) (session 12). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#narlikar); [Nature](https://doi.org/10.1038/35008158){target=_blank} πŸ”’ ## Problems in this section | Session | Problem | Kind | What it trains | | :-- | :-- | :-- | :-- | | 07 | [Weird Organism](https://tyson-swetnam.github.io/aosd/problems/weird-organism/index.md) | puzzle | Seeing past a stereotype (editors' reconstruction) | | 07 | [Rearranged Triangle](https://tyson-swetnam.github.io/aosd/problems/rearranged-triangle/index.md) | puzzle | Distrusting what the eye reports | | 07 | [Dominoes (group lab)](https://tyson-swetnam.github.io/aosd/problems/dominoes-lab/index.md) | lab | Pooling failed attempts until the reason for failure shows (editors' reconstruction) | | 08 | [The Barometer Story](https://tyson-swetnam.github.io/aosd/problems/barometer-story/index.md) | case study | Noticing the unstated expectation behind a question | | 08 | [Telltale Number](https://tyson-swetnam.github.io/aosd/problems/telltale-number/index.md) | puzzle | Finding the fact a puzzle never states | | 08 | [Tying Knots](https://tyson-swetnam.github.io/aosd/problems/tying-knots/index.md) | puzzle | Separating the rules given from the rules assumed | | 09 | [Mercury's Mysterious Hidden Hemisphere](https://tyson-swetnam.github.io/aosd/problems/mercurys-hidden-hemisphere/index.md) | case study | Questioning what every textbook repeats | | 10 | [Taboo Questions](https://tyson-swetnam.github.io/aosd/problems/taboo-questions/index.md) | discussion | Asking the question you were not supposed to ask | | 10 | [Paths Through Mazes](https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/index.md) | puzzle | Changing strategy, and checking a count a second way | | 11 | [Walking Through Walls](https://tyson-swetnam.github.io/aosd/problems/walking-through-walls/index.md) | thought experiment | Restating the problem before solving it | | 12 | [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md) | puzzle | Independent derivations that check one another | See the [problem index](https://tyson-swetnam.github.io/aosd/problems/index.md) for every problem in the course. --- *The goal is not to eliminate all blocks (impossible), but to recognize them when they occur and have strategies for working around them.* ---8<--- https://tyson-swetnam.github.io/aosd/section3/ --- title: "Section 3: Observations and Questions" description: "Sessions 13 to 18 of Winfree's course: pooled outdoor and chemical observations, noticing what isn't there, being fooled by your own senses, and asking the question an observation can actually answer, with the readings and fourteen linked problems." type: Lesson tags: [course, student-facing, section-3, observation, questions] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Section 3: Observations and Questions Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *Six sessions of looking hard, pooling what you see, and asking questions an observation can answer* ## Overview The syllabus names this section "Observations and Questions". Among the course's aims it lists practice in "recognizing ignorance" and "posing questions". Section 3 opens by discussing the class's outdoor observations at a pedestrian crosswalk. It then runs a chemical pattern-formation lab across three sessions. Much of this work cannot be done alone. In Winfree's words: "Class meetings will also prove essential for some problems in which no one individual can collect enough data, but if we pool data, reality will come into focus." The paper problems in between test the same habits. What have you never seen, and why did you never notice? When do your eyes deceive you? When does a careful drawing or a stack of filters overrule a confident guess? Session 04's "facts before explanations of facts" now meets real data. The syllabus gives most problems only a title. Each problem page says how sure the editors are of what Winfree meant. ## The sessions ### Session 13: Crosswalk and bridges **Readings due:** none listed. Discuss [Ant Walk](https://tyson-swetnam.github.io/aosd/problems/ant-walk/index.md) and [Seven Bridges of KΓΆnigsberg](https://tyson-swetnam.github.io/aosd/problems/seven-bridges/index.md). Seven Bridges asks whether a walk can cross every bridge exactly once. Ant Walk cannot be identified from its title; its page is the editors' reconstruction of a similar walk along wires. Discuss observations outdoors on the [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md). This lab straddles the sections: it began in session 12, the last session of [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md). Its original text is lost, so its page is also a reconstruction. Start the [Chemical Pattern-Formation Lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md), which runs through session 15. ### Session 14: What isn't there **Readings due:** Judson, Chapter 4: *Chance*; Anderson on research strategy. The lab continues with "More chemical observations". Discuss [What Isn't There (Surprisingly Hard)](https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/index.md); the parenthesis is the syllabus's own. ### Session 15: Evidence and the senses **Readings due:** Judson, Chapter 8: *Evidence*. The session takes up [Mother Nature as Magician; Hallucinations](https://tyson-swetnam.github.io/aosd/problems/mother-nature-as-magician/index.md). The chemical lab ends with "Last chemical observations". Discuss [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md), Winfree's own puzzle about the Moon inside a rainbow's colour band. ### Session 16: Hairy People, Green Stars, Escher **Readings due:** Ehrlich, Chapter 2: "More Guns Means Less Crime". Discuss [Hairy People](https://tyson-swetnam.github.io/aosd/problems/hairy-people/index.md), [Green Stars](https://tyson-swetnam.github.io/aosd/problems/green-stars/index.md) and [Escher Print Gallery](https://tyson-swetnam.github.io/aosd/problems/escher-print-gallery/index.md). ### Session 17: Zygotes and honeycombs **Readings due:** Ehrlich, Chapter 3: "AIDS Is Not Caused by HIV". Discuss [Zygotes](https://tyson-swetnam.github.io/aosd/problems/zygotes/index.md) and [Martian HoneyCombs](https://tyson-swetnam.github.io/aosd/problems/martian-honeycombs/index.md). Neither can be identified from its title, so both pages are the editors' reconstructions. ### Session 18: Cevians and filters **Readings due:** Ehrlich, Chapter 4: "Sun Exposure Is Beneficial". Discuss [Cevians](https://tyson-swetnam.github.io/aosd/problems/cevians/index.md) and [Superposed Filters](https://tyson-swetnam.github.io/aosd/problems/superposed-filters/index.md). ## Key ideas ### Observation is not interpretation A notebook entry should separate what you saw from what you concluded. "Four people stepped out before the car had stopped" is an observation. "People here trust drivers" is an inference. The crosswalk and the chemical dish both reward writing the first kind of sentence before the second. That is session 04's "distinguishing things we know vs only imagine", applied to data you collected yourself. ### Absences are data Some absences are obvious and some are not. Winfree made this point in a later column, [SAS08](https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html){target=_blank} πŸ”“: some things that never happen are easy to notice, and others are hard. What Isn't There, Green Stars and Rainbow Moon each start from something nobody sees. The work is to turn "never seen" into a prediction that could fail. ### Honest senses, false conclusions In an illusion, the eye reports faithfully and the inference goes wrong. In a hallucination, the percept has no outside cause at all. Escher's loops add a third trap: every local piece passes inspection, and the error shows only when you follow the whole. ### Questions that observations can answer These are the editors' categories, not Winfree's. They are useful prompts for these problems: - **Descriptive**: What exactly happens, in what order, and how often? (crosswalk, chemical dish) - **Absence**: What never happens here, and would I have noticed if it did? (What Isn't There, Green Stars) - **Counting and constraint**: What property decides whether a route or an answer can exist at all? (Seven Bridges, Ant Walk, Hairy People) - **Test**: What would I see if my first guess were wrong? (Cevians, Superposed Filters, Rainbow Moon) ### Pooled observation One observer at a crossing sees one corner for one stretch of time. A class sees many. Pooling only works if each record is legible to others: time, place, what was counted, and what was guessed. ## GamesWorth focus for this section - Before explaining the crosswalk or the chemical dish, write a plain log of what happened, with times. - Over a week, list things that never happen, then mark which ones you would actually have noticed. - For Rainbow Moon, Cevians and Superposed Filters, write your prediction down before you check it. Record how far off it was. - For Hairy People and Seven Bridges, note the moment you stopped trying cases and started counting. - For the problems whose original statements are lost (Ant Walk, Pedestrian Crosswalk Mystery, Zygotes, Martian HoneyCombs), list every assumption your version of the problem depends on. ## Readings for this section - **Horace Freeland Judson**, *The Search for Solutions*, Chapter 4: *Chance* (session 14). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#judson); [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* - **Anderson on research strategy** (session 14). The syllabus names no title; the editors identify it as Philip W. Anderson, *More Is Different*, *Science* 177, 393-396 (1972). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#anderson); [PDF](https://www.rpgroup.caltech.edu/embl_pboc_2023/assets/pdfs/anderson1972.pdf){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions*, Chapter 8: *Evidence* (session 15). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#judson); [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True*, Chapter 2: "More Guns Means Less Crime" (session 16). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 3: "AIDS Is Not Caused by HIV" (session 17). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 4: "Sun Exposure Is Beneficial" (session 18). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ ## Problems in this section | Session | Problem | Kind | What it trains | | :-- | :-- | :-- | :-- | | 12-13 | [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md) | lab | Logging a real street scene before explaining it; pooling the class's records (editors' reconstruction) | | 13 | [Ant Walk](https://tyson-swetnam.github.io/aosd/problems/ant-walk/index.md) | puzzle | Counting wires at each corner instead of tracing routes (editors' reconstruction) | | 13 | [Seven Bridges of KΓΆnigsberg](https://tyson-swetnam.github.io/aosd/problems/seven-bridges/index.md) | puzzle | Keeping only the connections and asking what decides whether a walk exists | | 13-15 | [Chemical Pattern-Formation Lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md) | lab | Recording chemical waves over three sessions before explaining them | | 14 | [What Isn't There (Surprisingly Hard)](https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/index.md) | discussion | Noticing absences and turning them into testable expectations | | 15 | [Mother Nature as Magician; Hallucinations](https://tyson-swetnam.github.io/aosd/problems/mother-nature-as-magician/index.md) | discussion | Separating honest perception from false inference | | 15 | [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md) | puzzle | Reading a failed prediction as evidence about your assumptions | | 16 | [Hairy People](https://tyson-swetnam.github.io/aosd/problems/hairy-people/index.md) | puzzle | Proving what nobody could observe, and checking the upper bound it rests on | | 16 | [Green Stars](https://tyson-swetnam.github.io/aosd/problems/green-stars/index.md) | puzzle | Asking whether a missing colour lies in the stars or in the eye | | 16 | [Escher Print Gallery](https://tyson-swetnam.github.io/aosd/problems/escher-print-gallery/index.md) | puzzle | Checking the whole loop, not just each local step | | 17 | [Zygotes](https://tyson-swetnam.github.io/aosd/problems/zygotes/index.md) | puzzle | Counting the unseen from the seen, and naming the assumptions (editors' reconstruction) | | 17 | [Martian HoneyCombs](https://tyson-swetnam.github.io/aosd/problems/martian-honeycombs/index.md) | thought experiment | Sorting "facts" into mathematics, physics, biology and the unmeasured (editors' reconstruction) | | 18 | [Cevians](https://tyson-swetnam.github.io/aosd/problems/cevians/index.md) | puzzle | Trusting a careful measurement over a confident first guess | | 18 | [Superposed Filters](https://tyson-swetnam.github.io/aosd/problems/superposed-filters/index.md) | puzzle | Writing the prediction down, then looking | See the [problem index](https://tyson-swetnam.github.io/aosd/problems/index.md) for every problem in the course. --- *Good questions are worth more than quick answers, and careful observation is where every one of them starts.* ---8<--- https://tyson-swetnam.github.io/aosd/section4/ --- title: "Section 4: Patterns, Empirical Generalizations" description: "Sessions 19 to 24 of Winfree's course: finding regularities in counts, tables and games, pooling class data, and telling a pattern observed from a pattern explained, with the Cell Shapes lab, Eleusis, and the start of the Stacked Cantilevers lab." type: Lesson tags: [course, student-facing, section-4, patterns, generalization, data] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Section 4: Patterns, Empirical Generalizations Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *Six sessions of counting, pooling data, and asking whether a regularity is a law* ## Overview The [syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md) calls this section "Patterns, Empirical Generalizations". Sections 1 to 3 taught you to check facts, notice your blocks and ask good questions. Now the facts pile up, and the job is to find the rule inside them: in chords drawn across a circle, a table of paired numbers, a patch of cells, a row of accepted playing cards. A rule that fits the data is not yet a rule you understand. In [N Dots on the Rim of a Circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md) the obvious rule holds case after case and then fails. In [Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md) a pattern that most people expected to go away survived for thirty years. The syllabus calls its exercises practice in "cultivating multiple alternative solutions" and "eliminating rejectable candidate solutions". A pattern is a candidate like any other. Four activities happen in class: the Cell Shapes and Egg Pouches labs, the start of the Stacked Cantilevers lab, and a game of Eleusis. Some patterns only appear in more data than one person can gather. As the syllabus puts it: "Class meetings will also prove essential for some problems in which no one individual can collect enough data, but if we pool data, reality will come into focus." ## The sessions For most of these problems the syllabus gives only a name, and Winfree's problem sheets are lost. The problem pages are the editors' reconstructions, and each says how sure the identification is. ### Session 19: Pattern **Readings due:** Judson, Chapter 2: *Pattern*. Discuss [Presidents and States](https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/index.md). What Winfree meant by it is unknown; the page offers two regularities from American history that look like laws. Discuss [N Dots on the Rim of a Circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md), "connected to slice the disk": count the pieces, predict the next case, then check. ### Session 20: The Cell Shapes lab begins **Readings due:** Ehrlich, Chapter 5: "Low Doses of Nuclear Radiation Are Beneficial". "Start Cell Shapes lab in class." The [Cell Shapes Lab](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) runs for three sessions. No lab sheet survives; the page suggests counting the sides of cells in a flat froth or a leaf peel first, and theorizing later. ### Session 21: Pooled experiments, paired numbers **Readings due:** Ehrlich, Chapter 6: "The Solar System Has Two Suns". "Collaborative experiments on Cell Shapes", the second lab session. Deal with [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md); in a 2002 web copy of Winfree's handout this link points to a bookmark named `Keplers_Laws`, the clue the reconstruction follows. Deal with [Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md). ### Session 22: Alternative thinking languages **Readings due:** Adams, Chapter 6: Alternative thinking languages. "Further experiments on cell shapes", the lab's last scheduled session. Do the [Egg Pouches Lab](https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/index.md) in class; only its name survives, and the page reconstructs it as a lab in pooling counts. Deal with [Platonic Solids and Applications](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md), where a sketch or a straw model does work that words cannot. ### Session 23: Eleusis **Readings due:** Ehrlich, Chapter 7: "Oil, Coal, and Gas Have Abiogenic Origins". "Play Eleusis in class": in [Eleusis](https://tyson-swetnam.github.io/aosd/problems/eleusis/index.md) the dealer invents a secret rule and the players discover it by experiment. Deal with [The Mirror Mystery](https://tyson-swetnam.github.io/aosd/problems/mirror-mystery/index.md). ### Session 24: The Stacked Cantilevers lab begins **Readings due:** Ehrlich, Chapter 10: "There Was No Big Bang". "Start Stacked Cantilevers lab." The [Stacked Cantilevers Lab](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) begins in this last session of Section 4 and continues into [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), where it is listed with that section's problems: "Further collaborations" in session 25 and "Theory of stacking cantilevers, resolution of wagers" in session 26. ## Key ideas **Observed is not explained.** A rule that fits every case you have drawn is still a guess until you can say why it must hold. [N Dots on the Rim of a Circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md) is the classic warning; session 4's phrase "distinguishing things we know vs only imagine" applies to patterns too. **How pattern-finding goes wrong.** - *Seeing patterns in noise.* Search enough rules and some will fit by chance. Ask how many others you could have tried ([Presidents and States](https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/index.md)). - *Stopping at confirmations.* Five agreeing cases do not guarantee the sixth. Choose the test most likely to break the rule ([Eleusis](https://tyson-swetnam.github.io/aosd/problems/eleusis/index.md) answers only right or wrong, so the card you choose is the experiment). - *Overgeneralizing.* A rule true in its home range can fail outside it. Look for the boundary ([Platonic Solids and Applications](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md) asks you to build the solid where the rule fails). - *Believing what everyone believes.* Vary the observation before explaining it ([The Mirror Mystery](https://tyson-swetnam.github.io/aosd/problems/mirror-mystery/index.md)). - *Biased samples.* One specimen, or one group's counts, is an anecdote. Pool the data ([Cell Shapes Lab](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md), [Egg Pouches Lab](https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/index.md)). **Some patterns survive every attack.** The solar neutrino shortfall held for thirty years while most people assumed the calculation or the detectors were wrong ([Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md)). An empirical law can be right long before anyone can say why, as Kepler's rule was ([Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md)). **Separate what is forced from what is only usual.** In a froth some regularities follow from topology and some are merely typical. Counting first, then proving what you can, tells them apart. ## GamesWorth focus for this section - Before drawing or counting the next case, write your prediction down with the date and time, then record whether it held. - Keep a list of every rule you tried and discarded, not just the one that survived. The syllabus asks you to record "how you got into and out of blind alleys." - After each Eleusis round, log each hypothesis and the card you played to test it. - For every generalization, add one line: *what observation would break this?* - On a left-hand page, write a morning-after on N Dots on the Rim of a Circle or Paired Observations: when did you first believe the rule, and on what evidence? ## Readings for this section - **Horace Freeland Judson**, *The Search for Solutions*, Chapter 2: *Pattern* (session 19). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#judson); [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True*, Chapter 5: "Low Doses of Nuclear Radiation Are Beneficial" (session 20). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 6: "The Solar System Has Two Suns" (session 21). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ - **James L. Adams**, *Conceptual Blockbusting*, Chapter 6: Alternative thinking languages (session 22). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#adams); [publisher](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674059/?lens=basic-books){target=_blank} πŸ”’ - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 7: "Oil, Coal, and Gas Have Abiogenic Origins" (session 23). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ - **Robert Ehrlich**, *Nine Crazy Ideas in Science*, Chapter 10: "There Was No Big Bang" (session 24). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#ehrlich); [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ ## Problems in this section | Session | Problem | Kind | What it trains | | :-- | :-- | :-- | :-- | | 19 | [Presidents and States](https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/index.md) | discussion | Pattern, accident or law? (editors' reconstruction) | | 19 | [N Dots on the Rim of a Circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md) | puzzle | Pattern observed vs pattern explained | | 20-22 | [Cell Shapes Lab](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) | lab | Counting and pooling before theorizing | | 21 | [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md) | puzzle | Finding a law in a table, then testing it | | 21 | [Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md) | case study | A pattern that would not go away | | 22 | [Egg Pouches Lab](https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/index.md) | lab | Generalizing from pooled specimens (editors' reconstruction) | | 22 | [Platonic Solids and Applications](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md) | puzzle | Finding a rule's limits, then proving it | | 23 | [Eleusis](https://tyson-swetnam.github.io/aosd/problems/eleusis/index.md) | puzzle | Induction by experiment | | 23 | [The Mirror Mystery](https://tyson-swetnam.github.io/aosd/problems/mirror-mystery/index.md) | puzzle | Checking a belief everyone shares | See the [problem index](https://tyson-swetnam.github.io/aosd/problems/index.md) for every problem in the course. --- *A pattern is where an explanation starts, not where it ends: count, predict, try to break it, and only then ask why it holds.* ---8<--- https://tyson-swetnam.github.io/aosd/section5/ --- title: "Section 5: Inferences, Hypotheses, Explanations" description: "The last six sessions of Winfree's schedule (25-30): from pooled data to theory, reading Chamberlin, Platt, Judson and Feynman while the class holds rival hypotheses, settles wagers and discovers the laws of a toy universe." type: Lesson tags: [course, student-facing, section-5, inference, hypotheses, explanation] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Section 5: Inferences, Hypotheses, Explanations Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *Six sessions of building theories and trying to kill them* ## Overview Section 4 collected regularities. Section 5 asks why they hold. The syllabus titles it "Inferences, Hypotheses, Explanations", and its readings run in a line: Chamberlin on multiple working hypotheses, Platt on strong inference, Judson on strong predictions, Feynman on the character of physical law, and Judson again on theory. The syllabus describes the course's exercises as practice "of cultivating multiple alternative solutions, of eliminating rejectable candidate solutions", and asks you to "generate several alternative guesses, and test them for workability". The problems here have that shape: several explanations fit, and the work is finding the observation that separates them. Three of the activities are group labs that begin with experiments and end with a theory: Stacked Cantilevers, LoShu and the toy universe. They depend on the class, since for some problems no one person can collect enough data, "but if we pool data, reality will come into focus". ## The sessions The syllabus gives little more than a name for each problem, so every problem page in this section is the editors' reconstruction of the exercise. For The Miracle of FujiYama and Antigen Invasions even the subject is a guess. ### Session 25: Multiple working hypotheses **Readings due:** Chamberlin, *The Method of Multiple Working Hypotheses*. "Further collaborations on Stacked Cantilevers". The [Stacked Cantilevers Lab](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) began in session 24, the last session of [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), and closes in session 26. The class pools measurements of how far a pile of blocks can lean out over a table edge. ### Session 26: Theory and wagers **Readings due:** listed in the schedule as TBA. "Theory of stacking cantilevers, resolution of wagers", closing the lab. Then [Summing a Series](https://tyson-swetnam.github.io/aosd/problems/summing-a-series/index.md): the block theory produces 1 + 1/2 + 1/3 + ... + 1/n, and the question is whether it has a limit. Then [Stalactites](https://tyson-swetnam.github.io/aosd/problems/stalactites/index.md): why dripping water leaves stone hanging from a cave ceiling, why it has that shape, and how old it is. ### Session 27: Strong inference **Readings due:** Platt, *Strong Inference*. "Start LoShu lab experiments in class". The [LoShu Lab](https://tyson-swetnam.github.io/aosd/problems/loshu-lab/index.md) runs two experiments, a pick-three-to-make-15 card game and 3 x 3 magic squares, and asks for one theory that explains both. The Thanksgiving break falls between this session and the next. ### Session 28: Strong predictions **Readings due:** Judson, Chapter 7: *Strong Predictions*. Deal with [The Miracle of FujiYama](https://tyson-swetnam.github.io/aosd/problems/miracle-of-fujiyama/index.md), reconstructed as a row of eight coupled reactors that stores three patterns: where must the row settle? Then "Finish complete theory of LoShu." Then "Start discovering laws of a toy universe in class": the [Laws of a Toy Universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md) lab, which continues into session 29. ### Session 29: The character of physical law **Readings due:** Feynman, *The Character of Physical Law*. Deal with [Antigen Invasions](https://tyson-swetnam.github.io/aosd/problems/antigen-invasions/index.md), reconstructed as a record of repeated invasions by foreign substances, and [Martian DNA](https://tyson-swetnam.github.io/aosd/problems/martian-dna/index.md), a picture captioned as Martian hereditary material. "Finish collaborative discovery of The Laws." The syllabus adds: "All GamesWorth books collected for inspection". ### Session 30: Theory **Readings due:** Judson, Chapter 9: *Theory*. Deal with [Bacterial Hybrids](https://tyson-swetnam.github.io/aosd/problems/bacterial-hybrids/index.md): two strains that cannot grow alone give colonies when mixed. It is the last problem in the schedule. The final exam follows in exam week. ## Key ideas **Multiple working hypotheses.** Chamberlin, a geologist, traces how a favoured explanation hardens into a ruling theory that bends every new fact to fit. His remedy is to keep several hypotheses alive together, so that fondness for one cannot decide the question. Stalactites and Antigen Invasions both reward writing the whole list before choosing. **Strong inference.** Platt turns Chamberlin's attitude into a routine: devise alternative hypotheses; devise a crucial experiment whose possible outcomes each exclude one or more of them; run it cleanly; then recycle with whatever survives. A card game or a toy universe is a cheap place to practise the cycle, because a test takes minutes. **Strong predictions and silent zones.** A useful theory names the outcome before the run, so one run can embarrass it. The FujiYama reconstruction adds the complement: find the cases where your theory predicts nothing, since those are where it cannot be tested. **No finite data forces one law.** A class can agree on The Laws and still be wrong about a case its runs never visited, and a series can look settled for a thousand terms. This is session 04's "distinguishing things we know vs only imagine", now applied to theories. **Where the claim came from.** Martian DNA asks which conclusions came from the picture and which from its caption, the section's version of session 04's "facts before explanations of facts". ## GamesWorth focus for this section - Before the cantilever theory is worked out, write your wager and the reasoning behind it in the notebook, dated. After session 26, do a morning-after on why it won or lost. - For Stalactites, Antigen Invasions and Bacterial Hybrids, list every explanation you can before judging any, then write the one observation that would separate your top two. - In the LoShu and toy-universe labs, record each hunch as a testable rule, the test you ran, and whether the rule died. - For Summing a Series, note the point where your numbers stopped being evidence and you needed an argument. - For Martian DNA, split a page in two: what the picture shows, and what you took from the caption. The syllabus explains that the name GamesWorth alludes to "how much thought it takes to play one game of serious chess". Winfree's own puzzles in that spirit are introduced in his [*Adventures in Discovery*](https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html){target=_blank} πŸ”“ column. ## Readings for this section - **T. C. Chamberlin**, *The Method of Multiple Working Hypotheses* (1890; reprinted in *Science*, 1965) (session 25). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#chamberlin); [PDF](https://www.whoi.edu/cms/files/chamberlin65sci_72744.pdf){target=_blank} πŸ”“ - **John R. Platt**, *Strong Inference*, *Science* (1964) (session 27). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#platt-strong-inference); [PDF](https://www.whoi.edu/cms/files/platt64sci_72743.pdf){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions*, Chapter 7: *Strong Predictions* (session 28). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#judson); [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* - **Richard P. Feynman**, *The Character of Physical Law* (1964 Messenger Lectures; MIT Press, 1965) (session 29). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#feynman-character-of-physical-law); [lectures at the Internet Archive](https://archive.org/details/the-messenger-lectures){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions*, Chapter 9: *Theory* (session 30). [Reading list entry](https://tyson-swetnam.github.io/aosd/readings/index.md#judson); [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* ## Problems in this section | Session | Problem | Kind | What it trains | | :-- | :-- | :-- | :-- | | 24-26 | [Stacked Cantilevers Lab](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) | lab | Pooling measurements, placing wagers, then building the theory that settles them | | 26 | [Summing a Series](https://tyson-swetnam.github.io/aosd/problems/summing-a-series/index.md) | puzzle | Not trusting the first thousand terms; deciding by argument | | 26 | [Stalactites](https://tyson-swetnam.github.io/aosd/problems/stalactites/index.md) | puzzle | Multiple working hypotheses; finding the hidden assumption in an age estimate | | 27-28 | [LoShu Lab](https://tyson-swetnam.github.io/aosd/problems/loshu-lab/index.md) | lab | Experiments first, then one theory that explains two different games | | 28 | [The Miracle of FujiYama](https://tyson-swetnam.github.io/aosd/problems/miracle-of-fujiyama/index.md) | puzzle | Making a strong prediction and finding where it goes silent (editors' reconstruction) | | 28-29 | [Laws of a Toy Universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md) | lab | Inferring hidden laws and designing the experiment that could refute them | | 29 | [Antigen Invasions](https://tyson-swetnam.github.io/aosd/problems/antigen-invasions/index.md) | puzzle | Stating rules from a record, inventing rival mechanisms, finding the deciding observation (editors' reconstruction) | | 29 | [Martian DNA](https://tyson-swetnam.github.io/aosd/problems/martian-dna/index.md) | puzzle | Separating what the picture shows from what the caption suggests | | 30 | [Bacterial Hybrids](https://tyson-swetnam.github.io/aosd/problems/bacterial-hybrids/index.md) | puzzle | Listing every explanation, reading linkage from counts, killing the rival | See the [problem index](https://tyson-swetnam.github.io/aosd/problems/index.md) for every problem in the course. --- *At the final exam you choose problems and "exhibit as many distinct approaches as you can, and as many cross-checking distinct solutions as you can."* ---8<--- https://tyson-swetnam.github.io/aosd/problems/ # Problems Every problem, lab and discussion named in Professor Winfree's [syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md), one page each, grouped by section and listed in the order the course met them. Each page gives the statement, why the problem is in the course, where it comes from, hints, and sources. The course grades effort rather than answers, so resolutions are folded away until you choose to open them. A few of Winfree's problems are known only by the name in his schedule. Where the editors could not identify one with confidence, the page says so and offers the likeliest interpretations; the listing below marks those. ## [Section 1: Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes](https://tyson-swetnam.github.io/aosd/section1/index.md) * Session 01 β€” [13 Nails](https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/index.md) (puzzle) - Balance thirteen loose nails on the head of one upright nail (or explain a celt, the top that reverses its own spin): the first of the course's two contrasting challenges, a hands-on puzzle that exposes unspoken assumptions. * Session 01 β€” [Triangle Problem](https://tyson-swetnam.github.io/aosd/problems/triangle-problem/index.md) (puzzle) *(identification uncertain)* - The day-one triangle trap: a familiar question gets a confident, unanimous answer that turns out to be nonsense, and the course begins by asking why. * Session 02 β€” [Conscious Machines](https://tyson-swetnam.github.io/aosd/problems/conscious-machines/index.md) (thought-experiment) *(identification probable)* - Could a machine be conscious, and how would you tell? Winfree's second opening challenge: a question with no apparatus and no checkable answer, posed as the deliberate opposite of the 13 nails. * Session 03 β€” [Square Windows](https://tyson-swetnam.github.io/aosd/problems/square-windows/index.md) (puzzle) *(identification probable)* - Halve the light of a square window three feet high and three feet wide while keeping it square, three feet high and three feet wide: a one-paragraph puzzle whose only obstacle is an assumption you were never given. * Session 04 β€” [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md) (puzzle) - A worm bores from the first page of Volume I to the last page of Volume III on a shelf; how far does it travel? A classic trick puzzle about checking what you know against what you only imagine. * Session 04 β€” [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md) (case-study) - Fontenelle's story of the Silesian boy's golden tooth: learned books explained it before anyone checked it, and the lesson is facts before explanations of facts. * Session 04 β€” [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md) (case-study) - A short parable about a retired biochemist and a retired geneticist who try to learn how cars work from a hill above a factory; read to separate what was observed from what was inferred. * Session 05 β€” [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md) (case-study) - Blondlot's imaginary radiation of 1903-1904, exposed by R. W. Wood's blind tests and dissected in Langmuir's Pathological Science: a case study in detecting nonsense before explaining it. * Session 05 β€” [Phone Cord Problem](https://tyson-swetnam.github.io/aosd/problems/phone-cord-problem/index.md) (puzzle) *(identification probable)* - A coiled telephone cord whose plugs never turn still twists itself into a tangle with a loop where the spiral reverses: a five-minute exercise in facts before explanations. * Session 05 β€” [Stockholm Restrooms](https://tyson-swetnam.github.io/aosd/problems/stockholm-restrooms/index.md) (discussion) *(identification uncertain)* - An unidentified travel anecdote from the pathological-science session, reconstructed as an exercise in writing down what you saw before you explain it. * Session 06 β€” [Collective Reproduction](https://tyson-swetnam.github.io/aosd/problems/collective-reproduction/index.md) (puzzle) *(identification uncertain)* - An unidentified group effort closing Section 1, offered here as a reconstructed puzzle: can a small, ancient valley of three-parent elves have family trees with no repeated ancestors? It trains checking the hidden assumptions in a plausible story. * Session 06 β€” [Evaporated Gold](https://tyson-swetnam.github.io/aosd/problems/evaporated-gold/index.md) (puzzle) *(identification uncertain)* - An unidentified session-6 item, reconstructed here as an order-of-magnitude estimate of a vacuum-deposited gold film, where the wrong answer comes from an assumption nobody noticed making. ## [Section 2: Creative Blocks](https://tyson-swetnam.github.io/aosd/section2/index.md) * Session 07 β€” [Dominoes (group lab)](https://tyson-swetnam.github.io/aosd/problems/dominoes-lab/index.md) (lab) *(identification uncertain)* - An undocumented group lab from Winfree's session on perceptual blocks, reconstructed here as the cut-chessboard puzzle done with plain blocks, with a toppling-block experiment as the rival reading. * Session 07 β€” [Rearranged Triangle](https://tyson-swetnam.github.io/aosd/problems/rearranged-triangle/index.md) (puzzle) *(identification probable)* - Four pieces fill a 13-by-5 right triangle; rearrange the same four and one unit square is left over. Where did it go? The Section 2 exercise in perceptual blocks. * Session 07 β€” [Weird Organism](https://tyson-swetnam.github.io/aosd/problems/weird-organism/index.md) (puzzle) *(identification uncertain)* - A lost perceptual-block handout from Winfree's session on Adams's Chapter 2, reconstructed as an exercise in describing a familiar living thing so plainly that a stereotype hides it. * Session 08 β€” [Telltale Number](https://tyson-swetnam.github.io/aosd/problems/telltale-number/index.md) (puzzle) - Write a ten-digit number whose first digit counts its zeros, whose second counts its ones, and so on: ten billion candidates collapse to one as soon as you notice the fact the puzzle never states. * Session 08 β€” [The Barometer Story](https://tyson-swetnam.github.io/aosd/problems/barometer-story/index.md) (case-study) - Calandra's parable of the student who measured a building with a barometer six ways, none of them the one the examiner had in mind: a lesson in hidden assumptions and emotional blocks. * Session 08 β€” [Tying Knots](https://tyson-swetnam.github.io/aosd/problems/tying-knots/index.md) (puzzle) *(identification probable)* - Hold one end of a rope in each hand and tie a knot in it without ever letting go: an impossible-looking task that becomes easy the moment you find the rule nobody stated. * Session 09 β€” [Mercury's Mysterious Hidden Hemisphere](https://tyson-swetnam.github.io/aosd/problems/mercurys-hidden-hemisphere/index.md) (case-study) *(identification probable)* - For 75 years every textbook said Mercury keeps one face to the Sun and hides a frozen hemisphere no one could ever see; radar in 1965 showed the planet turns three times for every two orbits. * Session 10 β€” [Paths Through Mazes](https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/index.md) (puzzle) *(identification probable)* - A path-counting puzzle: in how many ways can ABRACADABRA be read down a diamond of letters, and can you check the count a second way? * Session 10 β€” [Taboo Questions](https://tyson-swetnam.github.io/aosd/problems/taboo-questions/index.md) (discussion) *(identification probable)* - The session-10 discussion of the questions a scientist is not supposed to ask, reconstructed from the syllabus, with James Adams's ping-pong-ball-in-a-pipe exercise as the documented warm-up. * Session 11 β€” [Walking Through Walls](https://tyson-swetnam.github.io/aosd/problems/walking-through-walls/index.md) (thought-experiment) *(identification probable)* - Ask for a better door and you get a hinged slab; ask for a better way to get through a wall and the answers change - a drill in restating the problem, from Adams's Conceptual Blockbusting. * Session 12 β€” [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md) (puzzle) *(identification probable)* - Find the formula for 1 + 2 + ... + n, then find it again by as many independent routes as you can: an exercise whose point is not the answer but the way separate derivations lock together like jig-saw pieces. ## [Section 3: Observations and Questions](https://tyson-swetnam.github.io/aosd/section3/index.md) * Session 12 β€” [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md) (lab) *(identification uncertain)* - Winfree's outdoor group lab: watch a real pedestrian crossing, record what happens before explaining it, pool the class's logs, and turn the oddities into questions that can be tested. * Session 13 β€” [Ant Walk](https://tyson-swetnam.github.io/aosd/problems/ant-walk/index.md) (puzzle) *(identification uncertain)* - Only the title survives in the syllabus; the editors' reconstruction puts an ant on the wires of a cube and an octahedron, trying to walk every wire exactly once, settled by counting the wires at each corner. * Session 13 β€” [Chemical Pattern-Formation Lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md) (lab) *(identification probable)* - A three-session group lab watching chemical waves spread through a petri dish, almost certainly the Belousov-Zhabotinsky reaction that Winfree spent his career on, run as an exercise in recording observations before explaining them. * Session 13 β€” [Seven Bridges of KΓΆnigsberg](https://tyson-swetnam.github.io/aosd/problems/seven-bridges/index.md) (puzzle) - Can a walker cross all seven bridges of KΓΆnigsberg exactly once? Euler's answer threw away the map and kept only the connections β€” the founding move of graph theory. * Session 14 β€” [What Isn't There (Surprisingly Hard)](https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/index.md) (discussion) *(identification probable)* - List the things that never happen in the world you observe, then ask why they were so hard to notice: an exercise in turning absences into expectations you can test. * Session 15 β€” [Mother Nature as Magician; Hallucinations](https://tyson-swetnam.github.io/aosd/problems/mother-nature-as-magician/index.md) (discussion) *(identification probable)* - A reconstructed session-15 discussion on evidence from the senses: illusions in which honest observation leads to a false inference, and hallucinations, where the percept has no outside cause at all. * Session 15 β€” [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md) (puzzle) - Winfree's own puzzle: why you have never seen the Moon sitting inside the colour band of a rainbow, when it can happen, and what a failed prediction says about your assumptions. * Session 16 β€” [Escher Print Gallery](https://tyson-swetnam.github.io/aosd/problems/escher-print-gallery/index.md) (puzzle) *(identification probable)* - Loops that pass inspection piece by piece: Escher's Print Gallery, which closes on itself around a blank centre, and the Penrose impossible triangle and staircase, whose error lives in no single corner. * Session 16 β€” [Green Stars](https://tyson-swetnam.github.io/aosd/problems/green-stars/index.md) (puzzle) *(identification probable)* - Stars look red, orange, yellow, white or blue, and the Sun's light peaks in the green, yet nobody sees a green star: a question about noticing an absence and about whether the answer lies in the stars or in the eye. * Session 16 β€” [Hairy People](https://tyson-swetnam.github.io/aosd/problems/hairy-people/index.md) (puzzle) *(identification probable)* - Must two people in Tucson have exactly the same number of hairs on their heads? A four-century-old wager that proves a fact nobody could ever observe, and shows why the whole proof rests on an honest upper bound. * Session 17 β€” [Martian HoneyCombs](https://tyson-swetnam.github.io/aosd/problems/martian-honeycombs/index.md) (thought-experiment) *(identification uncertain)* - Winfree's statement is lost; this reconstruction asks which features of a bee's honeycomb a Martian comb would have to share, and so which of our honeycomb 'facts' rest on mathematics, physics, Earth biology, or nothing measured at all. * Session 17 β€” [Zygotes](https://tyson-swetnam.github.io/aosd/problems/zygotes/index.md) (puzzle) *(identification uncertain)* - Winfree's statement is lost; an editors' reconstruction asks how many twin pairs came from one zygote when all you can see is whether the twins share a sex. * Session 18 β€” [Cevians](https://tyson-swetnam.github.io/aosd/problems/cevians/index.md) (puzzle) *(identification probable)* - Join each corner of a triangle to the one-third point of the opposite side: what fraction of the area is the small triangle in the middle? A puzzle about trusting a measurement over a confident first guess. * Session 18 β€” [Superposed Filters](https://tyson-swetnam.github.io/aosd/problems/superposed-filters/index.md) (puzzle) - Stack polarizing filters, write down what you expect, then look: what happens when a third absorbing filter is slid between two crossed ones, and does the order of the filters matter? ## [Section 4: Patterns, Empirical Generalizations](https://tyson-swetnam.github.io/aosd/section4/index.md) * Session 19 β€” [N Dots on the Rim of a Circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md) (puzzle) - Join every pair of n dots on a circle by chords and count the pieces of the disk: a lesson in the difference between a pattern observed and a pattern explained. * Session 19 β€” [Presidents and States](https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/index.md) (discussion) *(identification uncertain)* - Winfree's own statement of this exercise is lost; this reconstruction takes two famous regularities from American history, the zero-year presidents and the bellwether states, and asks whether a run of confirmations makes a pattern, an accident or a law. * Session 20 β€” [Cell Shapes Lab](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) (lab) *(identification probable)* - A three-session group lab in counting the sides of cells in a flat soap froth, a leaf peel or a drawn mosaic, pooling the class's data and hunting for the empirical rules hidden in it. * Session 21 β€” [Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md) (case-study) *(identification probable)* - An editors' reconstruction of the solar neutrino problem, in which every detector counted too few neutrinos from the Sun for thirty years: state the pattern, sort the explanations, and find the measurement that decides between them. * Session 21 β€” [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md) (puzzle) *(identification probable)* - An editors' reconstruction of a lost Winfree exercise filed under Kepler's laws: find the rule hidden in six pairs of numbers, one distance and one period per planet, then test it on planets and moons it never saw. * Session 22 β€” [Egg Pouches Lab](https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/index.md) (lab) *(identification uncertain)* - An in-class lab known only by its name, reconstructed here as counting and pooling data along a worm-like chain of egg pouches such as a whelk egg-case string, to find out what a whole class can generalize that one specimen cannot. * Session 22 β€” [Platonic Solids and Applications](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md) (puzzle) *(identification probable)* - Count the corners, edges and faces of the five regular solids, find the rule that ties them together, test it until it breaks, and use it to explain soccer balls, viruses and radiolarian skeletons. * Session 23 β€” [Eleusis](https://tyson-swetnam.github.io/aosd/problems/eleusis/index.md) (puzzle) - Play Robert Abbott's card game Eleusis, in which a dealer writes a secret rule and the players must discover it by playing cards and being told only right or wrong: induction from data, practised with your own hands. * Session 23 β€” [The Mirror Mystery](https://tyson-swetnam.github.io/aosd/problems/mirror-mystery/index.md) (puzzle) *(identification probable)* - Why does a mirror seem to swap left and right but not up and down? A household observation that everyone believes, and a lesson in checking the facts before explaining them. ## [Section 5: Inferences, Hypotheses, Explanations](https://tyson-swetnam.github.io/aosd/section5/index.md) * Session 24 β€” [Stacked Cantilevers Lab](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) (lab) *(identification probable)* - A three-session group lab on how far a pile of identical blocks can lean out over a table edge, moving from pooled measurements and wagers to a theory built on the harmonic series. * Session 26 β€” [Stalactites](https://tyson-swetnam.github.io/aosd/problems/stalactites/index.md) (puzzle) *(identification probable)* - Why does dripping water leave stone hanging from a cave ceiling, why is it shaped as it is, and how old is it? A reconstructed exercise in multiple working hypotheses and hidden assumptions. * Session 26 β€” [Summing a Series](https://tyson-swetnam.github.io/aosd/problems/summing-a-series/index.md) (puzzle) *(identification probable)* - The theory of stacked blocks produces the sum 1 + 1/2 + 1/3 + ... + 1/n: does it have a limit? A reconstructed exercise in not trusting the first thousand terms. * Session 27 β€” [LoShu Lab](https://tyson-swetnam.github.io/aosd/problems/loshu-lab/index.md) (lab) *(identification probable)* - A two-session lab built around the Lo Shu magic square: experiment with a pick-three-to-make-15 card game and with 3 x 3 magic squares, then build one theory that explains both. * Session 28 β€” [Laws of a Toy Universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md) (lab) - A class lab: watch a grid of cells change generation by generation, work out the hidden law that drives it, then design an experiment that could prove your law wrong. * Session 28 β€” [The Miracle of FujiYama](https://tyson-swetnam.github.io/aosd/problems/miracle-of-fujiyama/index.md) (puzzle) *(identification uncertain)* - Winfree's statement is lost; an editors' reconstruction asks where a row of eight coupled reactors that stores three patterns must settle, and where that strong prediction goes silent. * Session 29 β€” [Antigen Invasions](https://tyson-swetnam.github.io/aosd/problems/antigen-invasions/index.md) (puzzle) *(identification uncertain)* - A lost Section 5 problem, reconstructed: from a record of repeated invasions by foreign substances, state the rules the body follows, invent rival mechanisms, and find the observation that decides between them. * Session 29 β€” [Martian DNA](https://tyson-swetnam.github.io/aosd/problems/martian-dna/index.md) (puzzle) *(identification probable)* - Winfree's write-up is lost; this reconstruction hands you a picture captioned as Martian hereditary material and asks which of your conclusions came from the picture and which came from the caption. * Session 30 β€” [Bacterial Hybrids](https://tyson-swetnam.github.io/aosd/problems/bacterial-hybrids/index.md) (puzzle) *(identification probable)* - Two strains that cannot grow alone give colonies when mixed: list every explanation, read the linkage hidden in a table of colony counts, and design the experiment that kills the rival. An editors' reconstruction; Winfree's own problem sheet is lost. ---8<--- https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/ --- title: "13 Nails" description: "Balance thirteen loose nails on the head of one upright nail (or explain a celt, the top that reverses its own spin): the first of the course's two contrasting challenges, a hands-on puzzle that exposes unspoken assumptions." type: Activity tags: [course, student-facing, problem, section-1, mechanics, centre-of-gravity, hands-on, rattleback] status: stable problem: section: 1 session: 1 identification: confident kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: wesleyan-1j20-25 resource: "https://physicsdemos.site.wesleyan.edu/home/mechanics/1j/1j20-25-twelve-nails-on-one" title: "1J20.25 Twelve Nails On One (physics demonstration)" author: "Wesleyan University Department of Physics" - id: spangler-balancing-nails resource: "https://stevespangler.com/experiments/balancing-nail-puzzle/" title: "Balancing Nails" author: "Steve Spangler Science" - id: protradecraft-13-nails resource: "https://www.protradecraft.com/construction-phase/tools/video/55181668/jobsite-magic-trick-balance-13-nails-on-the-head-of-one" title: "Jobsite Magic Trick: Balance 13 Nails on the Head of One" author: "Patrick Roehrman, ProTradeCraft" - id: pleacher-balancing-nails resource: "https://www.pleacher.com/mp/puzzles/tricks/mobnail.html" title: "Balancing Nails (puzzle page)" author: "David Pleacher" - id: anu-nail-balancing resource: "https://science.anu.edu.au/engagement/science-lab-experiments-home-school/try-nail-balancing-challenge" title: "Nail balancing challenge" author: "ANU College of Science and Medicine" - id: ricks-hobby-garage-nails resource: "https://rickshobbygarage.com/the-balancing-nails-puzzle/" title: "The Balancing Nails Puzzle" author: "Rick Simper (Rick's Hobby Garage)" - id: walker-1895 resource: "" title: "On a curious dynamical property of celts (no digitised copy located)" author: "G. T. Walker" - id: wikisource-walker resource: "https://en.wikisource.org/wiki/Author:Gilbert_Thomas_Walker" title: "Author page: Gilbert Thomas Walker (citation for \"On a dynamical top\", 1896)" author: "Wikisource" - id: linda-hall-gilbert-walker resource: "https://www.lindahall.org/about/news/scientist-of-the-day/gilbert-walker/" title: "Gilbert Walker (Scientist of the Day)" author: "Linda Hall Library" - id: jearl-walker-1979 resource: "https://www.scientificamerican.com/article/the-amateur-scientist-1979-10/" title: "The Amateur Scientist: The mysterious 'rattleback': a stone that spins in one direction and then reverses" author: "Jearl Walker" - id: bondi-1986 resource: "https://doi.org/10.1098/rspa.1986.0052" title: "The rigid body dynamics of unidirectional spin" author: "Hermann Bondi" - id: garcia-hubbard-1988 resource: "https://doi.org/10.1098/rspa.1988.0078" title: "Spin reversal of the rattleback: theory and experiment" author: "A. Garcia and M. Hubbard" - id: pippard-1990 resource: "https://doi.org/10.1088/0143-0807/11/1/112" title: "How to make a Celt or rattleback" author: "A. B. Pippard" - id: wikipedia-rattleback resource: "https://en.wikipedia.org/wiki/Rattleback" title: "Rattleback" author: "Wikipedia contributors" - id: iowa-1m40-90 resource: "https://instructional-resources.physics.uiowa.edu/1m4090-celts-or-rattle-backs" title: "1M40.90 Celts or Rattle Backs (lecture demonstration)" author: "University of Iowa Department of Physics and Astronomy" - id: jagla-rojo-2023 resource: "https://arxiv.org/abs/2309.06154" title: "The chiral knife edge: a simplified rattleback to illustrate spin inversion" author: "E. A. Jagla and A. G. Rojo" - id: franti-2012 resource: "https://arxiv.org/abs/1202.6506" title: "On the rotational dynamics of the Rattleback" author: "Lasse Franti" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # 13 Nails Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 1. Set alongside the [Triangle Problem](https://tyson-swetnam.github.io/aosd/problems/triangle-problem/index.md); its contrasting partner, [Conscious Machines](https://tyson-swetnam.github.io/aosd/problems/conscious-machines/index.md), follows in session 2.* !!! abstract "The problem" **The 13 nails problem.** Drive one large common nail upright into a block of wood, so that it stands firmly with its head a few centimetres clear. Balance thirteen more nails of the same kind, all at once, on the head of that upright nail. When you let go they may touch only each other and that nail head: nothing may rest on the block, the table or your hands, and there is no glue, tape, bending or notching. **The celts problem (the alternative).** A celt, or rattleback, is a small boat-shaped object with a smoothly curved underside. Spin it gently on a smooth table, first one way and then the other. One way it turns like an ordinary top; the other way it rocks end to end, the spin dies, and it starts turning the opposite way untouched. Say what you observe, decide whether any law of physics is broken, and explain the reversal. Either way, record how you attacked it, what you assumed without noticing, and what you tried that did not work. *Reconstruction. Winfree's own wording has not survived; the syllabus gives only the names. The set-up and the constraints above are paraphrased from standard published descriptions of the two demonstrations, and the ban on glue and bending is an editorial tightening, not a documented rule of his.* ![Thirteen loose nails lying on a wooden block, beside one nail of the same kind driven upright into the block with its head a few centimetres clear of the surface](https://tyson-swetnam.github.io/aosd/assets/images/problems/thirteen-nails.svg){ width="560" } *Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 1 is the introductory meeting of Section 1, "Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes". After the [Triangle Problem](https://tyson-swetnam.github.io/aosd/problems/triangle-problem/index.md) has done its job of "Demonstrating the need", the syllabus sets the "First of two contrasting challenges": this one, or, "alternatively, the celts problem". The second challenge, in session 2, is [Conscious Machines](https://tyson-swetnam.github.io/aosd/problems/conscious-machines/index.md). The contrast is the point. This challenge is small and hands-on, with an answer you can hold in your hand; the other is enormous and may have no answer at all. The course is about how you work on both. The nails fit the section's theme of false assumptions. Nearly everyone starts by piling nails on the fixed nail's head, or assumes a balanced object must carry its weight above its support, and the puzzle gives way the moment those assumptions do. The celt catches nonsense from the other side: its reversal looks like a broken conservation law. Winfree writes that the puzzles exist "to slow you down for a few minutes so you can examine the working of your own mind", and an object on the table does that in the first hour. ## Where it comes from **Balancing nails.** No author and no date are known. The puzzle circulates as a jobsite or bar bet, and it is catalogued as a physics demonstration of centre of mass and equilibrium (Wesleyan 1J20.25, "Twelve Nails On One"). The count varies with the teller: six at ANU, eleven on a twelfth at Steve Spangler, twelve at Wesleyan, thirteen at ProTradeCraft, fourteen at Pleacher, sixteen at Rick's Hobby Garage. Thirteen is the number in Winfree's title. The one hobbyist who went looking for its history reports "zero luck on finding any history on it"; his bored-carpenter story is offered as his own guess, and nothing better has been found. **Celts.** "Celt" (soft c) is the antiquarians' word for a prehistoric stone or bronze axe or chisel; some such stones, being slightly asymmetric, spin happily in one direction only. The first scientific treatment was a demonstration by the Cambridge mathematician Gilbert Thomas Walker, "On a curious dynamical property of celts", given to the Cambridge Philosophical Society in 1895 with J. J. Thomson presiding. His analysis "On a dynamical top" (1896) treated the celt as a rigid body rolling without slipping and showed that no law of mechanics is broken. The subject then drew only occasional papers until Jearl Walker's "Amateur Scientist" column of October 1979, after which plastic rattlebacks became a standard science toy and a research literature grew. ??? tip "Hints" - Nails: the fixed nail is the only support, but nothing says the thirteen must be assembled on it. - Nails: think of a nail's head as a hook. Where does its weight hang once hooked over something? - Nails: a hanging object is stable when its centre of gravity sits below the point it hangs from. Can you get the bundle's below that nail head? - Celts: write down, in order, what it does when spun each way, keeping observation apart from explanation. Then rock it end to end without spinning it, and see whether the curved underside lines up with the long axis of the mass or is twisted from it. ??? success "Resolution" **Nails.** Lay one nail flat on the table as a "rail", and lay eleven more across it, perpendicular and packed close, with their heads projecting past the rail on alternate sides. Lay the thirteenth on top, parallel to the rail but pointing the opposite way, in the groove between the crossed nails. Pinch the two parallel nails together at their middles and lift: the cross nails swing down and hook under them by their heads, locking the bundle. Set the middle of the rail on the fixed nail's head. The cross nails hang below the rail, so the bundle's centre of gravity lies below its support: it is stable, and swings back if nudged. ![Side view of the finished bundle: a rail nail resting across the head of the fixed nail, cross nails hanging from it with their heads alternating sides, a top nail laid head-to-tail to lock them, and the centre of gravity marked below the support point](https://tyson-swetnam.github.io/aosd/assets/images/problems/thirteen-nails-solution.svg){ width="560" } *Drawn for this site (CC BY 4.0).* **Celts.** No law is broken: angular momentum about the vertical is not conserved, because the table's friction and normal force do not act through the celt's centre of mass. The principal axes of curvature of its underside are slightly rotated, in the horizontal plane, from its principal axes of inertia, and that skew couples spin to the pitching and rolling oscillations. Spun the "wrong" way, the spin feeds the pitching oscillation, which grows into the rattle while the spin dies; the coupling then turns that oscillation back into spin the other way, the direction in which it is stable. ## Sources - **Wesleyan University Department of Physics**, "1J20.25 Twelve Nails On One" (physics demonstration) β€” [physicsdemos.site.wesleyan.edu](https://physicsdemos.site.wesleyan.edu/home/mechanics/1j/1j20-25-twelve-nails-on-one){target=_blank} πŸ”“ - **Steve Spangler Science**, "Balancing Nails" β€” [stevespangler.com](https://stevespangler.com/experiments/balancing-nail-puzzle/){target=_blank} πŸ”“ - **Patrick Roehrman**, "Jobsite Magic Trick: Balance 13 Nails on the Head of One", *ProTradeCraft* (2016) β€” [protradecraft.com](https://www.protradecraft.com/construction-phase/tools/video/55181668/jobsite-magic-trick-balance-13-nails-on-the-head-of-one){target=_blank} πŸ”“ - **David Pleacher**, "Balancing Nails" (puzzle page) β€” [pleacher.com](https://www.pleacher.com/mp/puzzles/tricks/mobnail.html){target=_blank} πŸ”“ - **ANU College of Science and Medicine**, "Nail balancing challenge" β€” [science.anu.edu.au](https://science.anu.edu.au/engagement/science-lab-experiments-home-school/try-nail-balancing-challenge){target=_blank} πŸ”“ - **Rick Simper**, "The Balancing Nails Puzzle", *Rick's Hobby Garage* (2022) β€” [rickshobbygarage.com](https://rickshobbygarage.com/the-balancing-nails-puzzle/){target=_blank} πŸ”“ - **G. T. Walker**, "On a curious dynamical property of celts", *Proceedings of the Cambridge Philosophical Society* (1895) β€” no digitised copy located; secondary bibliographies give 8: 305–306, which the editors have not checked πŸ”’ - **G. T. Walker**, "On a dynamical top", *Quarterly Journal of Pure and Applied Mathematics* 28: 175–184 (1896) β€” citation confirmed at the [Wikisource author page](https://en.wikisource.org/wiki/Author:Gilbert_Thomas_Walker){target=_blank} πŸ”“, which hosts no text of the paper - **Linda Hall Library**, "Gilbert Walker", *Scientist of the Day* β€” [lindahall.org](https://www.lindahall.org/about/news/scientist-of-the-day/gilbert-walker/){target=_blank} πŸ”“ - **Jearl Walker**, "The Amateur Scientist: The mysterious 'rattleback': a stone that spins in one direction and then reverses", *Scientific American* 241(4): 172–184 (October 1979) β€” [scientificamerican.com](https://www.scientificamerican.com/article/the-amateur-scientist-1979-10/){target=_blank} πŸ”’ - **Hermann Bondi**, "The rigid body dynamics of unidirectional spin", *Proceedings of the Royal Society A* 405: 265–274 (1986) β€” [doi:10.1098/rspa.1986.0052](https://doi.org/10.1098/rspa.1986.0052){target=_blank} πŸ”’ - **A. Garcia and M. Hubbard**, "Spin reversal of the rattleback: theory and experiment", *Proceedings of the Royal Society A* 418: 165–197 (1988) β€” [doi:10.1098/rspa.1988.0078](https://doi.org/10.1098/rspa.1988.0078){target=_blank} πŸ”’ - **A. B. Pippard**, "How to make a Celt or rattleback", *European Journal of Physics* 11: 63–64 (1990) β€” [doi:10.1088/0143-0807/11/1/112](https://doi.org/10.1088/0143-0807/11/1/112){target=_blank} πŸ”’ - **Wikipedia contributors**, "Rattleback" (overview, accessed 2026) β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Rattleback){target=_blank} πŸ”“ - **University of Iowa Department of Physics and Astronomy**, "1M40.90 Celts or Rattle Backs" (lecture demonstration) β€” [instructional-resources.physics.uiowa.edu](https://instructional-resources.physics.uiowa.edu/1m4090-celts-or-rattle-backs){target=_blank} πŸ”“ - **E. A. Jagla and A. G. Rojo**, "The chiral knife edge: a simplified rattleback to illustrate spin inversion" (2023) β€” [arXiv:2309.06154](https://arxiv.org/abs/2309.06154){target=_blank} πŸ”“ - **Lasse Franti**, "On the rotational dynamics of the Rattleback" (2012) β€” [arXiv:1202.6506](https://arxiv.org/abs/1202.6506){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (PDF) β€” [aosd_syllabus.pdf](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/triangle-problem/ --- title: "Triangle Problem" description: "The day-one triangle trap: a familiar question gets a confident, unanimous answer that turns out to be nonsense, and the course begins by asking why." type: Activity tags: [course, student-facing, problem, section-1, geometry, false-assumptions, detecting-nonsense] status: stable problem: section: 1 session: 1 identification: unknown kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: arnold-2004 resource: "https://www.imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf" title: "Problems for children from 5 to 15" author: "V. I. Arnold; transl. V. Goryunov and S. Gusein-Zade" - id: arnold-1997 resource: "https://www.karlin.mff.cuni.cz/~spurny/doc/articles/arnold.htm" title: "On teaching mathematics" author: "V. I. Arnold" - id: winfree-sas05 resource: "https://web.archive.org/web/20030114051703/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS05/SAS05.html" title: "A Personal Encounter with Non-Euclidean Space (SAS Adventures in Discovery column no. 5)" author: "Arthur T. Winfree" - id: winfree-handout-2001 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "ECOL 479/579 The Art of Scientific Discovery: course handout and retrospective syllabus (Spring 2001)" author: "Arthur T. Winfree" - id: khovanova-2009 resource: "https://blog.tanyakhovanova.com/2009/05/an-experiment-inspired-by-vladimir-arnold/" title: "An Experiment Inspired by Vladimir Arnold" author: "Tanya Khovanova" - id: talwalkar-2016 resource: "https://mindyourdecisions.com/blog/2016/02/28/evil-geometry-problem-sunday-puzzle/" title: "Evil Geometry Problem: Sunday Puzzle" author: "Presh Talwalkar" - id: peterson-2025 resource: "https://www.themathdoctors.org/vacuous-solutions-correct-but-not-really/" title: "Vacuous Solutions: Correct, But Not Really" author: "Dave Peterson" - id: brilliant-thales-triangle resource: "https://brilliant.org/wiki/can-a-right-triangle-with-hypotenuse-10-have-a/" title: "In a right triangle with hypotenuse 10, can the altitude perpendicular to the hypotenuse be 6?" author: "Brilliant.org wiki" - id: wikipedia-thales resource: "https://en.wikipedia.org/wiki/Thales%27s_theorem" title: "Thales's theorem" author: "Wikipedia" - id: mathworld-thales resource: "https://mathworld.wolfram.com/ThalesTheorem.html" title: "Thales' Theorem" author: "Eric W. Weisstein, MathWorld" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Triangle Problem Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 1. Assigned together with [13 Nails](https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/index.md).* !!! abstract "The problem" The exact triangle problem Winfree used on the first day is not recorded. The syllabus gives only the phrase "Demonstrating the need: triangle problem". Below are the two candidates the evidence cannot rank. Both are written for this page. Neither is a transcription of his assignment. **Candidate A: a triangle between the stars.** This is Winfree's own; he used it with a class later in the semester. Pick any three stars on a clear night and imagine straight lines joining them, or stretch a taut string from one to the next. Do the three interior angles of the triangle you see add up to 180 degrees? Write down your answer and your reason. Then consider a triangle whose corners are a star on the southern horizon, a star on the western horizon, and a star directly overhead. **Candidate B: the impossible right triangle.** The hypotenuse of a right-angled triangle is 10 inches; the altitude dropped onto it is 6 inches. Find the area of the triangle. V. I. Arnold printed this as problem 6 of his *Problems for children from 5 to 15*, with a story attached. The question came from a standard American examination. American school students had been coping with it successfully for over a decade. Then Russian school students arrived from Moscow, and none of them was able to solve it as their American peers had, giving 30 square inches. Why? Take either question as you would any textbook exercise: write down your answer and your reasoning before reading further. Then ask yourself what you assumed. ## Why it is in the course The syllabus puts this item first, before any reading has been discussed, under the label "Demonstrating the need". The need is the one the course description spells out: the exercises are "practice scrimmages" in "recognizing ignorance", "eliminating rejectable candidate solutions", and "spotting and taking advantage of your own mistakes". A booby-trapped triangle demonstrates that need in five minutes. Everyone in the room already knows the answer, everyone gives the same one, and the confident, unanimous answer is wrong. The lesson is that fluency with a familiar fact is not the same as thinking, and that the first job with any problem is to ask whether its premises can even be true. That is the agenda of Section 1, "Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes", and of the reading assigned for the same day, Adams's preface and Chapter 1. The syllabus pairs the triangle with [13 Nails](https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/index.md) as the "first of two contrasting challenges". The triangle is a closed trap that punishes haste; the nails problem is an open construction that rewards persistence. The contrast is itself part of the day's message. A wrong answer here is not a failure but the raw material of the course. The syllabus says of mistakes that "they are often the most available doors to discovery", and the GamesWorth notebook exists to record exactly how one walked through this one. ## Where it comes from The celestial triangle is Winfree's own. In his "Adventures in Discovery" column for the Society for Amateur Scientists of 7 December 2001, "A Personal Encounter with Non-Euclidean Space", he describes asking a classroom of university seniors, the week before, to write down whether a triangle traced between stars has angles summing to 180 degrees. Everyone said yes. Then he offered a star on the southern horizon, one on the western horizon, and one at the zenith: three right angles. Objections followed about the curvature of the sky and about angles seen in perspective, so the class went out into the hall to look at converging floor tiles. His comment, that "we are all so brainwashed by a 10th grade encounter with Euclid that we have trouble even seeing the blatantly different behavior of lines in our visual space", is a compact statement of what the first section of the course is about. The right-triangle version is best known from Vladimir I. Arnold's brochure *Problems for children from 5 to 15* (Moscow: MCCME, 2004), 77 problems he put onto paper in Paris in spring 2004. Problem 6 presents it as a question from a standard American examination that American school students had been coping with successfully for over a decade. His earlier essay *On teaching mathematics* (1997) does not contain the anecdote, so the 2004 brochure is its first appearance in his writing; the problem itself, on his account, was in American use from roughly the early 1990s, well before Winfree's course. It has since become a staple of classroom experiments on checking premises. Tanya Khovanova (2009) gave it to her own students and reported answers of 30, of 24 (from assuming a 6-8-10 triangle), and one negative number under a square root. The mathematics behind the resolution is ancient: Thales's theorem, that the angle in a semicircle is a right angle. ??? tip "Hints" - Star triangle: choose the three stars in extreme positions, two on the horizon a quarter-turn apart and one straight overhead, and estimate each corner angle separately. - Star triangle: what surface are those "straight" lines actually drawn on, and what counts as a straight line on that surface? - Right triangle: before you compute anything, try to draw the figure to scale. Can you actually construct a right triangle with these two measurements? - Right triangle: fix the hypotenuse as a segment of length 10. Where can the vertex with the right angle lie? An old theorem about angles in a semicircle answers this. - Right triangle: given the answer to the previous hint, what is the largest the altitude to the hypotenuse could possibly be? Compare it with 6. - Right triangle, if you prefer algebra: call the legs a and b, so that a^2 + b^2 = 100 and (from the area computed two ways) ab = 60. What do (a + b)^2 and (a - b)^2 come out to? ??? success "Resolution" **Star triangle.** The three stars are directions, and the "straight" lines between them are great circles on the celestial sphere. The triangle is spherical, and the sum of its angles exceeds 180 degrees by an amount proportional to its area. Winfree's example (one star on the southern horizon, one on the western horizon, one at the zenith) has three right angles, a sum of 270 degrees. The unanimous "must be 180" is the Euclidean assumption imported unexamined from tenth-grade geometry. **Right triangle.** No such triangle exists, so it has no area. By Thales's theorem the right-angle vertex of a right triangle lies on the circle whose diameter is the hypotenuse; the distance from any point of that circle to the diameter is at most the radius, here 5. An altitude of 6 to a hypotenuse of 10 is therefore impossible, and the largest area any right triangle with hypotenuse 10 can have is (1/2)(10)(5) = 25. ![A right triangle on a hypotenuse of length 10, with the semicircle on which its right-angle vertex must lie and a hollow point six units above the hypotenuse lying outside that semicircle](https://tyson-swetnam.github.io/aosd/assets/images/problems/triangle-problem.svg){ width="560" } *The right-angle vertex of any triangle on hypotenuse AB lies on the dashed semicircle, so the altitude to AB can never exceed the radius. Drawn for this site (CC BY 4.0).* Algebra says the same thing: with legs a and b, a^2 + b^2 = 100 and ab = 60 give (a + b)^2 = 220 but (a - b)^2 = 100 - 120 = -20, which no real a and b satisfy. The answer 30 comes from applying (1/2)(base)(height) without asking whether the figure exists; the answer 24 comes from assuming a 6-8-10 triangle, but there the altitude to the hypotenuse is 4.8, not 6. Arnold's Moscow students "failed" because they checked the premises first. ## Sources - **V. I. Arnold**, *Problems for children from 5 to 15* (Moscow: MCCME, 2004; English translation by V. Goryunov and S. Gusein-Zade), problem 6 β€” [PDF at imaginary.org](https://www.imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf){target=_blank} πŸ”“ - **V. I. Arnold**, "On teaching mathematics", address at the Palais de la DΓ©couverte, 7 March 1997; *Russian Mathematical Surveys* 53:1 (1998), 229–236 β€” [online text](https://www.karlin.mff.cuni.cz/~spurny/doc/articles/arnold.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, "A Personal Encounter with Non-Euclidean Space", SAS Adventures in Discovery column no. 5 (7 December 2001) β€” [Internet Archive](https://web.archive.org/web/20030114051703/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS05/SAS05.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, ECOL 479/579 The Art of Scientific Discovery: course handout and retrospective syllabus (Spring 2001) β€” [Internet Archive](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Tanya Khovanova**, "An Experiment Inspired by Vladimir Arnold" (2009) β€” [blog post](https://blog.tanyakhovanova.com/2009/05/an-experiment-inspired-by-vladimir-arnold/){target=_blank} πŸ”“ - **Presh Talwalkar**, "Evil Geometry Problem: Sunday Puzzle", Mind Your Decisions (2016) β€” [blog post](https://mindyourdecisions.com/blog/2016/02/28/evil-geometry-problem-sunday-puzzle/){target=_blank} πŸ”“ - **Dave Peterson**, "Vacuous Solutions: Correct, But Not Really", The Math Doctors (2025) β€” [article](https://www.themathdoctors.org/vacuous-solutions-correct-but-not-really/){target=_blank} πŸ”“ - **Brilliant.org wiki**, "In a right triangle with hypotenuse 10, can the altitude perpendicular to the hypotenuse be 6?" β€” [wiki page](https://brilliant.org/wiki/can-a-right-triangle-with-hypotenuse-10-have-a/){target=_blank} πŸ”“ - **Wikipedia**, "Thales's theorem" β€” [article](https://en.wikipedia.org/wiki/Thales%27s_theorem){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Thales' Theorem", MathWorld β€” [entry](https://mathworld.wolfram.com/ThalesTheorem.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure. The syllabus gives the name, the session (day one, with Adams's preface and Chapter 1), the label "Demonstrating the need", and the neighbour, 13 Nails. Winfree's archived course handout repeats that line and nothing more. A search of his whole archived lab site turns up no problem sheet and only one piece of triangle content anywhere: the star demonstration in his column of 7 December 2001. The editors weighed three readings. - **Winfree's triangle between the stars.** Genuinely his, genuinely a triangle used to demonstrate the need to test an assumption everyone shares, and documented in his own words. But the column says "last week", placing the demonstration in late November 2001, week 14 of the semester whose day one was 21 August, and inside a run of columns on the apparent curvature of lines in the sky. Within that semester the two are separate occasions, and the handout describes itself as a retrospective of the Spring 2001 offering with the dates changed. Low to medium confidence. - **The impossible right triangle** (hypotenuse 10 inches, altitude 6 inches). Fits the name, the day-one purpose, the Section 1 theme, and the course's preference for elementary mathematics that needs no special knowledge; Arnold reports it as American examination material from well before 2001. But no document of any kind connects Winfree to it, and Arnold's brochure postdates the course by three years. Low to medium confidence. - **A "how many triangles are in this figure" counting puzzle.** A common classroom opener that fits "error checking", but nothing in the syllabus or on Winfree's pages points to it, and "demonstrating the need" fits a booby-trapped question better. Low confidence. The evidence does not rank the first two above each other, so this page presents both. Whichever it was, the session was about the same thing: a familiar-looking problem whose confident answer collapses when the premises are examined. Former students who remember the first day are invited to correct the record. --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/conscious-machines/ --- title: "Conscious Machines" description: "Could a machine be conscious, and how would you tell? Winfree's second opening challenge: a question with no apparatus and no checkable answer, posed as the deliberate opposite of the 13 nails." type: Activity tags: [course, student-facing, problem, section-1, consciousness, turing-test, error-checking, thought-experiment] status: stable problem: section: 1 session: 2 identification: probable kind: thought-experiment generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: turing-1950 resource: "https://www.cs.ox.ac.uk/activities/ieg/e-library/sources/t_article.pdf" title: "Computing Machinery and Intelligence" author: "A. M. Turing" - id: searle-1980 resource: "https://doi.org/10.1017/S0140525X00005756" title: "Minds, Brains, and Programs" author: "John R. Searle" - id: chalmers-1995 resource: "https://consc.net/papers/puzzle.html" title: "The Puzzle of Conscious Experience" author: "David J. Chalmers" - id: sep-turing-test resource: "https://plato.stanford.edu/entries/turing-test/" title: "The Turing Test" author: "Graham Oppy and David Dowe (Stanford Encyclopedia of Philosophy)" - id: sep-chinese-room resource: "https://plato.stanford.edu/entries/chinese-room/" title: "The Chinese Room Argument" author: "David Cole (Stanford Encyclopedia of Philosophy)" - id: feynman-1974-cargo-cult resource: "https://calteches.library.caltech.edu/51/2/CargoCult.htm" title: "Cargo Cult Science" author: "Richard P. Feynman" - id: koch-tononi-2008 resource: "https://spectrum.ieee.org/can-machines-be-conscious" title: "Can Machines Be Conscious?" author: "Christof Koch and Giulio Tononi" - id: platt-1962-excitement-of-science resource: "https://archive.org/details/excitementofscie0000plat" title: "The Excitement of Science (contains The Art of Creative Thinking)" author: "John Rader Platt" - id: wikipedia-mechanical-turk resource: "https://en.wikipedia.org/wiki/Mechanical_Turk" title: "Mechanical Turk" author: "Wikipedia" - id: protradecraft-13-nails resource: "https://www.protradecraft.com/jobsite-magic-trick-balance-13-nails-head-one" title: "Jobsite Magic Trick: Balance 13 Nails on the Head of One" author: "ProTradeCraft" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Conscious Machines Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 2. The "second of two contrasting challenges"; the first, in session 1, was [13 Nails](https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: it records only the title of this challenge. The wording below is the editors' reconstruction from the title, the pairing with the 13 nails problem, and the readings due the same day. On Tuesday the challenge was thirteen nails: balance them all on the head of one, and you can see at once whether you have succeeded. Today's challenge has no apparatus at all. **Could a machine be conscious?** Notice what kind of question this is. With the nails, you could tell at once whether an attempt had worked. Here, suppose someone wheels a machine into the room and claims it is conscious. What observation would convince you? What would convince you of the opposite? If you cannot name one, is this still a question about the world, or only about how you would like to use a word? Spend a GamesWorth on it in your notebook: 1. Write down what you mean by *machine* and by *conscious*, then check whether those definitions already decided your answer. 2. How do you know that anyone *else* in the room is conscious, and would that method work on a machine? 3. Try to design a test that a skeptic and a believer could both agree on in advance. Then look for the loophole each side would use to reject the result. 4. Compare with the nails. Which of the two could you have got *wrong*, and how would you have found out? There is no answer key. What is graded is the record of how you tried. ![Cutaway engraving of Kempelen's chess automaton: a turbaned figure leans over a chessboard on a cabinet, and inside the cabinet, among gears and rollers, a man sits at a second board working the machine](https://tyson-swetnam.github.io/aosd/assets/images/problems/conscious-machines-turk.jpg){ width="560" } *One of Joseph Racknitz's guesses at how the "thinking machine" worked. He was right that a person was hidden inside, wrong about where and how. Joseph Racknitz, 1789, via Humboldt University Library / Wikimedia Commons. Public domain.* ## Why it is in the course The syllabus labels the first two exercises "contrasting challenges". The 13 nails problem is the kind of problem the course is mostly made of: silly, concrete, and checkable. A wrong idea announces itself by falling on the table. Conscious machines is its opposite. Nothing falls. On day two of a section called Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes, the challenge lets you feel what it is like to argue confidently about something you have no way to check, and then to notice that. It rehearses what session 4 calls "distinguishing things we know vs only imagine". The same day's readings supply the tools. Feynman's *Cargo Cult Science* is about activities with the outward form of science but not the part that makes it work. Platt's *The Art of Creative Thinking* is the source of the GamesWorth habit. Adams's first chapter is about attacking ill-defined problems. Turing's move in 1950, designing a test instead of answering "Can machines think?", is the same habit: turn a question about words into a question about what you could observe, and be honest when you cannot. ## Where it comes from Kempelen's chess-playing Turk (1770) was exhibited as a thinking automaton for 84 years. It was in fact worked by a chess player hidden in the cabinet. People suspected as much early and argued it in print: Joseph Racknitz in 1789, Robert Willis in 1821, Edgar Allan Poe in 1836. But each guessed wrong about the mechanism, and nobody described it accurately until 1857, three years after the machine burned in Philadelphia. The argument was settled not by debate but by someone finally looking inside. Alan Turing set the modern form in "Computing Machinery and Intelligence" (1950): he judged "Can machines think?" to be "too meaningless to deserve discussion" and replaced it with the imitation game. Searle's Chinese room (1980) argued that a system can pass any behavioural test while understanding nothing; Chalmers (1995) named the residue the "hard problem"; Koch and Tononi (2008) proposed a criterion based on integrated information. No test has yet been accepted by both sides. The question was still open when Winfree set it in August 2001, and it is still open now. ??? tip "Hints" - Treat your definitions as suspects: does your definition of "conscious" quietly contain the answer, for instance by requiring biology, or by requiring only behaviour? - Count your data. You have examined exactly one conscious system from the inside. Everything else is inferred from behaviour and resemblance to yourself. Does that inference stop, on principle, at machines? - Ask what observation would change your mind. If nothing could, your position is not a hypothesis about the world; it is a decision about how to use a word. That is not wrong, but it should be labelled. - Compare with the nails. What is the difference between not yet knowing the answer and there being no way to find out? ## What happened There is no solution to publish, and that is the point of the pairing. What follows is interpretive. Turing refused the question as posed and substituted a test whose outcome could be observed. He also noted that demanding inner certainty would, if applied consistently, deny that other people think, so we adopt "the polite convention that everyone thinks". Searle showed that such a test can be passed by a system that understands nothing from the inside. Chalmers named what the substitution loses. Seventy-five years on, no test has been accepted in advance by both a skeptic and a believer. The lesson is not that the question is silly but that it is a different kind of question from the nails: one where no experiment corrects you, so confidence is cheap and mistakes go undetected. Recognising which kind of question you are holding, before you invest a GamesWorth in it, is the skill this session exercises. ## Sources - **A. M. Turing**, "Computing Machinery and Intelligence", *Mind* LIX(236), 433–460 (1950) β€” [PDF, Oxford CS e-library](https://www.cs.ox.ac.uk/activities/ieg/e-library/sources/t_article.pdf){target=_blank} πŸ”“ - **John R. Searle**, "Minds, Brains, and Programs", *Behavioral and Brain Sciences* 3(3), 417–424 (1980) β€” [doi:10.1017/S0140525X00005756](https://doi.org/10.1017/S0140525X00005756){target=_blank} πŸ”’ - **David J. Chalmers**, "The Puzzle of Conscious Experience", *Scientific American*, December 1995 β€” [author's copy](https://consc.net/papers/puzzle.html){target=_blank} πŸ”“ - **Graham Oppy and David Dowe**, "The Turing Test", *Stanford Encyclopedia of Philosophy* (2003, rev. 2021) β€” [plato.stanford.edu](https://plato.stanford.edu/entries/turing-test/){target=_blank} πŸ”“ - **David Cole**, "The Chinese Room Argument", *Stanford Encyclopedia of Philosophy* (2004, rev. 2024) β€” [plato.stanford.edu](https://plato.stanford.edu/entries/chinese-room/){target=_blank} πŸ”“ - **Richard P. Feynman**, "Cargo Cult Science", Caltech commencement address (1974) β€” [Caltech Engineering and Science](https://calteches.library.caltech.edu/51/2/CargoCult.htm){target=_blank} πŸ”“ - **Christof Koch and Giulio Tononi**, "Can Machines Be Conscious?", *IEEE Spectrum* (2008) β€” [spectrum.ieee.org](https://spectrum.ieee.org/can-machines-be-conscious){target=_blank} πŸ”“ - **John Rader Platt**, *The Excitement of Science* (1962; 1974 reprint), containing "The Art of Creative Thinking" β€” [Internet Archive](https://archive.org/details/excitementofscie0000plat){target=_blank} πŸ”“ *(borrow)* - **Wikipedia**, "Mechanical Turk" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Mechanical_Turk){target=_blank} πŸ”“ - **ProTradeCraft**, "Jobsite Magic Trick: Balance 13 Nails on the Head of One" β€” [protradecraft.com](https://www.protradecraft.com/jobsite-magic-trick-balance-13-nails-head-one){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus is the only record: the title, the session (02, Thu 23 August 2001) and the label "second of two contrasting challenges", paired with the 13 nails problem of the day before. Winfree's archived lab site says nothing about this challenge β€” no handout, no problem sheet, no column β€” so his exact wording in class is lost. Reading "contrasting" as checkable against uncheckable is the editors' inference from the pairing and from the section title, which is why the identification is "probable" and the statement above is labelled a reconstruction. Candidate readings, most likely first: 1. An open-ended discussion challenge: could a machine be conscious, and how would you know? Posed as the opposite kind of problem from the nails. 2. A design challenge in the spirit of Adams, assigned the same day: specify what a machine would have to do before you would call it conscious. A sharper version of the first, not a rival. 3. A nonsense-detection exercise on a specific published claim about conscious computers, using Feynman's criteria. Least likely: the syllabus assigns no reading on machine consciousness. --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/square-windows/ --- title: "Square Windows" description: "Halve the light of a square window three feet high and three feet wide while keeping it square, three feet high and three feet wide: a one-paragraph puzzle whose only obstacle is an assumption you were never given." type: Activity tags: [course, student-facing, problem, section-1, false-assumptions, geometry, lewis-carroll, dudeney] status: stable problem: section: 1 session: 3 identification: probable kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: carroll-picture-book-1899 resource: "https://archive.org/details/lewiscarrollpict00carruoft" title: "The Lewis Carroll Picture Book: a selection from the unpublished writings and drawings of Lewis Carroll (letter to Helen Feilden, Christ Church, 15 March 1873, pp. 212-215; puzzle on p. 214)" author: "Charles L. Dodgson (Lewis Carroll); Stuart Dodgson Collingwood (ed.)" - id: dudeney-puzzles-1931 resource: "https://archive.org/details/in.ernet.dli.2015.219121" title: "Puzzles and Curious Problems, no. 205 'The Square Window' (puzzle p. 62; solution in the answers section)" author: "Henry Ernest Dudeney (ed. Alice Dudeney)" - id: proofwiki-dudeney-geometrical resource: "https://proofwiki.org/wiki/Henry_Ernest_Dudeney/Puzzles_and_Curious_Problems/Geometrical_Problems" title: "Henry Ernest Dudeney / Puzzles and Curious Problems / Geometrical Problems (list of puzzles 204-205 with statements)" author: "ProofWiki" - id: dudeney-gardner-1967 resource: "https://archive.org/details/536puzzlescuriou0000henr" title: "536 Puzzles and Curious Problems (Martin Gardner's combined edition of Modern Puzzles and Puzzles and Curious Problems)" author: "Henry Ernest Dudeney; Martin Gardner (ed.)" - id: knuth-dudeney-index-2001 resource: "https://www-cs-faculty.stanford.edu/~knuth/dudeney-twd.txt" title: "Dudeney's columns in The Weekly Dispatch (index, entry 1896.10.04 P99 'A remarkable window: trick question')" author: "Donald E. Knuth" - id: wakeling-carroll-games-1992 resource: "https://archive.org/details/lewiscarrollsgam0000carr" title: "Lewis Carroll's Games and Puzzles" author: "Lewis Carroll; Edward Wakeling (ed.)" - id: wiseman-friday-puzzle-2012 resource: "https://richardwiseman.wordpress.com/2012/04/30/answer-to-the-friday-puzzle-153/" title: "Answer to the Friday Puzzle (no. 153): the square window" author: "Richard Wiseman" - id: winfree-site-wayback-2002 resource: "https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/" title: "A. T. Winfree faculty web site (Wayback Machine capture, 25 Dec 2002)" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Square Windows Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 3. The only problem the syllabus lists for this session, a solo problem discussed alongside Judson's chapter "The Rage to Know".* !!! abstract "The problem" Lewis Carroll set this puzzle in a letter to a young friend, Helen Feilden, from Christ Church, Oxford, on 15 March 1873 (printed in *The Lewis Carroll Picture Book*, 1899, now in the public domain): > "A gentleman (a nobleman let us say, to make it more interesting) had a sitting-room with only one window in it β€” a square window, 3 feet high and 3 feet wide. Now he had weak eyes, and the window gave too much light, so (don't you like 'so' in a story?) he sent for the builder, and told him to alter it, so as only to give half the light. Only, he was to keep it square β€” he was to keep it 3 feet high β€” and he was to keep it 3 feet wide. How did he do it? Remember, he wasn't allowed to use curtains, or shutters, or coloured glass, or anything of that sort." In plain terms: the builder must produce a new window that is (1) square, (2) 3 feet high, (3) 3 feet wide, and (4) admits exactly half as much light as the old one, using ordinary clear glass and solid wall and nothing else. Before you reach for an answer, write down every condition the puzzle actually states, and then every condition you have silently added. The problem is only hard while the two lists are confused. Dudeney's later version (*Puzzles and Curious Problems*, 1931, no. 205) is shorter: a man had a window a yard square that let in too much light; he blocked up one half of it and still had a square window a yard high and a yard wide. How? Winfree's own statement of the problem has not survived; the wording above comes from the historical sources, not from his course materials. ![A square window, three feet high and three feet wide, filled with clear glass](https://tyson-swetnam.github.io/aosd/assets/images/problems/square-windows.svg){ width="560" } *The window before the alteration. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 3 falls early in Section 1, "Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes", and the syllabus assigns it a single task: "discuss Square Windows solo problem". Read as the Carroll and Dudeney puzzle, it is about the cleanest demonstration of a false assumption there is. Nothing in the statement says the window's sides run horizontal and vertical. Nearly everyone imports that condition anyway, and then declares the problem impossible. The syllabus describes the purpose of its puzzles in Winfree's own words: "The purpose of the puzzles (many of them silly) is to slow you down for a few minutes so you can examine the working of your own mind." The square window takes a few minutes, needs no apparatus, and rewards exactly that kind of self-examination. It also sets up the next session, where [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md) is labelled "distinguishing things we know vs only imagine": the constraint that blocks you here is imagined, not given. The habit it rehearses recurs all term: separate the conditions that were actually stated from the ones you supplied, and test which of your own "facts" are really facts. (This reading is our reconstruction from the syllabus wording and the section title; Winfree left nothing on the problem beyond the one schedule line.) ## Where it comes from The earliest dated appearance found is Charles L. Dodgson's letter of 15 March 1873 to Helen Feilden, written at Christ Church, Oxford, and printed by his nephew Stuart Dodgson Collingwood in *The Lewis Carroll Picture Book* (1899). Dodgson ends the letter with a rueful story about reducing a small girl to tears with the fox-goose-corn puzzle at a dinner party, and hopes that the square window will not have any awful effect on his correspondent. The book prints the puzzle without an answer. Henry Ernest Dudeney included it as "The Square Window", no. 205, in *Puzzles and Curious Problems* (1931), the posthumous collection assembled by his widow Alice Dudeney, whose prefatory note of 24 April 1931 thanks the *Daily News* and the *Strand Magazine*. Donald Knuth's index of Dudeney's 1896-1904 *Weekly Dispatch* columns lists a puzzle of 4 October 1896 called "A remarkable window", marked "trick question", which may be an earlier printing; that has not been confirmed. The puzzle has circulated in popular collections ever since, usually without attribution. Richard Wiseman's Friday Puzzle no. 153 of April 2012 restates it with a window one metre by one metre and names neither Carroll nor Dudeney. Whatever the wording, the point is the same. "High" and "wide" measure the extent of the opening, not necessarily the length of its sides. ??? tip "Hints" - Write the four stated conditions (square; 3 feet high; 3 feet wide; half the light) on one side of a page, and on the other side everything you have assumed but were not told. Compare the lists. - "Three feet high" says how far the opening reaches from bottom to top; it does not say that a 3-foot edge runs along the bottom. - Ask what other squares could touch all four sides of the original 3 ft by 3 ft square, then ask how much glass each one contains. - Draw the original square and both of its diagonals. They cut it into eight equal pieces. How many must you keep, and can the ones you keep make a square? - Dudeney's phrasing gives a strong nudge: the builder "blocked up one half of it". Which half, and in what shape? ??? success "Resolution" Keep the same 3 ft by 3 ft opening but turn the square through 45 degrees. The new window is the square whose four corners are the midpoints of the old window's four sides, and the four corner triangles outside it are bricked up. Its diagonals are the old window's horizontal and vertical midlines, so it is still exactly 3 feet from side to side and 3 feet from top to bottom, and it is still a square. Each blocked corner triangle is one-eighth of the original area (its legs are half-sides), so the four together remove exactly half the glass. Equivalently, a square with a 3 ft diagonal has area 3 x 3 / 2 = 4.5 square feet, half of 9. ![Before: the square window. After: a square turned 45 degrees inside the same opening, corners at the side midpoints, the four corner triangles bricked up](https://tyson-swetnam.github.io/aosd/assets/images/problems/square-windows-solution.svg){ width="560" } *Before and after. Drawn for this site (CC BY 4.0).* Dudeney's 1931 solution is the same picture. Of his own diagram he writes: "After he had blocked out the four triangles indicated by the dotted lines, he still had a square window, as seen, measuring a yard in height and a yard in breadth." The lesson is not the geometry but the moment of noticing that "high" and "wide" were never promised to be side lengths. ## Sources - **Charles L. Dodgson (Lewis Carroll)**, letter to Helen Feilden, Christ Church, 15 March 1873, in S. D. Collingwood (ed.), *The Lewis Carroll Picture Book* (1899), pp. 212-215 β€” [Internet Archive](https://archive.org/details/lewiscarrollpict00carruoft){target=_blank} πŸ”“ - **Henry Ernest Dudeney** (ed. Alice Dudeney), *Puzzles and Curious Problems* (1931), no. 205 "The Square Window", p. 62 β€” [Internet Archive](https://archive.org/details/in.ernet.dli.2015.219121){target=_blank} πŸ”“ - **ProofWiki**, "Henry Ernest Dudeney / Puzzles and Curious Problems / Geometrical Problems" β€” [proofwiki.org](https://proofwiki.org/wiki/Henry_Ernest_Dudeney/Puzzles_and_Curious_Problems/Geometrical_Problems){target=_blank} πŸ”“ - **Henry Ernest Dudeney**, ed. Martin Gardner, *536 Puzzles and Curious Problems* (1967) β€” the standard modern edition of Dudeney's later puzzles; not checked for this one β€” [Internet Archive](https://archive.org/details/536puzzlescuriou0000henr){target=_blank} πŸ”“ *(borrow)* - **Donald E. Knuth**, "Dudeney's columns in The Weekly Dispatch" (index, 2001) β€” [Stanford](https://www-cs-faculty.stanford.edu/~knuth/dudeney-twd.txt){target=_blank} πŸ”“ - **Lewis Carroll**, ed. Edward Wakeling, *Lewis Carroll's Games and Puzzles* (1992) β€” the standard collection of Carroll's puzzles; not checked for this one β€” [Internet Archive](https://archive.org/details/lewiscarrollsgam0000carr){target=_blank} πŸ”“ *(borrow)* - **Richard Wiseman**, "Answer to the Friday Puzzle (no. 153)" (2012) β€” [richardwiseman.wordpress.com](https://richardwiseman.wordpress.com/2012/04/30/answer-to-the-friday-puzzle-153/){target=_blank} πŸ”“ - **Arthur T. Winfree**, faculty web site, Wayback Machine capture of 25 December 2002 β€” searched by the editors; it links the course handout but states this problem nowhere β€” [web.archive.org](https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the name and the session: "discuss Square Windows solo problem" on Tuesday 28 August, between the [Conscious Machines](https://tyson-swetnam.github.io/aosd/problems/conscious-machines/index.md) challenge and the [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md). That one line is the whole of the Winfree record. The copy of the same handout archived on his faculty web site adds nothing to it, and no other page there mentions the problem. The identification therefore rests on the name and the context, not on any statement of the problem by Winfree. The candidates the editors weighed: - **The Carroll / Dudeney "Square Window" puzzle** (the reading given above). The title matches Dudeney's exactly, it is a one-paragraph solo puzzle with no apparatus, and its whole point is a hidden assumption, which fits the section's "False Assumptions" theme. The plural "Windows" fits a problem that compares two windows meeting the same verbal specification. This is the likeliest reading. - **Dudeney's companion puzzle no. 204, "The Donjon Keep Window"**, which sits immediately before "The Square Window" in the 1931 book and also turns on something understood but not stated. Its name does not match and it needs more geometry, but Winfree may have handed out both together, which would also explain the plural. - **The de Havilland Comet**, the first jet airliner, whose square-cornered windows concentrated stress and contributed to fatal metal-fatigue break-ups in 1953-54. A genuine "cherishing mistakes" episode, but a "solo problem" to "discuss" points to a puzzle rather than a case history, and nothing in Winfree's materials mentions the Comet. --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/ --- title: "Bookworm's Journey" description: "A worm bores from the first page of Volume I to the last page of Volume III on a shelf; how far does it travel? A classic trick puzzle about checking what you know against what you only imagine." type: Activity tags: [course, student-facing, problem, section-1, recreational-mathematics, hidden-assumptions, observation] status: stable problem: section: 1 session: 4 identification: confident kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: dudeney-1917 resource: "https://www.gutenberg.org/files/16713/16713-h/16713-h.htm" title: "Amusements in Mathematics, Problem 420: The Industrious Bookworm" author: "Henry Ernest Dudeney" - id: loyd-1914 resource: "https://commons.wikimedia.org/wiki/File:Cyclopedia_of_Puzzles_by_Samuel_Loyd.pdf" title: "Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks and Conundrums" author: "Sam Loyd (compiled by Sam Loyd Jr.)" - id: singmaster-sources resource: "https://www.puzzlemuseum.com/singma/singma5/SOURCES/SOURCE3.DOC" title: "Sources in Recreational Mathematics, section 7.AT, Bookworm's Distance" author: "David Singmaster" - id: singmaster-chronology resource: "https://utenti.quipo.it/base5/introduz/singchro.htm" title: "Chronology of Recreational Mathematics" author: "David Singmaster" - id: northrop-1944 resource: "https://store.doverpublications.com/products/9780486780160" title: "Riddles in Mathematics: A Book of Paradoxes" author: "Eugene P. Northrop" - id: northrop-1945-archive resource: "https://archive.org/details/dli.ernet.247854" title: "Riddles in Mathematics: A Book of Paradoxes (1945 English Universities Press printing, Digital Library of India scan)" author: "Eugene P. Northrop" - id: proofwiki-bookworm resource: "https://proofwiki.org/wiki/Bookworm_Riddle" title: "Bookworm Riddle" author: "ProofWiki contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Bookworm's Journey Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 4. Discussed together with [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md) and [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md).* !!! abstract "The problem" A set of volumes of a learned work stands in its usual order on a bookshelf: Volume I at the left, then Volume II, and so on, spines facing outward. A bookworm starts at the first page of Volume I and bores in a straight line, perpendicular to the covers, until it reaches the last page of the last volume. Suppose there are three volumes. In each, the leaves (the paper, not counting covers) are together 3 inches thick, and each cover is 1/8 inch thick. **How long is the bookworm's tunnel?** Before you compute, decide what you actually *know* about where "the first page of Volume I" and "the last page of Volume III" sit on the shelf, and what you are only *imagining*. If you have three matching books to hand, you are allowed to go and look. *Variant for a larger shelf:* an encyclopedia has ten volumes, each 2 inches thick including two 1/8-inch covers. Same question. The three-volume dimensions are H. E. Dudeney's, from "The Industrious Bookworm", Problem 420 of *Amusements in Mathematics* (1917, public domain), in which Professor Rackbrane reports that a bookworm "has actually bored a hole straight through from the first page to the last" of three volumes on his shelf. The ten-volume variant is the common modern retelling, reconstructed here. Winfree's own numbers are not recorded. ![Professor Rackbrane pointing at three volumes, I, II and III, standing on a shelf](https://tyson-swetnam.github.io/aosd/assets/images/problems/bookworms-journey-dudeney.png){ width="560" } *Dudeney's illustration for Problem 420, "The Industrious Bookworm", from Amusements in Mathematics (1917). Public domain, via Project Gutenberg eBook #16713.* ## Why it is in the course The syllabus lists this puzzle for session 4 with the note "distinguishing things we know vs only imagine". It shares the session with the [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md) ("facts before explanations of facts") and the [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md), all inside Section 1, whose theme is detecting nonsense, error checking, false assumptions and cherishing mistakes. Dudeney's own solution opens with "the hasty reader", who answers at once and answers wrongly. The mistake is not arithmetical. It comes from a picture the imagination supplies without being asked, and that picture stands in, unnoticed, for a fact anyone could check in ten seconds by walking to a shelf. So the puzzle is a miniature of the section's theme, and of its neighbour the Golden Tooth, where a learned explanation was built on a fact nobody had verified. Before you explain or calculate, ask which parts of your mental picture are observations and which are assumptions you never examined. The syllabus says the puzzles in this course are "contrived much as the organizers of an Easter Egg Hunt do", and that their purpose "is to slow you down for a few minutes so you can examine the working of your own mind". This one only slows you down if you refuse to trust the first picture that comes to mind. ## Where it comes from According to David Singmaster's *Sources in Recreational Mathematics* (section 7.AT, "Bookworm's Distance") and his *Chronology of Recreational Mathematics*, the earliest known printed appearance is in *Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks and Conundrums* (1914), compiled by Loyd's son after his death. Loyd's version is a timing puzzle with two volumes: Volume I has 100 leaves and Volume II 150 leaves; a "destructive little bookworm (Ptinus brunneus)" bores one leaf a minute and takes an hour to get through a cover; how long does it take to get from the first page of Volume I to the last page of Volume II? Singmaster suspected an earlier nineteenth-century origin but could not document one. H. E. Dudeney published the three-volume distance version as Problem 420, "The Industrious Bookworm", in *Amusements in Mathematics* (1917), with Professor Rackbrane pointing at three volumes on his shelf. Singmaster lists more than a dozen later appearances between 1924 and 1971, under titles such as "Bookworm's pilgrimage" (1939) and "Way of a worm" (1943), with two, three or four volumes. Eugene P. Northrop put it on page 1 of *Riddles in Mathematics: A Book of Paradoxes* (1944), in his opening chapter "What is a Paradox?". He treats the puzzle as a specimen of reasoning led astray by hasty judgment, the wrong answer following from a failure to examine every part of the problem. A few pages later he calls it "the problem of the bookworm's journey" β€” the closest published match to Winfree's title, though a natural enough name for it, so it is corroboration rather than evidence of how the puzzle reached him. Today it is usually posed with a ten- or twenty-volume encyclopedia. ??? tip "Hints" - Do not reach for arithmetic yet. First ask: on a shelf, which physical side of Volume I is its first page on, the left or the right? Which side of the last volume is its last page on? - Picture pulling Volume I off the shelf and opening it to page 1. Which way did you have to turn it? Where was page 1 relative to the neighbouring volume while the book was still on the shelf? - Try the smallest case: just two volumes. How much paper does the worm actually cross? - If you have any three matching books in the room, line them up and poke a pencil in at page 1 of the first. This is the "go and look" step, and it is the whole lesson. - Once you have the geometry right, the arithmetic is one line: count the covers crossed and the complete volumes crossed. ??? success "Resolution" When the volumes stand in order on a shelf with spines outward, each book's front cover faces right, toward the next volume, and its back cover faces left. So page 1 of Volume I is at the right-hand edge of Volume I, just inside its front cover and next to Volume II. The last page of the final volume is at the left-hand edge of that volume, just inside its back cover. A straight tunnel from one to the other passes through the front cover of Volume I, all of the intermediate volumes (paper and both covers each), and the back cover of the last volume. It does not pass through the leaves of the first or last volume at all. ![Plan view of three volumes on a shelf showing that page 1 of Volume I lies at its right edge next to Volume II, and the last page of Volume III at its left edge, so the bookworm's straight tunnel crosses only the front cover of Volume I, the whole of Volume II and the back cover of Volume III](https://tyson-swetnam.github.io/aosd/assets/images/problems/bookworms-journey.svg){ width="560" } *The shelf seen from above. Drawn for this site (CC BY 4.0).* **Dudeney's three-volume case** (leaves 3 inches per volume, covers 1/8 inch): the worm penetrates four covers (1/2 inch) and the leaves of Volume II (3 inches), a total of **3 1/2 inches**, against the hasty answer of 9 inches. In Dudeney's words: "You will find, on examining any three consecutive volumes on your shelves, that the first page of Vol. I. and the last page of Vol. III. are actually the pages that are nearest to Vol. II." **Ten-volume variant** (each volume 2 inches thick including two 1/8-inch covers): two covers (1/4 inch) plus eight whole volumes (16 inches) makes **16 1/4 inches**, not 20. **Two-volume case** (Loyd's original, 1914): the worm crosses only two covers and no paper at all. At Loyd's rate of one hour per cover, "it requires but two hours". A caveat worth raising in class: the answer depends on a convention (Western books, spines out, volumes in left-to-right order). A right-to-left script or a differently arranged shelf changes the geometry. That is the point. The shelf is a fact to be checked, not a thing to be assumed. ## Sources - **Henry Ernest Dudeney**, *Amusements in Mathematics*, Problem 420, "The Industrious Bookworm" (1917) β€” [Project Gutenberg eBook #16713](https://www.gutenberg.org/files/16713/16713-h/16713-h.htm){target=_blank} πŸ”“ - **Sam Loyd** (compiled by Sam Loyd Jr.), *Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks and Conundrums*, pp. 327 and 383 (1914) β€” [scan on Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Cyclopedia_of_Puzzles_by_Samuel_Loyd.pdf){target=_blank} πŸ”“ - **David Singmaster**, *Sources in Recreational Mathematics*, section 7.AT, "Bookworm's Distance" (c. 2004) β€” [SOURCE3.DOC](https://www.puzzlemuseum.com/singma/singma5/SOURCES/SOURCE3.DOC){target=_blank} πŸ”“ - **David Singmaster**, *Chronology of Recreational Mathematics* (c. 2000) β€” [link](https://utenti.quipo.it/base5/introduz/singchro.htm){target=_blank} πŸ”“ - **Eugene P. Northrop**, *Riddles in Mathematics: A Book of Paradoxes*, Chapter One, p. 1 (D. Van Nostrand, 1944; Dover reprint, ISBN 9780486780160) β€” [Dover Publications](https://store.doverpublications.com/products/9780486780160){target=_blank} πŸ”’ - **Eugene P. Northrop**, *Riddles in Mathematics: A Book of Paradoxes*, 1945 English Universities Press printing β€” [Internet Archive, Digital Library of India scan](https://archive.org/details/dli.ernet.247854){target=_blank} πŸ”“ - **ProofWiki contributors**, "Bookworm Riddle" β€” [ProofWiki](https://proofwiki.org/wiki/Bookworm_Riddle){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/golden-tooth/ --- title: "Golden Tooth" description: "Fontenelle's story of the Silesian boy's golden tooth: learned books explained it before anyone checked it, and the lesson is facts before explanations of facts." type: Activity tags: [course, student-facing, problem, section-1, history-of-science, credulity, verification] status: stable problem: section: 1 session: 4 identification: confident kind: case-study generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: fontenelle-1687-maigron resource: "https://archive.org/details/histoiredesoracl00fontuoft" title: "Histoire des oracles (1687), Γ©dition critique par Louis Maigron" author: "Bernard le Bovier de Fontenelle (ed. Louis Maigron)" - id: fontenelle-1688-english resource: "https://archive.org/details/historyoforacles00font" title: "The history of oracles, and the cheats of the pagan priests. In two parts. Made English" author: "Bernard le Bovier de Fontenelle (transl. attributed to Aphra Behn)" - id: horst-1596 resource: "https://archive.org/details/hin-wel-all-00001238-001" title: "De aureo dente maxillari pueri Silesii ... denuo auctus liber" author: "Jakob Horst" - id: spielman-2009 resource: "https://dental.nyu.edu/content/dam/nyudental/documents/about/rarebooks/GoldenToothArticleJDR.pdf" title: "The boy with the golden tooth: a 1593 case report of the first molded gold crown" author: "Andrew I. Spielman" - id: jutte-2004 resource: "https://www.sehepunkte.de/2005/06/6962.html" title: "Ein Wunder wie der goldene Zahn: Eine 'unerhΓΆrte' Begebenheit aus dem Jahre 1593 macht Geschichte(n)" author: "Robert JΓΌtte" - id: ruel-kellermann-vons-2020 resource: "https://classiques-garnier.com/revue-d-histoire-litteraire-de-la-france-4-2020-120e-annee-n-4-varia-diagnosis-controversy-and-technique.html" title: "Diagnostic, controverse et technique: l'histoire de la dent d'or (1683)" author: "Micheline Ruel-Kellermann and Jacqueline Vons" - id: museum-of-hoaxes resource: "https://hoaxes.org/archive/permalink/the_boy_with_the_golden_tooth" title: "The Boy with the Golden Tooth (1593)" author: "Alex Boese, Museum of Hoaxes" - id: kavoussi-2009 resource: "https://sciencebasedmedicine.org/james-restons-tooth-of-gold/" title: "James Reston's Tooth of Gold" author: "Ben Kavoussi, Science-Based Medicine" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Golden Tooth Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 4. Discussed together with [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md) and [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md).* !!! abstract "The problem" This session's "problem" is a short historical reading rather than a puzzle. Read the passage below, which Fontenelle published in 1687 (here in the anonymous English translation of 1688, usually attributed to Aphra Behn; spelling modernised and the long s resolved), and stop at the marked point before reading the ending. > Let us be assured of the matter of fact, before we trouble ourselves > with enquiring into the cause. It is true that this method is too slow > and dull for the greatest part of mankind, who run naturally to the > cause and pass over the truth of the matter of fact; but for my part, > I will not be so ridiculous as to find out a cause for what is not. > > This kind of misfortune happened so pleasantly at the end of the last > age to some learned Germans, that I cannot forbear speaking of it. In > the year 1593 there was a report that the teeth of a child of Silesia, > of seven years old, dropped out, and that one of gold came in the > place of one of his great teeth. Horstius, a physician in the > University of Helmstad, wrote in the year 1595 the history of this > tooth, and pretends that it was partly natural and partly miraculous, > and that it was sent from God to this infant to comfort the Christians > who were then afflicted by the Turks. Now fancy to yourself what a > consolation this was, and what this tooth could signify, either to the > Christians or the Turks. In the same year (that this tooth might not > want for historians) one Rolandus wrote a book of it. Two years after, > Ingolsteterus, another learned man, wrote against the opinion of > Rolandus concerning this golden tooth; and Rolandus presently makes a > learned reply. Another great man, named Libavius, collected all that > had been said of this tooth, to which he added his own opinion. > > **[Stop here.]** Before reading on, write in your GamesWorth book: 1. Which statements in the story are *observations*, which are *reports of observations*, and which are *explanations*? Who actually saw the tooth? 2. What is the single cheapest, most decisive thing anyone could have done in 1593, and who was the right person to do it? 3. What did each successive author treat as evidence? What did the existence of four learned books prove about the tooth? 4. Think of one claim you currently believe on the strength of secondhand report. What would "consulting the goldsmith" look like for it? (The four questions are a reconstruction by the site's editors, not Winfree's; the syllabus gives only the title line.) Then read Fontenelle's ending under "What happened" below, and compare it with your answer to question 2. ![Title page of Jakob Horst's 1596 treatise on the golden tooth of the Silesian boy](https://tyson-swetnam.github.io/aosd/assets/images/problems/golden-tooth-horst.jpg){ width="560" } *Jakob Horst, De aureo dente maxillari pueri Silesii, Leipzig 1596, title page of the enlarged edition of the first learned treatise on the tooth. Wellcome Library, London, via Internet Archive and Wikimedia Commons. Public domain.* ## Why it is in the course The syllabus line for session 4 begins "Golden Tooth: facts before explanations of facts." Those five words are Fontenelle's maxim compressed: "Let us be assured of the matter of fact, before we trouble ourselves with enquiring into the cause." The session sits in Section 1, Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes. It shares its day with Salvation of Doug and Bookworm's Journey, which the syllabus labels "distinguishing things we know vs only imagine". That schedule line is everything Winfree left us about the golden tooth; the editors found no other mention of it in his surviving course pages, so we do not know which text he handed out. Fontenelle's own passage, quoted above, is the obvious choice. The golden tooth is the cleanest historical case of a scholarly literature built on an unexamined report. Book after book appeared before anyone made the one cheap, decisive observation, and each new book treated the earlier books, not the tooth, as the thing to discuss. The lesson is not that the scholars were stupid. It is that the natural order of the mind, run to the cause and skip the fact, has to be reversed on purpose, and that the decisive check is often humble and belongs to a craftsman rather than a professor. The story also sets up session 5, where [N-rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md) show the same pattern among trained physicists three centuries later. Its companions in session 4 drill the same habit on a small scale: separate what a problem states from what you imagined it said. (The syllabus says this of Bookworm's Journey; extending it to Salvation of Doug is our inference.) ## Where it comes from Around Easter 1593 word spread from the Silesian village of Weigelsdorf that a seven-year-old boy, Christoph MΓΌller, had lost his milk teeth and grown a gold molar. He was shown for money. Jakob Horst, professor of medicine at Helmstedt and himself a Silesian, examined the boy while in the region on private business, had the tooth tested with a touchstone, and in 1595 published *De aureo dente maxillari pueri Silesii*, enlarged in 1596. His verdict: the tooth was partly natural and partly miraculous, produced by an unusual planetary configuration at the boy's birth and sent to comfort Christians then at war with the Turks. Martin Ruland published his own account in 1595 as well. Johann Ingolstetter attacked Ruland in 1597, and Ruland replied. Andreas Libavius reviewed the whole controversy. Horst's Helmstedt colleague Duncan Liddel argued that the tooth must be man-made. Note the dates. The fraud was found out by 1596 at the latest: a letter of 31 December 1595 already reports it, and JΓΌtte dates the exposure to 1596. The learned quarrel therefore ran on for at least a year after the thing being explained had been shown not to exist. Anton van Dale retold the story near the end of *De oraculis ethnicorum* (1683), quoting Daniel Sennert. Fontenelle, adapting Van Dale, moved it to the head of his argument in the *Histoire des oracles* (Paris, 1687) and gave it its famous frame. It has been a standard exhibit in discussions of scientific method ever since. Kavoussi (2009) aims it at the acupuncture literature that grew up after 1971. ![Portrait of Bernard le Bovier de Fontenelle by Nicolas de LargilliΓ¨re](https://tyson-swetnam.github.io/aosd/assets/images/problems/golden-tooth-fontenelle.jpg){ width="560" } *Nicolas de LargilliΓ¨re, Portrait of Fontenelle, 18th century, MusΓ©e des Beaux-Arts de Chartres. Public domain, via Wikimedia Commons.* ??? tip "Hints" - Underline every sentence in the story that reports something someone saw or measured, and every sentence that explains. Count them. Notice which kind came first. - Ask what the cheapest decisive test would have been, and who was competent to make it. A touchstone and a goldsmith were available in every market town. - Notice that Ingolstetter argued against Ruland, and Libavius summarised everyone: each author took the previous books as the thing to be discussed, not the tooth. - Ask who benefited. The boy was being shown for money; the professors gained reputations. Motives do not settle facts, but they tell you where to look first. - Fontenelle's method "is too slow and dull for the greatest part of mankind". Find a claim in your own field that you accepted from a report, and write down what consulting the goldsmith would cost. ## What happened Fontenelle's ending (1688 translation, spelling modernised): > In fine there wanted nothing to so many famous works, but only the truth of its being a golden tooth. For when a goldsmith had examined it, he found that it was only a thin plate of gold fixed to the tooth with a great deal of art. Thus they first went about to compile books, and afterwards they consulted the goldsmith. > > Nothing is more natural than to do the same thing in all other cases. And I am not so convinced of our ignorance by the things that are, and of which the reasons are unknown, as by those which are not, and for which we yet find out reasons. That is to say, as we want those principles that lead us to truth, so we have those which agree exceeding well with error and falsehood. In the French: "Il ne manquoit autre chose Γ  tant de beaux Ouvrages, sinon qu'il fust vray que la dent estoit d'or. Quand un OrfΓ¨vre l'eut examinΓ©e, il se trouva que c'estoit une feuille d'or appliquΓ©e Γ  la dent avec beaucoup d'adresse; mais on commenΓ§a par faire des Livres, et puis on consulta l'OrfΓ¨vre." Two early accounts of the exposure survive, and they do not describe the same scene. In the account that reached Daniel Sennert from Breslau, the boy was shown to an assembly of learned men. A goldsmith rubbed the tooth on a touchstone; the streak vanished when he applied a reagent. A physician then found a small hole in the crown, worked an iron spatula under it, and lifted off a thin brass shell, perhaps gilded, fitted over an ordinary tooth. The other account is a letter of 31 December 1595, printed in the introduction to Duncan Liddel's treatise. There the plate is set so deep into the gum that the join goes unnoticed, until repeated rubbing on touchstones and ordinary chewing wear it thin. A drunken nobleman, refused a look at the tooth, stabs the boy in the cheek; the surgeon who stitches the wound finds the plate. In both accounts the man showing the boy fled. Horst's book survives, as Spielman (2009) notes, as the first description of a moulded gold dental crown. The lesson in Winfree's five words is the one Fontenelle drew: establish the fact, by the most direct means available, before spending an hour on its cause, and be most suspicious of explanations that come easily. ## Sources - **Bernard le Bovier de Fontenelle**, *Histoire des oracles* (1687), Γ©dition critique par Louis Maigron (Paris, 1908); the golden tooth is PremiΓ¨re Dissertation, ch. IV, pp. 31–33 β€” [Internet Archive](https://archive.org/details/histoiredesoracl00fontuoft){target=_blank} πŸ”“ - **Bernard le Bovier de Fontenelle**, *The history of oracles, and the cheats of the pagan priests. Made English* (London, 1688; translation attributed to Aphra Behn), first part pp. 21–23 β€” [Internet Archive](https://archive.org/details/historyoforacles00font){target=_blank} πŸ”“ - **Jakob Horst**, *De aureo dente maxillari pueri Silesii ... denuo auctus liber* (Leipzig, 1596; Latin) β€” [Internet Archive, Wellcome Library scan](https://archive.org/details/hin-wel-all-00001238-001){target=_blank} πŸ”“ - **Andrew I. Spielman**, "The boy with the golden tooth: a 1593 case report of the first molded gold crown", *Journal of Dental Research* 88(1): 8–11 (2009) β€” [author-hosted PDF](https://dental.nyu.edu/content/dam/nyudental/documents/about/rarebooks/GoldenToothArticleJDR.pdf){target=_blank} πŸ”“ - **Robert JΓΌtte**, *Ein Wunder wie der goldene Zahn* (Ostfildern: Thorbecke, 2004) β€” [review in sehepunkte 5 (2005)](https://www.sehepunkte.de/2005/06/6962.html){target=_blank} πŸ”“ (the review is open; the book is in print, not online) - **Micheline Ruel-Kellermann and Jacqueline Vons**, "Diagnostic, controverse et technique: l'histoire de la dent d'or (1683)", *Revue d'histoire littΓ©raire de la France* 120(4): 831–844 (2020) β€” [Classiques Garnier](https://classiques-garnier.com/revue-d-histoire-litteraire-de-la-france-4-2020-120e-annee-n-4-varia-diagnosis-controversy-and-technique.html){target=_blank} πŸ”’ - **Alex Boese**, "The Boy with the Golden Tooth (1593)", *Museum of Hoaxes* β€” [hoaxes.org](https://hoaxes.org/archive/permalink/the_boy_with_the_golden_tooth){target=_blank} πŸ”“ - **Ben Kavoussi**, "James Reston's Tooth of Gold", *Science-Based Medicine* (25 August 2009) β€” [sciencebasedmedicine.org](https://sciencebasedmedicine.org/james-restons-tooth-of-gold/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/ --- title: "Salvation of Doug" description: "A short parable about a retired biochemist and a retired geneticist who try to learn how cars work from a hill above a factory; read to separate what was observed from what was inferred." type: Activity tags: [course, student-facing, problem, section-1, genetics, inference, facts-vs-explanations, reading] status: stable problem: section: 1 session: 4 identification: confident kind: case-study generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: sullivan-lab-page resource: "https://sullivanlab.sites.ucsc.edu/salvation-of-doug/" title: "Salvation of Doug" author: "William T. Sullivan" - id: ucsc-review-both-stories resource: "https://review.ucsc.edu/spring04/bio-debate.html" title: "The Salvation of Doug: A Tale of Two Retired Scientists and Some Rope / The Demise of Bill" author: "William T. Sullivan; Douglas R. Kellogg" - id: ucsc-review-feature resource: "https://review.ucsc.edu/spring04/twoversions.html" title: "The Geneticist & the Biochemist: How a friendly rivalry illustrates the two cornerstones of biomedical research" author: "Tim Stephens" - id: ucsc-review-pdf resource: "https://review.ucsc.edu/spring04/UCSC_Review-spring04.pdf" title: "UC Santa Cruz Review, Vol. 41, No. 4 (March 2004), print PDF" author: "Tim Stephens (writer); Jim Burns (ed.)" - id: vincent-2016 resource: "https://doi.org/10.1039/c5ib00321k" title: "The appeasement of Doug: a synthetic approach to enhancer biology" author: "Ben J. Vincent, Javier Estrada, Angela H. DePace" - id: lau-2015 resource: "https://www.scienceandmathwithmrslau.com/2015/03/the-salvation-of-doug-and-the-demise-of-bill-what-genetics-and-biochemistry-are-all-about/" title: "The Salvation of Doug and the Demise of Bill: What Genetics and Biochemistry Are All About" author: "Mrs. Lau (high-school science teacher)" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Salvation of Doug Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 4. Discussed together with [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md) and [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md).* !!! abstract "The problem" This session's item is a reading, not a puzzle: William T. Sullivan's two-page parable **"The Salvation of Doug"**, free at the links under Sources. The *UC Santa Cruz Review* page prints Douglas Kellogg's reply, **"The Demise of Bill"**, beneath it. The parable is copyrighted, so it is summarised rather than reproduced here. **In brief (paraphrased).** Doug, a retired biochemist, and a retired geneticist watch an automobile factory from a hill: workers in each morning, cars out each afternoon. Neither knows how a car works. Doug grinds up a hundred cars, finds a car is about 10% glass, 25% plastic, 60% steel and 5% unidentified, then tries to mix the fractions back with a blowtorch to recover some "activity". The geneticist instead ties one worker's hands each morning and reads that afternoon's cars: no front or rear windows one day, though the side windows are there; no steering wheels the next, the cars piling up on the lawn at the first bend; and, when he ties a vice president's hands, no change at all. From each day he announces what he has learned, ending with the verdict that the vice president does nothing. Doug objects that several vice presidents may cover for one another. Next morning the geneticist heads down with a rope for every man in a suit, and Doug abandons his blowtorch and follows: his "salvation". As you read, keep two columns: what the two men actually *saw* each day, and what the geneticist *said he had learned*. (That exercise is the course editors' reconstruction; the syllabus gives only the title.) ![Workers entering a factory with one man's hands tied, cars leaving without a steering wheel, and a table separating each day's manipulation, observation and pronouncement](https://tyson-swetnam.github.io/aosd/assets/images/problems/salvation-of-doug.svg){ width="560" } *The geneticist's ledger. Only the middle column records what the two men saw. Drawn for this site (CC BY 4.0).* ![Men riding the bare chassis of a Model T as it comes off the assembly line at Ford's Highland Park plant](https://tyson-swetnam.github.io/aosd/assets/images/problems/salvation-of-doug-modelt.jpg){ width="560" } *"Model T's coming off the assembly line at the Highland Park plant." Library of Congress Prints and Photographs Division, LCCN 2011661047, via Wikimedia Commons. Public domain.* ## Why it is in the course Session 4 has three items in the syllabus: [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md), "facts before explanations of facts"; this reading, with no description at all; and [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md), "distinguishing things we know vs only imagine". The Golden Tooth is the story of learned men explaining a child's gold tooth before anyone checked that it was gold: explanation outrunning fact. Sullivan's parable is a laboratory version of the same error. Winfree left no note on how he used the reading; what follows is read from those two neighbours. Each afternoon delivers one plain observation; each evening the geneticist turns it into a confident causal claim. Some of those claims are sound, one overreaches, one is a classic false negative. Sorting them apart is the exercise. The ending shows how to settle the disagreement that follows: with a discriminating experiment, not a vote. ## Where it comes from William T. (Bill) Sullivan is a Drosophila geneticist at UC Santa Cruz. He says on his lab page that he uses the story in introductory genetics "to explain the rationale behind mutational analysis and to show younger students the basic differences between genetics and biochemistry". It took him about an hour to write, he told the *UC Santa Cruz Review*, against two years for a scientific paper. The Doug of the title is the biochemist Douglas R. Kellogg, who shared a laboratory with Sullivan at UC San Francisco: Sullivan a postdoctoral fellow, Kellogg a graduate student. Kellogg answered with "The Demise of Bill", in which the hand-tying geneticist declares seat belts "vestigial" because cars without them drive fine, then drives into a tree after a fruit fly crawls into his eye. Both parables were published in the 1990s and reprinted together by the *UC Santa Cruz Review* in 2004. Vincent, Estrada and DePace (2016) cite the original as *Generations* 1(3), 1993 β€” the only formal citation found, and one the editors could not check against the newsletter itself. The factory is the parable's engine: workers are genes, the parts they install are gene products, hand-tying is a loss-of-function mutation, and the cars on the lawn are a mutant phenotype. ??? tip "Hints" - Separate the manipulation (whose hands were tied), the observation (what the cars did) and the pronouncement (what he said he had learned). Only the middle one is a fact. - For the steering-wheel day, ask what else was missing from those cars. Did anyone look? - For the vice-president day, list every state of the world in which tying one suit's hands leaves the cars unchanged. Redundancy is one; a delayed effect is another. ## What happened There is no answer key; the reading is the lesson. **What was known.** One worker disabled, no front or rear windows but side windows as usual. Another disabled, no steering wheels and cars that fail the first turn. A vice president disabled, nothing changed. Everything else is inference, though "this worker is *needed* for the windows" follows closely. **The overreach.** "Steering wheels are responsible for turning the car" adds a mechanism to a correlation: a missing part upstream of the wheel would give the same lawn full of cars. A mutant phenotype tells you a gene is *required* for a process, not that it *performs* it. **The false negative.** "No effect, therefore the vice president does nothing" is the central mistake; Doug's redundancy objection is the standard rejoinder for knockouts with no phenotype. Tying all the suits at once is the right move: a multiple mutant rather than an argument. **Doug's side.** His percentages are real, but they answer what a car is made of, not how it works. Kellogg's reply gives the biochemist a better programme: take one car apart and trace the spark plug and its wiring. The moral of the session's placement is not that genetics beats biochemistry. It is: record the observation before the explanation, list the alternatives, then design the experiment that tells them apart. ## Sources - **William T. Sullivan**, "Salvation of Doug" (1993; author's lab page, undated) β€” [Sullivan Lab, UC Santa Cruz](https://sullivanlab.sites.ucsc.edu/salvation-of-doug/){target=_blank} πŸ”“ - **William T. Sullivan; Douglas R. Kellogg**, "The Salvation of Doug: A Tale of Two Retired Scientists and Some Rope" and "The Demise of Bill", *UC Santa Cruz Review* (Spring 2004 reprint of both stories) β€” [review.ucsc.edu](https://review.ucsc.edu/spring04/bio-debate.html){target=_blank} πŸ”“ - **Tim Stephens**, "The Geneticist & the Biochemist: How a friendly rivalry illustrates the two cornerstones of biomedical research", *UC Santa Cruz Review* (2004) β€” [review.ucsc.edu](https://review.ucsc.edu/spring04/twoversions.html){target=_blank} πŸ”“ - **Tim Stephens (writer); Jim Burns (ed.)**, *UC Santa Cruz Review*, Vol. 41, No. 4 (March 2004), print PDF β€” [review.ucsc.edu](https://review.ucsc.edu/spring04/UCSC_Review-spring04.pdf){target=_blank} πŸ”“ - **Ben J. Vincent, Javier Estrada, Angela H. DePace**, "The appeasement of Doug: a synthetic approach to enhancer biology", *Integrative Biology* 8(4), 475–484 (2016) β€” [doi:10.1039/c5ib00321k](https://doi.org/10.1039/c5ib00321k){target=_blank} πŸ”’ - **Mrs. Lau (high-school science teacher)**, "The Salvation of Doug and the Demise of Bill: What Genetics and Biochemistry Are All About" (2015) β€” [Science with Mrs. Lau](https://www.scienceandmathwithmrslau.com/2015/03/the-salvation-of-doug-and-the-demise-of-bill-what-genetics-and-biochemistry-are-all-about/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/n-rays/ --- title: "N-Rays" description: "Blondlot's imaginary radiation of 1903-1904, exposed by R. W. Wood's blind tests and dissected in Langmuir's Pathological Science: a case study in detecting nonsense before explaining it." type: Activity tags: [course, student-facing, problem, section-1, history-of-science, pathological-science, verification, perception] status: stable problem: section: 1 session: 5 identification: confident kind: case-study generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: wood-1904-nature resource: "https://doi.org/10.1038/070530a0" title: "The n-Rays" author: "R. W. Wood" - id: nature-70-1822-scan resource: "https://archive.org/details/dbc.wroc.pl.023949" title: "Nature, vol. 70, no. 1822 (29 September 1904), full issue scan containing Wood's letter on pp. 530-531" author: "R. W. Wood (letter); Macmillan (publisher)" - id: langmuir-1953-transcript resource: "https://www.cs.princeton.edu/~ken/Langmuir/langmuir.htm" title: "Pathological Science (colloquium at the Knolls Research Laboratory, 18 December 1953; transcribed by R. N. Hall)" author: "Irving Langmuir" - id: langmuir-1989-physics-today resource: "https://aip.brightspotcdn.com/PTO.v42.i10.36_1.online.pdf" title: "Pathological Science" author: "Irving Langmuir; edited by Robert N. Hall" - id: blondlot-1905 resource: "https://archive.org/details/nrayscollectiono00blonrich" title: "\"N\" Rays: a collection of papers communicated to the Academy of Sciences (translated by J. Garcin)" author: "R. Blondlot" - id: seabrook-1941 resource: "https://archive.org/details/doctor-wood" title: "Doctor Wood, Modern Wizard of the Laboratory (chapter 17)" author: "William Seabrook" - id: nye-1980 resource: "https://doi.org/10.2307/27757473" title: "N-Rays: An Episode in the History and Psychology of Science" author: "Mary Jo Nye" - id: klotz-1980 resource: "https://doi.org/10.1038/scientificamerican0580-168" title: "The N-Ray Affair" author: "Irving M. Klotz" - id: lagemann-1977 resource: "https://doi.org/10.1119/1.10643" title: "New light on old rays: N rays" author: "R. T. Lagemann" - id: wikipedia-n-ray resource: "https://en.wikipedia.org/wiki/N-ray" title: "N-ray" author: "Wikipedia contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # N-Rays Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 5. Read together with Langmuir's "Pathological Science"; discussed alongside [Phone Cord Problem](https://tyson-swetnam.github.io/aosd/problems/phone-cord-problem/index.md) and [Stockholm Restrooms](https://tyson-swetnam.github.io/aosd/problems/stockholm-restrooms/index.md).* !!! abstract "The problem" This is a case study rather than a puzzle. Read the situation as it looked in 1904, **before** looking at what happened, and decide what you would have done. **The situation as reported.** In March 1903 RenΓ© Blondlot, professor of physics at the University of Nancy, a corresponding member of the Institute of France and an experienced experimenter on electromagnetic waves, announced a new kind of radiation, which he named **N rays** after his town. He said the rays came from an X-ray tube, from a gas mantle, from a hot platinum wire, from the Sun, and later from hardened steel, unannealed glass and other bodies under strain; they passed through metals, wood and black paper but were stopped by wet paper and by water; they could be refracted by an aluminium prism into a spectrum of lines and focused by an aluminium lens. The detector was the human eye. A small electric spark, a tiny gas flame, or a screen of phosphorescent calcium sulphide was viewed in a nearly dark room; when the rays fell on it, it was said to glow a little more brightly. There was no meter and no photographic effect that survived scrutiny. Blondlot himself reported that a thermopile registered nothing: "The action of 'N' rays on this apparatus was absolutely nil, even in the most favorable conditions, though a candle placed 12 metres away from the thermopile gave a deflection of about 0.5 mm. on the scale." By the first half of 1904 nearly a hundred papers on N rays had appeared in the *Comptes rendus* of the Academy alone. Other workers reported N rays from muscle, nerves and the brain, from a vibrating tuning fork, from growing plants, even from a corpse; one claimed that metals could be anaesthetized with ether, after which they stopped emitting. Physicists in Germany, England and America tried to repeat the experiments and saw nothing. The Academy nevertheless awarded Blondlot a major prize for 1904. Blondlot's instructions for observers (1905 English edition of his papers, public domain) ran, in part: > "...in no way to try to fix the eye upon the luminous source, whose > variations in glow one wishes to ascertain. On the contrary, one > must, so to say, see the source without looking at it, and even > direct one's glance vaguely in a neighbouring direction. The observer > must play an absolutely passive part, under penalty of seeing > nothing. Silence should be observed as much as possible. Any smoke, > and especially tobacco smoke, must be carefully avoided... the > observer should accustom himself to look at the screen just as a > painter, and in particular an 'impressionist' painter, would look at > a landscape. To attain this requires some practice, and is not an > easy task. Some people, in fact, never succeed." **Your task.** 1. You are a visiting physicist, invited to spend one evening in Blondlot's darkened laboratory. You cannot see any of the effects yourself, and your host says that is because your eyes are not sensitive enough. **Design tests you could carry out on the spot, with the apparatus already in the room, that would distinguish a real radiation from an observer's expectation.** Say exactly what you would change, what the observers must not know, and what result would count as evidence either way. 2. List the features of the whole affair, as reported above, that should have raised suspicion **before** anyone travelled to Nancy. Which of them are general warning signs that could apply to a claim in your own field today? 3. Blondlot and most of his confirmers were honest, competent, and sincere. Explain how a laboratory full of such people can go on seeing something that is not there for more than a year, and what, in the way experiments were arranged, made it possible. Only then read "What happened" and compare your list with Wood's tests and with Langmuir's six symptoms. *Reconstructed for the course from Wood's 1904 letter to Nature, from Blondlot's own 1905 volume, and from Seabrook's 1941 memoir of Wood. The three numbered tasks are editorial framing, not Winfree's words: the syllabus gives only the title line.* ![Schematic of Blondlot's N-ray spectroscope: a lamp in an iron box with an aluminium window sends invisible rays through a slit to an aluminium prism (drawn dashed, because Wood secretly removed it) and on to a movable card with a phosphorescent line read by an observer in the dark.](https://tyson-swetnam.github.io/aosd/assets/images/problems/n-rays.svg){ width="560" } *Blondlot's N-ray spectroscope, drawn for this site (CC BY 4.0) from his own description in "N" Rays (1905) and from Wood's account in Nature, 29 September 1904.* ## Why it is in the course The syllabus lists the session-5 reading as "N-Rays" and Langmuir's *Pathological Science*, in the section headed Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes. The session before was "[Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md): facts before explanations of facts"; the session after brings "some ways to check for errors" and "hidden assumptions". N rays are the physicist's golden tooth: a year of increasingly elaborate explanations (refraction, dispersion into spectral lines, rays from hardened steel, from anaesthetized metals, from nerves and the brain) built on a "fact" that had never been checked. The episode shows three things. First, honest and skilled people are the easiest to deceive when the detector is their own expectation and the signal sits at the threshold of perception. Langmuir's six symptoms are a checklist for spotting such cases from outside. Second, error-checking is a procedure you can design. Wood's tests β€” blind trials, silent removal of the essential component, substitution of an inert object, and the sealed-screen photograph he proposed but never ran β€” are a model for the next session's "ways to check for errors". Third, there is a social cost to objecting. The syllabus describes class practice as "braving social opprobrium by blurting out nutty ideas, and risking devastating counter-attack by publicly objecting to nonsense blurted by others"; dozens of scientists confirmed the rays instead, and it took an outsider with nothing at stake to say so in print. ## Where it comes from Prosper-RenΓ© Blondlot (1849-1930) came upon the supposed rays while trying to detect polarization of X rays with a small spark as the detector, and announced them on 23 March 1903 in a note titled "On a New Species of Light". He published about twenty papers over the next two years; Augustin Charpentier added about twenty more and Jean Becquerel about ten. Later tallies give some 300 papers by about 120 authors in all. Blondlot collected his own in *"N" Rays* (Paris 1904; English translation by J. Garcin, 1905), which closes with a section headed "How the Action of 'N' Rays should be observed". Heinrich Rubens in Berlin could see nothing; by Wood's account the Kaiser had commanded him to demonstrate the rays at Potsdam, and after two weeks he had to confess he could not. At the British Association meeting in Cambridge, Rubens urged Robert W. Wood of Johns Hopkins to go to Nancy in his place. Wood's letter, written from Brussels and printed in *Nature* on 29 September 1904, named neither Blondlot nor the laboratory. The *Revue scientifique* ran an inquiry; of about forty published replies only about half a dozen backed Blondlot. Accounts of his prize differ β€” the Prix Leconte of 50,000 francs in one telling, the Lalande prize and a gold medal in Wood's β€” but both agree that the citation read at the December 1904 meeting honoured his life's work rather than N rays. Foreign belief evaporated by 1905, and Blondlot stepped down from his Nancy chair in 1910. Seabrook's claim that the exposure drove Blondlot mad is not supported by later historians; he died at Nancy in 1930, aged 81. On 18 December 1953 Irving Langmuir, Nobel laureate in chemistry, gave a colloquium at General Electric's Knolls Research Laboratory on what he called "pathological science", "the science of things that aren't so", using the Davis-Barnes effect, N rays, mitogenetic rays, the Allison effect, ESP and flying saucers as examples. R. N. Hall's transcript circulated as a GE report in 1968 and was published in *Physics Today* in October 1989. The syllabus names only "N-Rays" and "Langmuir, *Pathological Science*"; which text Winfree photocopied for the first item is not recorded. ![Blondlot's 1904 photographs of a small electric spark without and with N rays, his Figs. 6 and 7](https://tyson-swetnam.github.io/aosd/assets/images/problems/n-rays-blondlot-spark.png){ width="560" } *Blondlot's Figs. 6 and 7, from the English edition of his collected papers: a spark photographed "without" and "with" N rays, the rays said to come first from a Nernst lamp and then from two large files. This is the kind of evidence Wood criticised as built from hand-alternated exposures. R. Blondlot, 1904, public domain, via Wikimedia Commons.* ??? tip "Hints" - Ask what the detector is. If every reported effect is "the screen looked a little brighter to the observer", the observer, not the screen, is the instrument, and it is the observer who must be calibrated. - Every test you propose should have a version in which the observer does not know whether the "cause" is present. What could you silently remove, cover or replace in a dark room? - Blondlot claimed to resolve spectrum lines less than 0.1 mm apart through a slit 2-3 mm wide. Before thinking about new physics, ask whether the numbers are even consistent with the apparatus. - Notice the asymmetry: ten bricks give no more effect than one, a stronger source gives no stronger signal, and every failure to see the rays is explained by the observer's insensitivity or fatigue. What kind of claim can never be refuted this way? - Ask who was confirming the rays and who was not, and what each group had to gain or lose. Then ask why it took an outsider to run the decisive test. ## What happened Wood spent about three hours in Blondlot's laboratory in September 1904. His own account, from the *Nature* letter (public domain): > "I suggested that the attempt be made to announce the exact moments at which I introduced my hand into the path of the rays, by observing the screen. In no case was a correct answer given, the screen being announced as bright and dark in alternation when my hand was held motionless in the path of the rays, while the fluctuations observed when I moved my hand bore no relation whatever to its movements." On the spectroscope, maxima less than 0.1 mm apart were being read from a ray bundle 3 mm wide. Wood said he was surprised, and "was told that this was one of the inexplicable and astounding properties of the rays": > "...I subsequently found that the removal of the prism (we were in a dark room) did not seem to interfere in any way with the location of the maxima and minima in the deviated (!) ray bundle. I then suggested that an attempt be made to determine by means of the phosphorescent screen whether I had placed the prism with its refracting edge to the right or the left, but neither the experimenter nor his assistant determined the position correctly in a single case (three trials were made). This failure was attributed to fatigue." A steel file was supposed to emit N rays that made a dimly lit clock face easier to see: "the substitution of a piece of wood of the same size and shape as the file in no way interfered with the experiment. The substitution was of course unknown to the observer." Wood concluded that all the changes in luminosity "are purely imaginary", and proposed a properly blinded test: a dozen photographs taken through two sealed aluminium screens, one holding wet paper and one dry, by a person who never knew which was in use. In Seabrook's biography Wood added a detail he left out of the letter. The assistant, grown suspicious, asked to repeat a reading himself; Wood walked audibly toward the prism but did not touch it, and the assistant announced that he saw no spectrum because "the American has made some derangement". Wood says there that he put the prism back before the lights came up, so Langmuir's retelling, in which Wood pockets it, is an embellishment. Why did they see it? Every effect sat at the threshold of visibility, where, as Langmuir put it, "you really don't know whether you are seeing it or not". The observers always knew when the rays were supposed to be on, and no one had run a blind trial. The photographs were no better: they were built from five-second exposures alternated by hand by an experimenter who knew which image was which, on a spark whose brightness Wood estimated varied by 25 per cent on its own. Langmuir's symptoms of pathological science (Table I of the Hall transcript; items 3 to 5 are his exact wording, the others lightly condensed): 1. The maximum effect is produced by a causative agent of barely detectable intensity, and the magnitude of the effect is substantially independent of the intensity of the cause. 2. The effect remains close to the limit of detectability, or many measurements are needed because of the very low statistical significance of the results. 3. Claims of great accuracy. 4. Fantastic theories contrary to experience. 5. Criticisms are met by ad hoc excuses thought up on the spur of the moment. 6. The ratio of supporters to critics rises to somewhere near 50 percent and then falls gradually to oblivion. Langmuir stressed that "these are cases where there is no dishonesty involved", but where people are "led astray by subjective effects, wishful thinking or threshold interactions". Every symptom can be checked against the story: "Ten bricks didn't have any more effect than one" (1), the 0.1 mm maxima read through a slit a few millimetres wide (3), rays from anaesthetized metals and from a corpse (4), "your eyes are not sensitive enough" and "fatigue" (5), the collapse after 1904 (6). The lesson for the course: a fact must be established before it is explained, and the way to establish it is a test whose outcome the observer cannot anticipate. Wood did not argue about the theory of N rays at all. He changed one thing the observers could not see and watched whether their reports changed with it. ## Sources - **R. W. Wood**, "The n-Rays", *Nature* 70 (no. 1822): 530–531 (29 September 1904) β€” [doi:10.1038/070530a0](https://doi.org/10.1038/070530a0){target=_blank} πŸ”’ (the text itself is public domain; full scan below) - **Nature, vol. 70, no. 1822 (29 September 1904)**, full issue scan containing Wood's letter, Lower Silesian Digital Library β€” [Internet Archive](https://archive.org/details/dbc.wroc.pl.023949){target=_blank} πŸ”“ - **Irving Langmuir**, "Pathological Science", colloquium at the Knolls Research Laboratory, 18 December 1953, transcribed by R. N. Hall (GE report 68-C-035, 1968) β€” [full transcript, Princeton](https://www.cs.princeton.edu/~ken/Langmuir/langmuir.htm){target=_blank} πŸ”“ - **Irving Langmuir, edited by Robert N. Hall**, "Pathological Science", *Physics Today* 42(10): 36–48 (October 1989) β€” [free PDF](https://aip.brightspotcdn.com/PTO.v42.i10.36_1.online.pdf){target=_blank} πŸ”“ - **R. Blondlot**, *"N" Rays: a collection of papers communicated to the Academy of Sciences*, translated by J. Garcin (Longmans, Green, 1905) β€” [Internet Archive](https://archive.org/details/nrayscollectiono00blonrich){target=_blank} πŸ”“ - **William Seabrook**, *Doctor Wood, Modern Wizard of the Laboratory* (Harcourt, Brace, 1941), chapter 17 β€” [Internet Archive](https://archive.org/details/doctor-wood){target=_blank} πŸ”“ - **Mary Jo Nye**, "N-Rays: An Episode in the History and Psychology of Science", *Historical Studies in the Physical Sciences* 11(1): 125–156 (1980) β€” [doi:10.2307/27757473](https://doi.org/10.2307/27757473){target=_blank} πŸ”’ - **Irving M. Klotz**, "The N-Ray Affair", *Scientific American* 242(5): 168–175 (May 1980) β€” [doi:10.1038/scientificamerican0580-168](https://doi.org/10.1038/scientificamerican0580-168){target=_blank} πŸ”’ - **R. T. Lagemann**, "New light on old rays: N rays", *American Journal of Physics* 45(3): 281–284 (1977) β€” [doi:10.1119/1.10643](https://doi.org/10.1119/1.10643){target=_blank} πŸ”’ - **Wikipedia contributors**, "N-ray" (overview) β€” [Wikipedia](https://en.wikipedia.org/wiki/N-ray){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/phone-cord-problem/ --- title: "Phone Cord Problem" description: "A coiled telephone cord whose plugs never turn still twists itself into a tangle with a loop where the spiral reverses: a five-minute exercise in facts before explanations." type: Activity tags: [course, student-facing, problem, section-1, topology, everyday-physics, hidden-assumptions, observation] status: stable problem: section: 1 session: 5 identification: probable kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2001 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery, EEB 479/479H/579 course handout (retrospective syllabus, Spring 2001)" author: "A. T. Winfree" - id: darwin-1865 resource: "https://www.gutenberg.org/ebooks/2485" title: "The Movements and Habits of Climbing Plants" author: "Charles Darwin" - id: goriely-tabor-1998 resource: "https://doi.org/10.1103/PhysRevLett.80.1564" title: "Spontaneous Helix Hand Reversal and Tendril Perversion in Climbing Plants" author: "Alain Goriely and Michael Tabor" - id: mcmillen-goriely-2002 resource: "https://doi.org/10.1007/s00332-002-0493-1" title: "Tendril Perversion in Intrinsically Curved Rods" author: "T. McMillen and A. Goriely" - id: baltimore-sun-1998 resource: "https://www.baltimoresun.com/1998/04/22/science-and-math-with-a-twist-kinks-researchers-are-studying-the-characteristics-that-make-materials-live-and-inorganic-develop-natural-coils-and-twists/" title: "Science and math, with a twist: Kinks" author: "Baltimore Sun" - id: science-news-1998 resource: "https://www.sciencenews.org/archive/mathematicians-describe-tendril-perversion" title: "Mathematicians Describe Tendril Perversion" author: "M. N. Jensen (Science News)" - id: amsi-stavely-2015 resource: "https://rhed.amsi.org.au/phone-cords-get-tangled/" title: "Why do phone cords get tangled?" author: "Will Stavely (AMSI Research and Higher Education, Monash University)" - id: wikipedia-tendril-perversion resource: "https://en.wikipedia.org/wiki/Tendril_perversion" title: "Tendril perversion" author: "Wikipedia" - id: silva-2016 resource: "https://doi.org/10.1038/srep23413" title: "Perversions with a twist" author: "P. E. S. Silva, J. L. Trigueiros, A. C. Trindade, R. Simoes, R. G. Dias, M. H. Godinho, F. Vistulo de Abreu" - id: anandtech-2004 resource: "https://forums.anandtech.com/threads/topology-of-phone-cord-twist.1452427/" title: "Topology of Phone cord twist (forum thread)" author: "AnandTech Forums users" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Phone Cord Problem Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 5. Discussed together with [Stockholm Restrooms](https://tyson-swetnam.github.io/aosd/problems/stockholm-restrooms/index.md), in the session whose readings are [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md) and Langmuir's "Pathological Science".* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's schedule says only "discuss phone cord problem". The statement below is built from that name, from the session's theme, and from the everyday form of the puzzle. An old-fashioned desk telephone joins its handset to its base by a coiled cord: a long helical spring of wire, made so that it "wants" to lie in a neat spiral of, say, right-handed turns. One plug is fixed in the base, the other in the handset, and neither plug can rotate in its socket. Over a few weeks of ordinary use the cord changes. It no longer hangs as a tidy spiral. Its coils wrap around one another, it shortens and stiffens, and somewhere along its length a peculiar loop appears: on one side of it the spiral winds one way, on the other side the opposite way. Left alone, the tangle only gets worse. Everyone who has used such a phone "knows" why this happens. Before accepting any explanation, separate what you actually know from what you merely imagine, and then work out: 1. Neither plug ever turns in its socket. How, physically, can any twist get into the cord at all? Trace the handset through one complete call, from cradle to ear to cradle, and keep track of its orientation. 2. Why does the cord answer with loops and knots rather than simply twisting up uniformly along its length? 3. Why does a loop form at which the handedness of the spiral reverses? Count the turns on either side of such a loop. Is anything conserved? Do the reversals come singly or in pairs? 4. Design a test of your explanation that you could carry out with a real cord in five minutes (a helical spring, a Slinky, or a coiled cord from a thrift shop will do). Predict, before you look, which way the accumulated twist will run for a right-handed user, and what will happen if one end is unplugged and left to hang free. 5. What part of the tangle can be removed without unplugging either end, and what part cannot? Say which, and why, before you try it. ![Schematic of a coiled phone cord between base and handset with right-handed coils on the left, left-handed coils on the right, and a short straight kinked segment, the reversal, between them](https://tyson-swetnam.github.io/aosd/assets/images/problems/phone-cord-problem.svg){ width="560" } *The reversal loop, and the constraint that produces it. Drawn for this site (CC BY 4.0).* ![A coiled telephone handset cord lying on a floor, with two kinked loops where the spiral's handedness reverses](https://tyson-swetnam.github.io/aosd/assets/images/problems/phone-cord-problem-perversion.jpg){ width="560" } *A real one, with two reversals. Photo "Touch me and I end up singing" by Daniel Oines (Flickr), CC BY 2.0, via Wikimedia Commons.* ## Why it is in the course Session 5 belongs to the section on "Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes". Its readings are the N-ray affair and Langmuir's "Pathological Science": cases in which competent observers saw what they expected to see. The session before asked for "facts before explanations of facts" and for "distinguishing things we know vs only imagine". The phone cord drills exactly that. Everyone has a confident story about why the cord tangles ("I twist it when I pick up the phone"), yet almost nobody has looked at a cord closely enough to know which way the coil winds, where the twist accumulates, whether the reversal loops come singly or in pairs, or whether the story predicts anything checkable. The problem also carries a hidden false assumption: that a cord must have been twisted before it can kink. It need not have been. A reversal loop is what a cord that wants to be a helix does when its ends are held and it carries no net twist at all. The problem offers error-checking by conservation too: Darwin's observation that the turns on the two sides of a reversal are equal is a bookkeeping check of the kind the course prizes. It also suits the syllabus's rule that exercises should "depend as little as possible on knowledge of any particular subject area". The whole investigation takes five minutes with a real cord, a pen line drawn along it, and a tally of how the handset is handled. ## Where it comes from The phenomenon is older than the telephone. In *The Movements and Habits of Climbing Plants* (1865), Charles Darwin reported that a tendril which has caught a support "invariably becomes twisted in one part in one direction, and in another part in the opposite direction; the oppositely turned spires being separated by a short straight portion. This curious and symmetrical structure has been noticed by several botanists, but has not been sufficiently explained." He gathered "ten attached tendrils of the Bryony, the longest with 33, and the shortest with only 8 spiral turns", and found the number of turns in one direction "in every case the same (within one) as in the opposite direction". Reversals need not come singly: he had "seen a tendril with the spires alternately turning five times in opposite directions, with straight pieces between them". He then explained the structure mechanically, with a bundle of strings wound round a stick, a line painted along twining stems, and paper vanes fixed to tendril tips. According to the Wikipedia overview, the word "perversion" for the passage from one handedness to the other goes back to the topologist J. B. Listing and was used by James Clerk Maxwell in *A Treatise on Electricity and Magnetism* (1873); that attribution rests on the overview alone and has not been checked against Maxwell's text. The coiled handset cord, in use from about the middle of the twentieth century, put Darwin's tendril on every office desk, and the reversal loop became a household nuisance. The mathematics behind it, the conservation of a ribbon's linking number as the sum of twist and writhe, was worked out in the 1960s and 1970s and is now standard in the study of DNA supercoiling. On 16 February 1998 Alain Goriely, of the Universite Libre de Bruxelles, and Michael Tabor, of the University of Arizona's Program in Applied Mathematics, published "Spontaneous Helix Hand Reversal and Tendril Perversion in Climbing Plants" in *Physical Review Letters*. Modelling a tendril as a thin elastic rod with intrinsic curvature whose ends cannot rotate, they showed the reversal to be "a paradigm for curvature induced morphogenesis in which symmetry breaking is constrained by a global invariant", and coined the phrase "tendril perversion". *Science News* reported the work on 28 February 1998 and the *Baltimore Sun* in April, both with the kinked phone cord as the everyday example. A fuller treatment followed from McMillen and Goriely in 2002. Winfree's Spring 2001 course listed the phone cord problem three years later, beside N-rays and pathological science. ??? tip "Hints" - Do not start with a theory. Get a real coiled cord (or a helical spring or Slinky) and record what you see: which way does the spiral wind? Where is the kink? How many turns lie on each side of it? Draw a straight pen line along the relaxed cord, then handle it, and watch what the line does. - Track the handset through one complete call. Does it return to the cradle in the same orientation it left, or has it been turned about the cord's axis? What if the same small turn happens at every call for a right-handed user? - Think of the cord as a ribbon whose ends are held so they cannot rotate. Then one quantity is fixed: the sum of twist (rotation of the ribbon about its centreline) and writhe (the centreline coiling and looping in space). What does a spring do when you force twist into it? - Darwin's demonstration: hold a bundle of parallel strings in one hand and turn them round and round with the other, and they do not become twisted; but hold a stick among them so that they wind spirally around it, and "they will inevitably become twisted". So a helix cannot form from a straight segment held at both ends without either twisting the wire along its length or letting one end spin once per turn. What third possibility remains? - Try the experiment with no net twist at all: stretch a coiled cord straight between two hands that you do not allow to rotate, then let it relax. What does it do, and does it need any stored twist to do it? - If a reversal loop is the cord's way of keeping zero net twist, what would remove one, and what would merely move it somewhere else? Does it matter whether two neighbouring loops wind the same way or opposite ways? ??? success "Resolution" Two separate things are going on, and the usual confident explanation covers only one of them. **The net twist, and where it goes.** The plugs do not turn, but the handset does. During a call it is lifted, brought to one ear, passed to the other hand, tucked against a shoulder and set down, often rotated about the cord's axis relative to how it was picked up. Because the base is fixed, each such rotation feeds a turn of twist into the cord, and a user with consistent habits adds turns of the same sign call after call, so they accumulate instead of cancelling. In the spirit of the course this is a hypothesis, not a fact, and it predicts things you can check: cords used by consistently right- and left-handed users should carry twist of opposite signs, and a handset end unplugged and left to dangle should spin as the cord sheds its stored turns. Once the twist is in, it has nowhere cheap to go. With both ends held, the sum of twist and writhe is fixed, and twisting the wire is expensive, so the cord trades twist for writhe: the coils wrap around one another and the cord bunches and knots, exactly as supercoiled DNA does (the explanation given on the AMSI page). That is the tangle. **The reversal loop, which needs no twist at all.** This is the part the folk explanation misses, and the part Darwin analysed in 1865 and Goriely and Tabor modelled in 1998. A coiled cord has *intrinsic curvature*: relaxed, it wants to be a helix. Stretch a section straight, as a long call across the desk does, then let it go while both ends are prevented from rotating. It cannot recoil into a helix all of one handedness, because forming each turn of a helix requires the free end to make one full rotation, or else the filament must twist about its own axis. Darwin verified the rotation by "affixing little paper vanes to the extreme points of the tendrils", and noted that for a tendril of thirty spires the alternative twisting "would burst the tendril before the thirty turns were completed". Instead the cord recoils into two helices of opposite handedness joined by a short straight or kinked segment, so the turns cancel and the net twist stays zero. Darwin: "there are as many turns in the one direction as in the other". Tabor called the result a "twistless spring": a spring that starts by coiling one way, reverses, and so has a net twist of zero. So a perversion is not evidence that anyone twisted the cord. **Fixing it.** The two parts come apart here too, which is the answer to question 5. Reversals are made in cancelling pairs, so two neighbouring loops of opposite handedness can annihilate each other and vanish without either plug being touched. That is why they so often appear in pairs, and why a lone loop is so stubborn. Net twist is different: it cannot leave a cord whose ends are both fixed, only move about or convert between twist and writhe. To be rid of it, an end must be allowed to rotate. The Baltimore Sun's 1998 account gives the practical version, "Just unplug one end of the cord and retwist the coil, from the kink out, reversing its twist." A swivel connector, or handling the handset the same way every time, prevents recurrence. The point is less the answer than the method: the first confident explanation ("I twist it when I pick it up") explains the tangle but not the kink, and both halves can be tested in minutes by counting turns, drawing a line along the cord, and stretching a cord straight and letting it go. ## Sources - **A. T. Winfree**, *The Art of Scientific Discovery, EEB 479/479H/579 course handout*; the schedule on it is "a retrospective syllabus of Spring 2001" β€” [Wayback Machine, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Charles Darwin**, *The Movements and Habits of Climbing Plants* (1865; 2nd ed. 1875), chapter IV β€” [Project Gutenberg #2485](https://www.gutenberg.org/ebooks/2485){target=_blank} πŸ”“ - **Alain Goriely and Michael Tabor**, "Spontaneous Helix Hand Reversal and Tendril Perversion in Climbing Plants", *Physical Review Letters* 80, 1564–1567 (1998) β€” [doi:10.1103/PhysRevLett.80.1564](https://doi.org/10.1103/PhysRevLett.80.1564){target=_blank} πŸ”’ - **T. McMillen and A. Goriely**, "Tendril Perversion in Intrinsically Curved Rods", *Journal of Nonlinear Science* 12, 241–281 (2002) β€” [doi:10.1007/s00332-002-0493-1](https://doi.org/10.1007/s00332-002-0493-1){target=_blank} πŸ”’ - **Baltimore Sun**, "Science and math, with a twist: Kinks" (22 April 1998) β€” [baltimoresun.com](https://www.baltimoresun.com/1998/04/22/science-and-math-with-a-twist-kinks-researchers-are-studying-the-characteristics-that-make-materials-live-and-inorganic-develop-natural-coils-and-twists/){target=_blank} πŸ”“ - **M. N. Jensen**, "Mathematicians Describe Tendril Perversion", *Science News* 153 no. 9, 134 (28 February 1998) β€” [sciencenews.org](https://www.sciencenews.org/archive/mathematicians-describe-tendril-perversion){target=_blank} πŸ”“ *(the landing page prompts for a subscription; the scanned page it embeds is served openly)* - **Will Stavely** (AMSI Research and Higher Education, Monash University), "Why do phone cords get tangled?" (2015) β€” [rhed.amsi.org.au](https://rhed.amsi.org.au/phone-cords-get-tangled/){target=_blank} πŸ”“ - **Wikipedia**, "Tendril perversion" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Tendril_perversion){target=_blank} πŸ”“ - **P. E. S. Silva and others**, "Perversions with a twist", *Scientific Reports* 6, 23413 (2016), CC BY 4.0 β€” [doi:10.1038/srep23413](https://doi.org/10.1038/srep23413){target=_blank} πŸ”“ - **AnandTech Forums users**, "Topology of Phone cord twist" (thread, 2004) β€” [forums.anandtech.com](https://forums.anandtech.com/threads/topology-of-phone-cord-twist.1452427/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the name, "discuss phone cord problem", in session 5 beside N-rays, Langmuir's "Pathological Science" and the unidentified Stockholm Restrooms. Winfree's archived lab pages, including the course handout and his "Adventures in Discovery" columns, were searched and say nothing more. That the item concerns the twisted coiled handset cord is probable from the name alone; the emphasis on the handedness-reversal loop is the editors' elaboration. It rests on two things: the cord is a no-prerequisites puzzle of the kind the syllabus describes, and the 1998 paper that named the phenomenon came from Michael Tabor on Winfree's own campus, with the press reporting it through the kinked phone cord. Nothing in Winfree's own writing connects him to it. Candidate readings: - Explain, from observation, why a coiled cord fixed at both ends becomes twisted and tangled and develops a reversal loop, and how to test the explanation (medium confidence; the reading used above). - A pencil-and-paper topology puzzle: a cord has accumulated many turns of twist while both ends stay plugged in; can the twist be removed without unplugging, and if not, why not? (medium confidence; largely a facet of the first reading.) - Dirac's belt trick demonstrated with a phone cord: a 720-degree twist between fixed ends can be undone without rotating either end, a 360-degree twist cannot (low confidence; no link to the session's theme and no source connects it to the course). --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/stockholm-restrooms/ --- title: "Stockholm Restrooms" description: "An unidentified travel anecdote from the pathological-science session, reconstructed as an exercise in writing down what you saw before you explain it." type: Activity tags: [course, student-facing, problem, section-1, hidden-assumptions, observation, pathological-science] status: stable problem: section: 1 session: 5 identification: unknown kind: discussion generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/479H/579) course handout with retrospective Spring 2001 syllabus, archived copy of 20 April 2002" author: "Arthur T. Winfree" - id: winfree-lab-page-2002 resource: "https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/" title: "A. T. Winfree lab home page (archived), with links to the course, student comments and the 'Adventures in Discovery' columns" author: "Arthur T. Winfree" - id: langmuir-1953-princeton resource: "https://www.cs.princeton.edu/~ken/Langmuir/langmuir.htm" title: "Pathological Science (transcript of the colloquium of 18 December 1953, GE report 68-C-035, 1968)" author: "Irving Langmuir, transcribed by R. N. Hall" - id: langmuir-1953-virginia resource: "https://galileo.phys.virginia.edu/~rjh2j/misc/Langmuir.pdf" title: "Pathological Science (PDF copy of the Langmuir transcript with background note)" author: "Irving Langmuir" - id: wikipedia-pathological-science resource: "https://en.wikipedia.org/wiki/Pathological_science" title: "Pathological science (overview)" author: "Wikipedia" - id: ghent-restroom-queues-2017 resource: "https://phys.org/news/2017-07-lengths-restroom.html" title: "Researchers study lengths of restroom queues" author: "Ghent University (Phys.org news release)" - id: sciam-drain-direction-2001 resource: "https://www.scientificamerican.com/article/can-somebody-finally-sett/" title: "Does water flowing down a drain spin in different directions depending on which hemisphere you're in? (Ask the Experts, 28 January 2001)" author: "Brad Hanson, Fred Decker, Robert Ehrlich and Thomas Humphrey (Scientific American, Ask the Experts)" - id: the-local-unisex-toilets-2015 resource: "https://www.thelocal.se/20150213/swedes-confused-over-gender-neutral-toilets" title: "Swedes confused over gender neutral toilets" author: "The Local (Sweden)" - id: travelwriter-unisex-toilets-2016 resource: "https://accidentaltravelwriter.com/lgbt-news-unisex-toilet-become-increasingly-popular-in-sweden/" title: "Unisex Toilets, an Idea Whose Time Has Come?" author: "Accidental Travel Writer (blog)" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Stockholm Restrooms Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 5. Discussed together with [Phone Cord Problem](https://tyson-swetnam.github.io/aosd/problems/phone-cord-problem/index.md), in the session devoted to [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md) and Langmuir's "Pathological Science".* !!! abstract "The problem" Reconstructed from the syllabus: the original handout has not been found, and what follows is a modest reconstruction from the name, the session and the section theme, not Winfree's text. The syllabus lists "discuss Stockholm restrooms" for the session on N-rays and pathological science. The item appears to have been a short observation-and-explanation exercise in the spirit of that session: an everyday oddity noticed by a traveller, to be explained by first establishing the facts and only then proposing rival explanations and ways to test them. A faithful version of the exercise runs like this. 1. Take one of these travellers' reports about public restrooms in Stockholm. All three are things visitors have actually reported; which one, if any, Winfree used is not known. - (a) In Stockholm's shopping centres, cafes and even at Arlanda airport, a visitor from the United States finds that many public toilets are single rooms with no "Men" or "Women" on the door, and no urinals anywhere. - (b) Where there are separate rooms, the doors read "Herrar" and "Damer". - (c) At a conference reception the queue for the women's restroom is long while the men's has none. 2. Write down exactly what was observed, and separate it from what was assumed. Is "there is no men's room" an observation, or an inference from not finding a sign you expected? 3. Give at least three explanations, including one in which the observer is simply mistaken. For each, say what further observation would count against it. 4. Only then decide which explanation you believe, and how strongly. The exercise is about method, not about Swedish plumbing: notice how fast an explanation arrives before the facts are settled, and how much of the "mystery" your own assumptions manufactured. ## Why it is in the course Session 5 sits in Section 1, "Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes". Its readings were the N-ray story and Langmuir's "Pathological Science": honest scientists who reported effects near the threshold of detectability and then defended them against all evidence. The session before had made a related point with historical cases: "[Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md): facts before explanations of facts" and "[Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md): distinguishing things we know vs only imagine". A small travel anecdote is a natural everyday-scale companion to those grand cases. It lets a student catch themselves supplying an explanation before the facts are in, and see how much of a puzzle can be created by an assumption imported from home. That fits Winfree's own account of the course puzzles in the syllabus: "The purpose of the puzzles (many of them silly) is to slow you down for a few minutes so you can examine the working of your own mind." But that reading is an inference. The syllabus gives the name and the session, and nothing else. ## Where it comes from No published source for a problem called "Stockholm restrooms" has been found, and the syllabus is the only Winfree document that names it. The archived copy of his course handout carries that same schedule, with the retrospective Spring 2001 dates shifted to the Fall 2001 calendar, and lists the item for session 05 with no explanation. Captures of the handout from 2002 to 2005 all carry the same text. Nothing else he left online mentions it. The editors searched the archived pages of his University of Arizona lab site that could be retrieved: the course handout, the lab home page, the student comments, and his surviving "Adventures in Discovery" columns for the Society for Amateur Scientists (2001-2002). None contains "Stockholm", "restroom", "toilet" or "Sweden". A complete listing of the archived site could not be obtained, so the search is thorough but not exhaustive. The item was most likely a personal anecdote, handed out on paper and now lost. The everyday facts behind the three candidate readings below are real and sourced at the end of this page, whether or not any of them was Winfree's: unisex single-occupancy toilets are the Swedish norm and surprise American visitors; the longer queue at women's restrooms is a real effect with a mundane multi-causal explanation, measured by queueing theorists at Ghent University in 2017; and hemisphere-dependent drain swirl is a misconception, the Coriolis acceleration at mid-latitudes being far too weak, as Scientific American's "Ask the Experts" panel spelled out. ??? tip "Hints" - Before explaining anything, list what was actually seen and separate it from what was inferred. "There is no men's room" may be an inference from not finding a sign you expected. - Ask what a visitor from Tucson assumes at a restroom door in Stockholm: how many doors there should be, what the words mean, what fixtures are inside. Test each assumption. - Include the possibility that the observer is mistaken among your rival explanations, and say for each what observation would rule it out. - Compare with the session's readings. Which of Langmuir's symptoms of pathological science (an effect near the limit of detectability, claims of great accuracy, ad hoc excuses, a ratio of supporters to critics that rises and then falls) would fit someone who refused to give up their first explanation? ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery (ECOL 479/479H/579) course handout with retrospective Spring 2001 syllabus* (2001), archived copy of 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *A. T. Winfree lab home page*, with links to the course, student comments and the "Adventures in Discovery" columns (archived 2002) β€” [Wayback Machine](https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/){target=_blank} πŸ”“ - **Irving Langmuir**, transcribed by R. N. Hall, *Pathological Science* (colloquium of 18 December 1953; GE report 68-C-035, 1968), online edition by Theo Pavlidis and Ken Steiglitz β€” [Princeton](https://www.cs.princeton.edu/~ken/Langmuir/langmuir.htm){target=_blank} πŸ”“ - **Irving Langmuir**, *Pathological Science* (PDF copy of the transcript with background note) β€” [University of Virginia](https://galileo.phys.virginia.edu/~rjh2j/misc/Langmuir.pdf){target=_blank} πŸ”“ - **Wikipedia**, "Pathological science" (overview) β€” [Wikipedia](https://en.wikipedia.org/wiki/Pathological_science){target=_blank} πŸ”“ - **Ghent University** (Phys.org news release), "Researchers study lengths of restroom queues" (2017) β€” [Phys.org](https://phys.org/news/2017-07-lengths-restroom.html){target=_blank} πŸ”“ - **Scientific American** (Ask the Experts: Brad Hanson, Fred Decker, Robert Ehrlich and Thomas Humphrey), "Does water flowing down a drain spin in different directions depending on which hemisphere you're in?" (28 January 2001) β€” [Scientific American](https://www.scientificamerican.com/article/can-somebody-finally-sett/){target=_blank} πŸ”“ - **The Local (Sweden)**, "Swedes confused over gender neutral toilets" (13 February 2015) β€” [The Local](https://www.thelocal.se/20150213/swedes-confused-over-gender-neutral-toilets){target=_blank} πŸ”“ - **Accidental Travel Writer** (blog), "Unisex Toilets, an Idea Whose Time Has Come?" (2016) β€” [Accidental Travel Writer](https://accidentaltravelwriter.com/lgbt-news-unisex-toilet-become-increasingly-popular-in-sweden/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not at all: the identification is **unknown**. The syllabus gives the name, the session and its neighbour in the same line, the phone cord problem, which is also unidentified. The editors read the archived handout, Winfree's lab page and his surviving columns, and searched on the exact name and every plausible form of the puzzle; nothing turned up. The statement above is a reconstruction, and these three readings are inferences from the name and the session theme, all of low confidence and none tied to Winfree by any source. - **A hidden-assumption story**: an American visitor is baffled by unisex single-occupancy toilets, or misreads "Herrar" and "Damer", until a false assumption is dropped. This fits the section's theme. - **A queue exercise**: why queues form at women's restrooms and not at men's, with rival hypotheses ranked and tested. A vivid effect for which everyone has a ready explanation, few of them checked. - **A drain-swirl exercise**: the folk belief about the Coriolis effect, closest to Langmuir's first symptom, an effect at the limit of detectability, though that belief is usually framed as a comparison across the equator rather than as a Stockholm story. If the original handout ever surfaces, from a former student's binder for instance, it should replace this reconstruction. --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/collective-reproduction/ --- title: "Collective Reproduction" description: "An unidentified group effort closing Section 1, offered here as a reconstructed puzzle: can a small, ancient valley of three-parent elves have family trees with no repeated ancestors? It trains checking the hidden assumptions in a plausible story." type: Activity tags: [course, student-facing, problem, section-1, hidden-assumptions, error-checking, ancestry, population-growth] status: stable problem: section: 1 session: 6 identification: unknown kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: winfree-faculty-page-2002 resource: "https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/" title: "Art Winfree's faculty home page (archived), linking the course handout, student testimonials and the 'Adventures in Discovery' columns" author: "Arthur T. Winfree" - id: wikipedia-pedigree-collapse resource: "https://en.wikipedia.org/wiki/Pedigree_collapse" title: "Pedigree collapse" author: "Wikipedia contributors" - id: penrose-1959 resource: "https://doi.org/10.1038/scientificamerican0659-105" title: "Self-Reproducing Machines" author: "L. S. Penrose" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Collective Reproduction Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 6, alongside [Evaporated Gold](https://tyson-swetnam.github.io/aosd/problems/evaporated-gold/index.md). The syllabus reads "Start group effort on Collective Reproduction".* !!! abstract "The problem" **A reconstruction.** No statement of Winfree's exercise survives. This one was written for this site, not by Winfree. Its elves come from the one clue to the original (see "Where it comes from"); the numbers are illustrative choices. Someone describes a hidden valley of elves and makes four claims: - (i) Every elf has exactly three parents. - (ii) No elf's parents are related, and no elf appears twice in any family tree. - (iii) The valley has never held more than 1,000 elves at one time. - (iv) Elves have lived in the valley, and only there, for 40 generations. 1. **Alone first.** How many great-great-grandparents does one elf have? How many ancestors 40 generations back? 2. **Pool.** Compare answers as a class. Where you disagreed, was it arithmetic or a different assumption? 3. **Find the fault.** Can all four claims be true together? Say which claims are involved. 4. **Repair the story** with the smallest change. What must then be true about elf family trees? 5. **Compare with people.** Does the same argument apply to your own ancestors, with two parents each, 30 generations back? ![A tree fanning upward from one elf to 3 parents, 9 grandparents and 27 great-grandparents, showing ancestor numbers tripling each generation.](https://tyson-swetnam.github.io/aosd/assets/images/problems/collective-reproduction.svg){ width="560" } *Three generations of one elf's family tree under claim (i). Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 6 closes Section 1, "Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes". Its topics include "hidden assumptions", and it starts the first group effort in the schedule. The syllabus says the class meets partly "to work jointly for a while on bigger problems or puzzles", including ones that "need lots of data-collecting, best pooled from many sources". If the leading reading in the note at the end is right (an inference), the lesson is that a harmless-sounding story can hide an impossibility that only arithmetic and a check of assumptions expose. ## Where it comes from The syllabus line is the only text of Winfree's that names the exercise. The schedule is "a retrospective syllabus of Spring 2001, with dates changed to reflect the future", so it was presumably run in Spring 2001. In the [HTML version of Winfree's handout](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank}, the link on this item points to a bookmark named `Elves_Collective_Reproduction`. The target is not in the handout, and its links to other course files point to paths on Winfree's own PC, archived only as failed captures. The name is the only trace of the content. No other checked page of his [archived site](https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/){target=_blank} mentions elves. ??? tip "Hints" - Draw one elf's family tree three generations back before calculating. Have a neighbour count your drawing. - List every assumption you make about the tree, including unstated ones. Which could fail? - Compare the size of the tree far back with something else the story tells you about the valley. Do the two fit? - If two claims clash, which does the least work in the story? Drop it and redraw the tree. ??? success "Resolution" For the reconstruction only; Winfree's own problem is unknown. With three parents and no repeats, generation *k* back holds 3^*k*^ ancestors: 3, 9, 27, then 81 great-great-grandparents. Forty generations back that is 3^40^ = 12,157,665,459,056,928,801, about 1.2 Γ— 10^19^. But a valley of at most 1,000 elves over 40 generations can have held only about 1,000 Γ— 40 = 40,000 different elves, far too few to fill that tree. The four claims cannot all be true. The cheapest repair is to drop claim (ii): the same elves fill many places in each tree, so parents are often related. Keeping (ii) would need far more elves, or immigrants, contradicting (iii) or (iv). This is pedigree collapse, and it applies to humans: 2^30^, roughly a billion, ancestor slots 30 generations back far exceeds the population of the time. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery*: course syllabus, 2001 β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: course handout, HTML version β€” [Wayback Machine, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, faculty home page β€” [Wayback Machine, 25 December 2002](https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/){target=_blank} πŸ”“ - **Wikipedia contributors**, "Pedigree collapse"; background for the reconstruction β€” [Wikipedia](https://en.wikipedia.org/wiki/Pedigree_collapse){target=_blank} πŸ”“ - **L. S. Penrose**, "Self-Reproducing Machines", *Scientific American* 200(6), 105–114 (June 1959) β€” [doi:10.1038/scientificamerican0659-105](https://doi.org/10.1038/scientificamerican0659-105){target=_blank} πŸ”’ !!! note "How sure are we that this is Winfree's problem?" Not sure. The bookmark shows only that elves figured in the item; no exercise about collectively reproducing elves could be found. Four readings remain, all low confidence, ordered by fit with "Elves": 1. **A reasoning puzzle about imaginary elves** whose reproduction is collective, for instance more than two parents each, with the lesson in hidden assumptions about ancestry. The reconstruction follows this. 2. **A "many hands" exercise**: the class, as a team of elves, copies a pattern piece by piece and checks where the joined result goes wrong. 3. **Self-reproduction as such**, with elves as the agents (compare Penrose 1959); nothing links it to Winfree. 4. **Class-wide replication**, formerly the lead reading: everyone reproduces the same result and the class pools the outcomes. The tension: "Elves" favours reading 1, but the syllabus's account of group efforts (data "best pooled from many sources") fits reading 4 better. Hence nothing rates above low. --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/evaporated-gold/ --- title: "Evaporated Gold" description: "An unidentified session-6 item, reconstructed here as an order-of-magnitude estimate of a vacuum-deposited gold film, where the wrong answer comes from an assumption nobody noticed making." type: Activity tags: [course, student-facing, problem, section-1, estimation, hidden-assumptions, error-checking, thin-films] status: stable problem: section: 1 session: 6 identification: unknown kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2001 resource: "https://web.archive.org/web/20030111064604/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery: course handout and retrospective syllabus (479/479H/579, Ecology and Evolutionary Biology, University of Arizona), Wayback Machine capture" author: "Arthur T. Winfree" - id: winfree-home-page resource: "https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/" title: "A. T. Winfree faculty home page (Wayback Machine capture, 25 Dec 2002)" author: "Arthur T. Winfree" - id: evaporation-deposition resource: "https://en.wikipedia.org/wiki/Evaporation_(deposition)" title: "Evaporation (deposition)" author: "Wikipedia contributors" - id: gold-wikipedia resource: "https://en.wikipedia.org/wiki/Gold" title: "Gold" author: "Wikipedia contributors" - id: faraday-1857 resource: "https://commons.wikimedia.org/wiki/File:The_Bakerian_Lecture-_Experimental_Relations_of_Gold_(and_Other_Metals)_to_Light_(IA_jstor-108616).pdf" title: "The Bakerian Lecture: Experimental Relations of Gold (and Other Metals) to Light. Philosophical Transactions of the Royal Society of London 147: 145-181" author: "Michael Faraday" - id: fermi-problem resource: "https://en.wikipedia.org/wiki/Fermi_problem" title: "Fermi problem" author: "Wikipedia contributors" - id: fritz-haber resource: "https://en.wikipedia.org/wiki/Fritz_Haber" title: "Fritz Haber" author: "Wikipedia contributors" - id: haber-1927 resource: "https://doi.org/10.1002/ange.19270401103" title: "Das Gold im Meerwasser. Angewandte Chemie (then Zeitschrift fuer angewandte Chemie) 40(11): 303-314" author: "Fritz Haber" - id: gold-in-sea-water resource: "https://www.911metallurgist.com/blog/how-much-gold-sea-water/" title: "How much Gold in Sea Water" author: "911 Metallurgist (reprinting an older mining-engineering article)" - id: ehrlich-nine-crazy-ideas resource: "https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science" title: "Nine Crazy Ideas in Science: A Few Might Even Be True (publisher page)" author: "Robert Ehrlich" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Evaporated Gold Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), session 6. The same session starts the group effort on [Collective Reproduction](https://tyson-swetnam.github.io/aosd/problems/collective-reproduction/index.md).* !!! abstract "The problem" Reconstructed from the syllabus. No statement for this problem survives: the syllabus gives the bare name, after the three skills listed for the day and before the group effort on Collective Reproduction. What follows is the editors' reading of that name β€” gold that has been *evaporated* onto something β€” not Winfree's wording. A physicist puts a bead of pure gold, mass 1.00 g, into a tungsten boat at the centre of an evacuated bell jar and heats it until the gold has entirely evaporated. A clean glass microscope slide, 2.5 cm by 7.5 cm, is fixed inside the jar 15 cm from the boat, facing it. When the jar is opened, the slide carries a gold-coloured film. 1. Before calculating, write down every assumption you will need, marking each as something you *know*, *assume*, or could *look up*. 2. Estimate the film's thickness, in metres and in atoms. 3. Check that answer by two independent routes that share no arithmetic. (One useful fact: a gram of gold beats out to about a square metre of leaf.) 4. Would the film look gold? Would light pass through it? What thickness would let light through, and what colour would it be? 5. Which assumptions, if wrong, would change the answer by more than a factor of ten? Which by less than two? You are graded on the care of your assumption list and your checks, not on the number you get. ![Schematic of a bell jar with a heated boat holding a gold bead at the bottom; dashed rays fan out in all directions, only a few reaching a small glass slide fixed 15 cm above, with a dotted circle of radius 15 cm showing the sphere over which the vapour spreads](https://tyson-swetnam.github.io/aosd/assets/images/problems/evaporated-gold.svg){ width="560" } *The reconstructed set-up. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 6 closes Section 1, "Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes". The readings due are a packet on valuing mistakes and Ehrlich's introduction on judging a crazy idea. The syllabus then names three skills for the hour β€” "Some ways to check for errors", "hidden assumptions", "what is 'understand'?" β€” and lists Evaporated Gold straight after them. That placement suggests it was the worked example. It is a placement, not evidence. The reconstruction above is built to exercise those three skills. List your assumptions before you compute, while you can still attack them. Cross-check by a route that shares no arithmetic with the first β€” harder than it sounds, as the resolution shows. Then ask whether you *understand* the number or have merely produced one. The same day the class begins [Collective Reproduction](https://tyson-swetnam.github.io/aosd/problems/collective-reproduction/index.md), where the checking has to be done on someone else's reasoning. ## Where it comes from We do not know. No puzzle of this name has been found outside Winfree's own schedule. The physics behind the likeliest reading is ordinary. Vacuum evaporation β€” heating a metal in a chamber pumped down to roughly 10^-4^ Pa, so atoms fly in straight lines and condense on whatever they hit β€” is how thin metal films are made, on electron-microscope specimens, mirrors and electrodes. Gold suits order-of-magnitude work because two of its numbers are famous: a density near 19.3 g/cm^3^, and a gram that beats out to about a square metre of leaf. Thin gold is also optically strange, which is where part 4 comes from: Michael Faraday's 1857 Bakerian Lecture examined how gold films thin enough to transmit light behave. ??? tip "Hints" - Write the assumptions down first, apart from the calculation; the ones you did not write down are the likeliest to be wrong. - Thickness is volume over area. The volume follows from the density; the area is the hard part. - Ask what would look different if the film were ten times thicker, or ten times thinner. An answer you can recognise as absurd is an answer you can check. - Atoms leaving a hot source in vacuum fly in straight lines in every direction. What fraction can reach a slide of that size, that far away? ??? success "Resolution" **This resolves the reconstruction above, not a recovered original.** *The hidden assumption.* At 19.3 g/cm^3^, 1.00 g of gold occupies about 0.052 cm^3^. If all of it landed on the slide (18.75 cm^2^), the film would be 28 micrometres thick, some 100,000 atoms. But the vapour leaves in every direction, spreading over roughly a sphere of radius 15 cm, area about 2,800 cm^2^, of which the slide intercepts 0.7 %. Mean thickness there: about 180 nanometres, roughly 600 atoms at gold's 0.29 nm spacing. A boat standing on the baseplate radiates into something nearer a hemisphere, so call the geometry good to a factor of two. The gap between that quibble and the factor-of-150 blunder is the lesson. *Which checks are really checks.* Gold leaf looks like an independent route and is not. A gram spread over a square metre is 52 nm thick; our sphere has 3.5 times less area, so 3.5 times the thickness, 180 nm. That is the same volume divided by an area β€” the answer re-derived, not tested. Counting atoms against Avogadro's number tells the same story: it checks the arithmetic, never the geometry, which is where the error was. Two checks do bite. Real specimen coatings run 5 to 20 nm and use far less than a gram, so a 28-micrometre film is absurd on its face. And ask where the rest went: if 99.3 % misses the slide, the jar must come out coated β€” a prediction the apparatus can falsify. *Appearance.* At 180 nm the film is opaque and mirror-bright. Gold turns semi-transparent only below a few tens of nanometres, and then transmits greenish-blue, as Faraday found in 1857 β€” so the colour of the slide tells you almost nothing beyond "thicker than about 50 nm". ## Sources - **Arthur T. Winfree**, "The Art of Scientific Discovery": course handout and retrospective syllabus (479/479H/579, Ecology and Evolutionary Biology, University of Arizona), archived capture (2001) β€” [Wayback Machine](https://web.archive.org/web/20030111064604/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, faculty home page, archived 25 Dec 2002 β€” [Wayback Machine](https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/){target=_blank} πŸ”“ - **Michael Faraday**, "The Bakerian Lecture: Experimental Relations of Gold (and Other Metals) to Light", *Philosophical Transactions of the Royal Society of London* 147, 145–181 (1857) β€” [public-domain scan at Wikimedia Commons](https://commons.wikimedia.org/wiki/File:The_Bakerian_Lecture-_Experimental_Relations_of_Gold_(and_Other_Metals)_to_Light_(IA_jstor-108616).pdf){target=_blank} πŸ”“ - **Wikipedia contributors**, "Evaporation (deposition)" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Evaporation_(deposition)){target=_blank} πŸ”“ - **Wikipedia contributors**, "Gold" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Gold){target=_blank} πŸ”“ - **Wikipedia contributors**, "Fermi problem" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Fermi_problem){target=_blank} πŸ”“ - **Wikipedia contributors**, "Fritz Haber" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Fritz_Haber){target=_blank} πŸ”“ - **Fritz Haber**, "Das Gold im Meerwasser", *Zeitschrift fΓΌr angewandte Chemie* (now *Angewandte Chemie*) 40(11), 303–314 (1927) β€” [doi:10.1002/ange.19270401103](https://doi.org/10.1002/ange.19270401103){target=_blank} πŸ”’ - **911 Metallurgist** (reprinting an older mining-engineering article), "How much Gold in Sea Water" β€” [911metallurgist.com](https://www.911metallurgist.com/blog/how-much-gold-sea-water/){target=_blank} πŸ”“ - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True* (2001) β€” [Princeton University Press](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not at all. The syllabus gives the name and nothing else, the archived handout carries no statement for it, Winfree's other archived pages say nothing about gold, and no published puzzle of this name turned up elsewhere. Three readings were weighed, and all three are low confidence. - **A thin-film thickness estimate**, used above. "Evaporated gold" is the standard laboratory phrase for vapour-deposited gold, and the syllabus promises exercises "made from elementary mathematics so as to require no lab setup". Two things in the syllabus cut against it. The exercises are meant to "depend as little as possible on knowledge of any particular subject area", and this one leans on physics. And the puzzles are "many of them silly", meant "to slow you down for a few minutes"; every Section 1 neighbour is short and whimsical, while this reconstruction is a multi-part calculation. - **Gold from seawater.** Old assays put tens of milligrams of gold in a tonne of seawater. Fritz Haber's 1920s scheme to extract it for German reparations found the true figure a thousand times smaller: the old numbers came from contaminated reagents. A fine "valuing mistakes" story β€” but the syllabus says nothing of the sea. - **A Fermi estimate**: spread all the gold ever mined over the Earth and say how thick the layer is. Nothing ties it to Winfree. If you know the original, the editors would like to hear from you. --- *Back to [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-1-detecting-nonsense-error-checking-false-assumptions-cherishing-mistakes)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/dominoes-lab/ --- title: "Dominoes (group lab)" description: "An undocumented group lab from Winfree's session on perceptual blocks, reconstructed here as the cut-chessboard puzzle done with plain blocks, with a toppling-block experiment as the rival reading." type: Activity tags: [course, student-facing, problem, section-2, perceptual-blocks, tiling, parity, insight] status: stable problem: section: 2 session: 7 identification: unknown kind: lab generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: black-critical-thinking-1946 resource: "https://archive.org/details/criticalthinking0000maxb" title: "Critical Thinking: An Introduction to Logic and Scientific Method" author: "Max Black" - id: golomb-polyominoes-1954 resource: "https://doi.org/10.1080/00029890.1954.11988548" title: "Checker Boards and Polyominoes" author: "Solomon W. Golomb" - id: gardner-maddening-puzzles-1957 resource: "https://doi.org/10.1038/scientificamerican0257-152" title: "Mathematical Games: An assortment of maddening puzzles" author: "Martin Gardner" - id: gardner-hexaflexagons-2008 resource: "https://archive.org/details/hexaflexagonspro0000gard" title: "Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games" author: "Martin Gardner" - id: mccarthy-mutilated-checkerboard resource: "http://www-formal.stanford.edu/jmc/creative/node2.html" title: "The mutilated checkerboard, from Creative Solutions to Problems" author: "John McCarthy" - id: kaplan-simon-insight-1990 resource: "https://doi.org/10.1016/0010-0285(90)90008-R" title: "In search of insight" author: "Craig A. Kaplan and Herbert A. Simon" - id: whitehead-domino-chain-reaction-1983 resource: "https://doi.org/10.1119/1.13456" title: "Domino \"chain reaction\"" author: "Lorne A. Whitehead" - id: pfeiffer-jones-structured-experiences-vi-1977 resource: "https://archive.org/details/handbookofstruct0000unse_g0d8" title: "A Handbook of Structured Experiences for Human Relations Training, Volume VI (exercise 202, Dominoes: A Communication Experiment)" author: "J. William Pfeiffer and John E. Jones (eds.)" - id: adams-conceptual-blockbusting resource: "https://archive.org/details/conceptualblockb00adam_4" title: "Conceptual Blockbusting: A Guide to Better Ideas" author: "James L. Adams" - id: wikipedia-mutilated-chessboard resource: "https://en.wikipedia.org/wiki/Mutilated_chessboard_problem" title: "Mutilated chessboard problem (overview and bibliography)" author: "Wikipedia contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Dominoes (group lab) Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 7. The same session discusses [Weird Organism](https://tyson-swetnam.github.io/aosd/problems/weird-organism/index.md) and [Rearranged Triangle](https://tyson-swetnam.github.io/aosd/problems/rearranged-triangle/index.md).* !!! abstract "The problem" This is a reconstruction, not a record. The schedule says only "Start group lab on Dominoes". The puzzle below is the best-known domino problem that fits a session on perceptual blocks. Other readings are listed at the end of this page. **The puzzle.** Remove two diagonally opposite corner squares from an 8 x 8 board, leaving 62. You have 31 plain blocks, each the size of two neighbouring squares. Can you lay them flat to cover every square, with no overlaps and nothing hanging off? If you can, show how. If you cannot, convince a skeptic that nobody can; failing is not enough. **As a group lab.** Work with real blocks and record every attempt in your GamesWorth book. Then remove two squares of your choosing, try smaller boards, pool the cases, and state what decides it. ![Chessboard with two opposite corner squares cut away, beside a plain two-square block](https://tyson-swetnam.github.io/aosd/assets/images/problems/dominoes-lab.svg){ width="560" } *The cut board and the plain block you have 31 of. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 7 opens Section 2 with Adams's chapter on perceptual blocks. If the lab was the cut board, it is a clean specimen of a perceptual block. Every attempt at laying blocks fails in a way that feels like bad luck. Progress comes only when somebody stops asking "where does the next block go?" and asks "what does any block always do?". The syllabus says the class meets partly to "work jointly for a while on bigger problems or puzzles that need a little equipment or need more diversity of approaches and crazy suggestions". Many failed tilings, pooled and argued over, are the evidence the group reasons from. ## Where it comes from The philosopher Max Black posed the question in *Critical Thinking* (1946) as an exercise in insight. Solomon Golomb discussed it in 1954, and Martin Gardner put it in his *Scientific American* column in 1957. In his first book of puzzles he treats it as a physical puzzle whose "props" are a chessboard and dominoes. In 1964 John McCarthy offered it as "a tough nut for first order theorem provers", because the colours the solution needs are not part of the problem as stated. In 1990 Craig Kaplan and Herbert Simon used it as the model insight problem in "In search of insight". ??? tip "Hints" - "Find a covering" and "decide whether one exists" are different questions. Failing at the first does not answer the second. - Look at one block. Wherever you put it, what is always true of the two squares underneath? - Shrink it: a 2 x 2 board with opposite corners removed, then a 4 x 4. Count something. - Your blocks are plain, but the board may not be. Would it matter if every square were the same colour, or if the missing squares were neighbours? - Once you have a rule, test it the other way: which boards *can* be covered? ??? success "Resolution" No covering exists. Every block covers one light and one dark square, so 31 blocks cover 31 of each. But diagonally opposite corners are the same colour (light or dark, depending on which diagonal you cut), so the cut board has 32 squares of one colour and 30 of the other. The same argument rules out removing any two squares of one colour. The converse is Gomory's theorem: remove one light and one dark square, anywhere, and the rest can always be covered. Draw a closed path through all 64 squares, stepping between neighbours. Colours alternate along it, so the cuts leave one or two even stretches, each coverable by blocks. Shmuel Winograd's proof avoids colour. For k from 1 to 7, the top k rows hold 8k - 1 squares, an odd number. Blocks lying inside those rows cover squares in pairs, so an odd number of vertical blocks must cross the boundary below row k. Each vertical block crosses exactly one of these seven boundaries, so the number of vertical blocks is a sum of seven odd numbers, which is odd. Columns give the same for horizontal blocks. Odd plus odd is even, but 31 is odd. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery (ECOL 479/579): course handout*, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Max Black**, *Critical Thinking: An Introduction to Logic and Scientific Method* (1946; Internet Archive copy is the 1952 second edition) β€” [archive.org](https://archive.org/details/criticalthinking0000maxb){target=_blank} πŸ”“ *(borrow)* - **Solomon W. Golomb**, "Checker Boards and Polyominoes", *American Mathematical Monthly* 61(10), 675–682 (1954) β€” [doi:10.1080/00029890.1954.11988548](https://doi.org/10.1080/00029890.1954.11988548){target=_blank} πŸ”’ - **Martin Gardner**, "Mathematical Games: An assortment of maddening puzzles", *Scientific American* 196(2), February 1957 β€” [doi:10.1038/scientificamerican0257-152](https://doi.org/10.1038/scientificamerican0257-152){target=_blank} πŸ”’ - **Martin Gardner**, *Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi* (Cambridge University Press, 2008), "Mutilated Chessboard" β€” [archive.org](https://archive.org/details/hexaflexagonspro0000gard){target=_blank} πŸ”“ *(borrow)* - **John McCarthy**, "The mutilated checkerboard", from *Creative Solutions to Problems* (1999) β€” [Stanford](http://www-formal.stanford.edu/jmc/creative/node2.html){target=_blank} πŸ”“ - **Craig A. Kaplan and Herbert A. Simon**, "In search of insight", *Cognitive Psychology* 22(3), 374–419 (1990) β€” [doi:10.1016/0010-0285(90)90008-R](https://doi.org/10.1016/0010-0285(90)90008-R){target=_blank} πŸ”’ - **Lorne A. Whitehead**, "Domino 'chain reaction'", *American Journal of Physics* 51(2), 182 (1983) β€” [doi:10.1119/1.13456](https://doi.org/10.1119/1.13456){target=_blank} πŸ”’ - **J. William Pfeiffer and John E. Jones (eds.)**, *A Handbook of Structured Experiences for Human Relations Training*, Vol. VI (1977), exercise 202, "Dominoes: A Communication Experiment" β€” [archive.org](https://archive.org/details/handbookofstruct0000unse_g0d8){target=_blank} πŸ”“ *(borrow)* - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas* β€” [archive.org](https://archive.org/details/conceptualblockb00adam_4){target=_blank} πŸ”“ *(borrow)* - **Wikipedia contributors**, "Mutilated chessboard problem" β€” [overview and bibliography](https://en.wikipedia.org/wiki/Mutilated_chessboard_problem){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure at all. The schedule line is Winfree's only text on it. No other archived Winfree page mentions dominoes. A full-text search of the assigned Adams book found no domino or chessboard exercise. In Winfree's handout, the link on this item points to a bookmark named `White_blocks`. Its target is lost. Winfree's bookmark names sometimes repeat the title and sometimes name a hidden subject (`Keplers_Laws` for "Paired Observations"). So `White_blocks` may describe plain blocks used as dominoes, or, loosely, the crux of the cut-board puzzle (the Resolution above says why that fit is loose). Both are inferences, and neither names a puzzle. The candidates: - **The cut chessboard**, as above. It needs only plain blocks and fits the day's reading. Against it, a short puzzle is a thin "lab". - **A toppling-block lab**: time the wave along rows of blocks set at different spacings, or topple a chain of ever-larger blocks (Whitehead 1983). A domino chain is a toy excitable medium, Winfree's research field. Against it, Section 2 is about blocks, not measurement, though the section does hold one other lab (Pedestrian Crosswalk, session 12). - **Counting tilings** of a 2 x n strip, though pattern-finding is Section 4's business. - **"Dominoes: A Communication Experiment"** (Pfeiffer and Jones, 1977), a communication exercise. Very unlikely. "Lab" need not mean apparatus: the exercises are "mostly made from elementary mathematics so as to require no lab setup". --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/rearranged-triangle/ --- title: "Rearranged Triangle" description: "Four pieces fill a 13-by-5 right triangle; rearrange the same four and one unit square is left over. Where did it go? The Section 2 exercise in perceptual blocks." type: Activity tags: [course, student-facing, problem, section-2, perceptual-blocks, dissection-paradox, recreational-mathematics, martin-gardner] status: stable problem: section: 2 session: 7 identification: probable kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: missing-square-wikipedia resource: "https://en.wikipedia.org/wiki/Missing_square_puzzle" title: "Missing square puzzle" author: "Wikipedia contributors" - id: gardner-magic-mystery-1956 resource: "https://archive.org/details/magicmathematics0000unse" title: "Mathematics, Magic and Mystery (Dover, 1956), chapter 8, \"Geometrical Vanishes, Part II\", section \"Curry Triangles\"" author: "Martin Gardner" - id: cut-the-knot-curry-paradox resource: "https://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml" title: "Curry's Paradox" author: "Alexander Bogomolny (Cut-the-Knot)" - id: mathworld-triangle-dissection resource: "https://mathworld.wolfram.com/TriangleDissectionParadox.html" title: "Triangle Dissection Paradox" author: "Eric W. Weisstein (MathWorld)" - id: loyd-cyclopedia-1914 resource: "https://archive.org/details/CyclopediaOfPuzzlesLoyd" title: "Sam Loyd's Cyclopedia of 5000 Puzzles, Tricks and Conundrums (1914), chessboard dissection p. 288" author: "Sam Loyd (ed. Sam Loyd Jr.)" - id: hooper-rational-recreations-1782 resource: "https://archive.org/details/rationalrecreat00davigoog" title: "Rational Recreations, vol. 4 (1782), \"Recreation CVI. The geometric money\", p. 286" author: "William Hooper" - id: chessboard-paradox-wikipedia resource: "https://en.wikipedia.org/wiki/Chessboard_paradox" title: "Chessboard paradox" author: "Wikipedia contributors" - id: hoopers-paradox-wikipedia resource: "https://en.wikipedia.org/wiki/Hooper%27s_paradox" title: "Hooper's paradox" author: "Wikipedia contributors" - id: sillke-jigsaw-paradox resource: "https://www.math.uni-bielefeld.de/~sillke/PUZZLES/jigsaw-paradox.html" title: "Jigsaw Paradox (annotated bibliography)" author: "Torsten Sillke" - id: math-doctors-disappearing-area resource: "https://www.themathdoctors.org/disappearing-area/" title: "Disappearing Area?" author: "The Math Doctors (Dave Peterson et al.)" - id: ballew-geometric-vanishes resource: "https://pballew.blogspot.com/2022/02/geometric-vanishes-little-history.html" title: "Geometric Vanishes, A Little History" author: "Pat Ballew" - id: adams-conceptual-blockbusting resource: "https://archive.org/details/conceptualblockb00adam_0" title: "Conceptual Blockbusting: A Guide to Better Ideas (Perseus, 2001 printing, 3rd ed.), chapter 2, \"Perceptual Blocks\"" author: "James L. Adams" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Rearranged Triangle Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 7. Discussed with [Weird Organism](https://tyson-swetnam.github.io/aosd/problems/weird-organism/index.md); the same session starts the group lab on [Dominoes](https://tyson-swetnam.github.io/aosd/problems/dominoes-lab/index.md).* !!! abstract "The problem" The syllabus gives only the name. No text of Winfree's describing this problem survives, so the statement below is an editors' reconstruction, following the standard version of the puzzle that Martin Gardner published in 1956. On graph paper, mark out a right triangle with a base of 13 units and a height of 5 units, the right angle at the lower left. Cut out four pieces: - **A**: a right triangle with legs 8 (horizontal) and 3 (vertical). - **B**: a right triangle with legs 5 (horizontal) and 2 (vertical). - **C**: an L of 8 unit squares: a row of 5 with a row of 3 on top of its left end. - **D**: an L of 7 unit squares: a row of 5 with a row of 2 under its right end. **Arrangement 1.** Put A in the lower left corner, its 8-unit leg along the base, so its sloping edge runs from the corner up to the point 8 across and 3 up. Put B above and to its right, so B's sloping edge carries on from (8, 3) to (13, 5). The space below B is a 5-by-3 rectangle; fill it with C and D. The figure looks like a 13-by-5 right triangle, completely covered. **Arrangement 2.** Swap the two small triangles. B now sits in the lower left corner, its edge running from (0, 0) to (5, 2), and A sits above and to its right, carrying on to (13, 5). The space below A is now an 8-by-2 rectangle; fit C and D into it. The outline looks like the same 13-by-5 triangle, but this time one unit square is left uncovered. The same four pieces cover the figure one way and leave a hole the other way. Where did the square go? Do not stop at "it is an optical illusion". Say exactly what is wrong with the picture, and check your answer with arithmetic. ![Two arrangements of the same four pieces, two right triangles and two L-shaped pieces, inside what looks like a 13-by-5 right triangle; in the second arrangement one unit square is left empty](https://tyson-swetnam.github.io/aosd/assets/images/problems/rearranged-triangle.svg){ width="560" } *The two arrangements. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 7 opens Section 2, "Creative Blocks", with Adams's Chapter 2 on *perceptual* blocks: seeing what you expect to see, failing to use all your senses, and not isolating the real problem. The rearranged triangle is a pure specimen of the first of those. Nothing in the drawing is a lie. The only false thing in the room is the reading your eye imposes on it. The cure is what Section 1 practised: stop looking and start counting. Add up the four pieces. Compare that with the area of the triangle you think you see. Write the two small triangles' slopes as fractions and compare them. Any one of those three checks kills the paradox, and none of them depends on the others. The syllabus calls the session-12 problem [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md) "like a jig-saw puzzle of cross-checks"; three independent checks on one figure work the same way. The session's other item, [Weird Organism](https://tyson-swetnam.github.io/aosd/problems/weird-organism/index.md), makes the same point in biology: a habit of perception decides what the observer believes is there. ## Where it comes from Dissection paradoxes, in which pieces are shuffled and area seems to appear or vanish, are centuries old. Edme-Gilles Guyot printed one in 1769, and William Hooper copied it, mis-drawn figure and all, as "The geometric money" in *Rational Recreations* (1774; corrected in 1782). The best-known ancestor is the chessboard paradox: an 8-by-8 square cut into four pieces that reassemble into a 5-by-13 rectangle, 64 apparently becoming 65. Oskar Schloemilch published it in 1868. Sam Loyd claimed to have shown it at a chess congress in 1858, a claim no record confirms, and it appears in his posthumous *Cyclopedia of Puzzles* (1914). The triangle form is much newer. Paul Curry, a New York amateur magician, devised it in 1953, and Martin Gardner popularised it in *Mathematics, Magic and Mystery* (1956), whose chapter 8, "Geometrical Vanishes, Part II", remains the standard history. By the late 1990s the puzzle was circulating online as two coloured drawings and a one-line challenge. That is a plausible but unproven guess at how a class in 2001 met it. ??? tip "Hints" - Before explaining anything, compute. What is the area of a genuine 13-by-5 right triangle? What is the total area of the four pieces? - Compare the two small triangles. Are they similar? Work out the slope of each sloping edge as a fraction. - Lay a straightedge along the long edge of each arrangement, or draw the true line from (0, 0) to (13, 5) on the graph paper and see which pieces cross it. - The numbers 2, 3, 5, 8, 13 are not an accident. Find out what family they belong to, and why consecutive members are so easy to confuse. ??? success "Resolution" Neither figure is a triangle. The two small triangles are not similar: A's sloping edge has slope 3/8 = 0.375 and B's has slope 2/5 = 0.400, while a true 13-by-5 hypotenuse would have slope 5/13 = 0.385 the whole way. The long edge is therefore a bent line, with its corner at (8, 3) in the first arrangement and at (5, 2) in the second. In Arrangement 1 the bend sags just below the true diagonal; in Arrangement 2 it bulges just above it. ![The two bent outlines drawn over the true straight line from (0,0) to (13,5), with the thin parallelogram between them shaded](https://tyson-swetnam.github.io/aosd/assets/images/problems/rearranged-triangle-resolution.svg){ width="560" } *Where the square goes. Drawn for this site (CC BY 4.0).* The two outlines differ by a long thin parallelogram with corners (0, 0), (5, 2), (13, 5) and (8, 3). Its area is exactly 1 square unit. That is the missing square. The arithmetic agrees. The pieces total 12 + 5 + 8 + 7 = 32 squares, while a real 13-by-5 triangle has area 32.5. The first outline encloses 32, half a square short of a real triangle, and the pieces fill it exactly. The second encloses 33, half a square over, so one square is left bare. The bend is only about 1.2 degrees, far too small to see, especially with thick lines drawn on the diagram. Why these numbers? 2, 3, 5, 8, 13 are consecutive Fibonacci numbers, and Cassini's identity guarantees a discrepancy of exactly one unit. Use larger Fibonacci numbers and the slopes come closer together, so the illusion gets better. ## Sources - **Martin Gardner**, *Mathematics, Magic and Mystery* (Dover, 1956), chapter 8, "Geometrical Vanishes, Part II" β€” [Internet Archive](https://archive.org/details/magicmathematics0000unse){target=_blank} πŸ”“ *(borrow)* - **Alexander Bogomolny**, "Curry's Paradox", Cut-the-Knot β€” [cut-the-knot.org](https://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Triangle Dissection Paradox", MathWorld β€” [mathworld.wolfram.com](https://mathworld.wolfram.com/TriangleDissectionParadox.html){target=_blank} πŸ”“ - **Wikipedia contributors**, "Missing square puzzle" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Missing_square_puzzle){target=_blank} πŸ”“ - **Wikipedia contributors**, "Chessboard paradox" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Chessboard_paradox){target=_blank} πŸ”“ - **Wikipedia contributors**, "Hooper's paradox" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Hooper%27s_paradox){target=_blank} πŸ”“ - **William Hooper**, *Rational Recreations*, vol. 4 (1782), "Recreation CVI. The geometric money", p. 286 β€” [Internet Archive](https://archive.org/details/rationalrecreat00davigoog){target=_blank} πŸ”“ - **Sam Loyd** (ed. Sam Loyd Jr.), *Cyclopedia of 5000 Puzzles, Tricks and Conundrums* (1914), chessboard dissection p. 288 β€” [Internet Archive](https://archive.org/details/CyclopediaOfPuzzlesLoyd){target=_blank} πŸ”“ - **Torsten Sillke**, "Jigsaw Paradox" (annotated bibliography) β€” [math.uni-bielefeld.de](https://www.math.uni-bielefeld.de/~sillke/PUZZLES/jigsaw-paradox.html){target=_blank} πŸ”“ - **The Math Doctors**, "Disappearing Area?" (2020) β€” [themathdoctors.org](https://www.themathdoctors.org/disappearing-area/){target=_blank} πŸ”“ - **Pat Ballew**, "Geometric Vanishes, A Little History" (2022) β€” [pballew.blogspot.com](https://pballew.blogspot.com/2022/02/geometric-vanishes-little-history.html){target=_blank} πŸ”“ - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, chapter 2, "Perceptual Blocks" (Perseus, 2001 printing, 3rd ed.; which edition the course used is not known) β€” [Internet Archive](https://archive.org/details/conceptualblockb00adam_0){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Art of Scientific Discovery*, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only a name and a session. Under Tuesday 11 September it lists "Adams Chapter 2: Perceptual blocks", then "Discuss Weird Organism", "Discuss Rearranged Triangle" and "Start group lab on Dominoes". The syllabus is the only Winfree source for this problem: nothing on his archived web site mentions the puzzle or any dissection paradox. The identification rests on the name, the pairing with Adams's chapter on perceptual blocks, and the puzzle's fame in 2001 β€” not on any text of Winfree's describing it. The candidates the editors weighed: - **Curry's triangle, the missing-square paradox** (the reading above). The name says a triangle is rearranged, the force of the puzzle is perceptual, and it needs nothing but graph paper. The likeliest reading. - **The chessboard version**, 64 apparently becoming 65. Same mechanism, same lesson, and older, so Winfree may have used it instead or as well. But the syllabus says "Triangle", not square. - **Dudeney's Haberdasher's Puzzle** (1902): an equilateral triangle cut into four pieces that rearrange into a square. Literally a rearranged triangle, but a construction problem rather than an illusion, so it does not illustrate a perceptual block. Unlikely. - **The coin-triangle inversion puzzle**: ten coins in a triangle, to be turned upside down by moving three. Also a triangle rearranged, and a stock creativity exercise, but it tests insight, not perception. Unlikely. If a Winfree handout ever turns up showing a different figure, this page should be revised. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/weird-organism/ --- title: "Weird Organism" description: "A lost perceptual-block handout from Winfree's session on Adams's Chapter 2, reconstructed as an exercise in describing a familiar living thing so plainly that a stereotype hides it." type: Activity tags: [course, student-facing, problem, section-2, perceptual-blocks, stereotyping, observation] status: stable problem: section: 2 session: 7 identification: unknown kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2003 resource: "https://web.archive.org/web/20030111064604/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (EEB 479/479H/579): course handout and retrospective syllabus (Internet Archive capture)" author: "Arthur T. Winfree" - id: adams-blockbusting-current resource: "https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674042/" title: "Conceptual Blockbusting: A Guide to Better Ideas (publisher page, Fifth Edition)" author: "James L. Adams" - id: adams-blockbusting-archive resource: "https://archive.org/details/conceptualblockb00jame" title: "Conceptual Blockbusting: A Guide to Better Ideas, 3rd edition (Internet Archive lending copy)" author: "James L. Adams" - id: platt-excitement-of-science resource: "https://archive.org/details/excitementofscie0000john" title: "The Excitement of Science (Internet Archive lending copy), containing the essay The Art of Creative Thinking" author: "John Rader Platt" - id: miner-nacirema-1956 resource: "https://doi.org/10.1525/aa.1956.58.3.02a00080" title: "Body Ritual among the Nacirema" author: "Horace Miner" - id: dallenbach-puzzle-picture-1951 resource: "https://doi.org/10.2307/1419008" title: "A Puzzle-Picture with a New Principle of Concealment" author: "Karl M. Dallenbach" - id: wikipedia-ambiguous-image resource: "https://en.wikipedia.org/wiki/Ambiguous_image" title: "Ambiguous image (overview)" author: "Wikipedia contributors" - id: wikipedia-platypus resource: "https://en.wikipedia.org/wiki/Platypus" title: "Platypus (overview)" author: "Wikipedia contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Weird Organism Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 7. Discussed together with [Rearranged Triangle](https://tyson-swetnam.github.io/aosd/problems/rearranged-triangle/index.md); the same session starts the group lab on [Dominoes](https://tyson-swetnam.github.io/aosd/problems/dominoes-lab/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: only the title survives, so the exercise below is the editors' own, in the form the evidence makes most likely β€” a familiar organism described so plainly that a stereotype hides it. A naturalist from somewhere very far away sends these notes on an organism they have never met, and asks you what it is. > It is among the largest organisms in its habitat, yet by mass almost > entirely dead: a living film a few cells thick lies beneath a coat of > dead armour, and nearly everything within is dead tissue the organism > keeps and builds upon. It grows only at its tips and in girth, never > lengthening in the middle. It eats air, using light to do so, and > drinks through the parts of itself buried in the ground. It never > moves from where it began life, and may live for centuries. Some > kinds recruit mobile animals to carry their offspring away. In your GamesWorth notebook, before class: 1. What is it? Name the class of things, not one species. 2. Mark every sentence that made you picture something exotic, and beside each write the assumption it violated. 3. Describe some living thing you see daily β€” yourself is allowed β€” so that a friend cannot identify it in under a minute. ## Why it is in the course Section 2 is six sessions on creative blocks, and session 7 is the perceptual-blocks session, read against Adams's Chapter 2. Winfree paired three exercises whose names all point at the gap between what is in front of you and what you perceive. What the other two were is an inference from those names: the Rearranged Triangle is probably a dissection figure in which the eye supplies a straight edge that is not there, and the Dominoes lab possibly the mutilated chessboard, where a hidden property of the board decides which tilings can exist. The Weird Organism puts a stereotype between you and a plain description: Adams's first perceptual block is seeing what you expect to see, and "organism" summons something animal-sized and mobile. The point was never the identification. Winfree wrote that the puzzles, "many of them silly", exist "to slow you down for a few minutes so you can examine the working of your own mind", and that "actually solving the practice problems is way less important than learning how to try". What you bring to class is the inventory of assumptions you caught yourself making. ## Where it comes from The problem is known from one document only: Winfree's course handout for EEB 479/479H/579, transcribed here as [the syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md), where "discuss Weird Organism" opens session 7. Solo problems went out a week ahead, so it was handed over at session 6. The sheet he handed out for the problem has never been found, on his archived pages or anywhere else, so the title is all that survives. Like most of the course's puzzles it was most likely his own, "contrived much as the organizers of an Easter Egg Hunt do". The title points at an old trick: describing the familiar as a stranger would see it, as in the Sphinx's riddle or Horace Miner's "Body Ritual among the Nacirema" (1956). Its visual cousin is the hidden figure, whose type specimen is a puzzle-picture published in the *American Journal of Psychology* in 1951 and commonly reproduced as the "Dallenbach cow": an animal most viewers cannot find, and then cannot un-see. ??? tip "Hints" - Separate observation from interpretation: underline every word that is already a conclusion β€” "eats", "armour", "dead" β€” and rewrite it as the bare thing an observer would see. - Notice what "organism" made you picture, then deliberately try the other kingdoms of life, and other scales of size and time. - Test the notes against the nearest living things: yourself, a pet, a houseplant, whatever grows on the wall outside. - Check any candidate against every sentence, especially the ones that fit badly; one that explains most of the notes is a hypothesis, not an answer. ??? success "Resolution" **For the reconstructed example only.** Winfree's own answer is not known. The notes describe a tree, and more broadly any woody perennial plant. The dead armour is bark, the living film is cambium and phloem, the wood within is mostly dead xylem the tree keeps and builds on; growth is at shoot and root tips and, in girth, at the cambium; eating air by light is photosynthesis; the recruited animals are pollinators and seed dispersers. Every clause is meant to be literally true of some ordinary tree, and the exotic creature came from the reader's stereotype of an organism as an animal. What belongs in the notebook is not the word "tree" but the assumptions that delayed it. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery (EEB 479/479H/579): course handout and retrospective syllabus* (2001) β€” [Internet Archive capture](https://web.archive.org/web/20030111064604/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, Chapter 2 "Perceptual Blocks" (Fifth Edition, 2019) β€” [publisher page](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674042/){target=_blank} πŸ”’ - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, 3rd edition (1986) β€” [Internet Archive](https://archive.org/details/conceptualblockb00jame){target=_blank} πŸ”“ *(borrow)* - **John Rader Platt**, *The Excitement of Science* (1962), containing "The Art of Creative Thinking" β€” [Internet Archive](https://archive.org/details/excitementofscie0000john){target=_blank} πŸ”“ *(borrow)* - **Horace Miner**, "Body Ritual among the Nacirema", *American Anthropologist* 58(3), 503–507 (1956) β€” [DOI](https://doi.org/10.1525/aa.1956.58.3.02a00080){target=_blank} πŸ”’ - **Karl M. Dallenbach**, "A Puzzle-Picture with a New Principle of Concealment", *American Journal of Psychology* 64(3), 431–433 (1951) β€” [DOI](https://doi.org/10.2307/1419008){target=_blank} πŸ”’ - **Wikipedia contributors**, "Ambiguous image" (overview) β€” [Wikipedia](https://en.wikipedia.org/wiki/Ambiguous_image){target=_blank} πŸ”“ - **Wikipedia contributors**, "Platypus" (overview, for the 1799 specimens) β€” [Wikipedia](https://en.wikipedia.org/wiki/Platypus){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not at all: the identification is **unknown**. The syllabus gives the name, the session and its two neighbours, and nothing else; searches on the exact phrase, on Adams's chapter, on puzzle collections and across Winfree's archived site found no text. Both session-mates are "what you see is not what is there" puzzles, which makes a perceptual-block exercise about an organism likely, but its form is open. The 2002 and 2007 captures of the handout are word for word the same, so nothing fuller is online. - **A written "identify this organism" puzzle**, the form reconstructed above. It fits Adams's stereotyping block and the course rule that exercises "depend as little as possible on knowledge of any particular subject area", but no text has been found. - **A hidden or ambiguous picture of an animal**, in the family of Dallenbach's cow. It would sit beside the Rearranged Triangle naturally, but a picture is hard to work on for a week. - **A "could such an organism exist?" thought experiment**, asking which assumptions about living things are necessary and which are habit. Nothing beyond the title supports it. - **A real organism that looks impossible**, the stock case being the platypus, whose first specimens in Britain were judged in 1799 to be a fake sewn together from several animals. Against it: the schedule lists readings and anecdotes bare β€” Golden Tooth, N-Rays, the Barometer Story β€” but puts "discuss" in front of the problems students worked on for a week, and this one carries it. If the handout ever surfaces, this page should be rewritten around it. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/telltale-number/ --- title: "Telltale Number" description: "Write a ten-digit number whose first digit counts its zeros, whose second counts its ones, and so on: ten billion candidates collapse to one as soon as you notice the fact the puzzle never states." type: Activity tags: [course, student-facing, problem, section-2, emotional-blocks, self-reference, arithmetic, recreational-mathematics] status: stable problem: section: 2 session: 8 identification: confident kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: fixx-solve-it-1978 resource: "https://archive.org/details/solveitperplexin00fixx" title: "Solve It! A Perplexing Profusion of Puzzles" author: "James F. Fixx" - id: openlibrary-search-inside resource: "https://openlibrary.org/search/inside?q=%22Telltale+Number+Write+a+ten-digit+number%22" title: "Full-text search inside Solve It! (Internet Archive via Open Library)" author: "Internet Archive / Open Library" - id: gardner-mathematical-circus-1979 resource: "https://archive.org/details/mathematicalcirc00gard" title: "Mathematical Circus (problem 7, pp. 128 and 135)" author: "Martin Gardner" - id: oeis-a046043 resource: "https://oeis.org/A046043" title: "A046043: Autobiographical numbers (or curious numbers)" author: "Robert Leduc (author); edited by N. J. A. Sloane, OEIS Foundation" - id: wikipedia-self-descriptive-number resource: "https://en.wikipedia.org/wiki/Self-descriptive_number" title: "Self-descriptive number" author: "Wikipedia contributors" - id: mathworld-self-descriptive resource: "https://mathworld.wolfram.com/Self-DescriptiveNumber.html" title: "Self-Descriptive Number" author: "Eric W. Weisstein, MathWorld" - id: khovanova-arxiv-2008 resource: "https://arxiv.org/abs/0803.0270" title: "Autobiographical Numbers (arXiv:0803.0270)" author: "Tanya Khovanova" - id: khovanova-blog-2007 resource: "https://blog.tanyakhovanova.com/2007/12/autobiographical-numbers/" title: "Autobiographical Numbers (blog post)" author: "Tanya Khovanova" - id: wikipedia-jim-fixx resource: "https://en.wikipedia.org/wiki/Jim_Fixx" title: "Jim Fixx" author: "Wikipedia contributors" - id: mathsisfun-10-digit resource: "https://www.mathsisfun.com/puzzles/10-digit-number.html" title: "10-digit Number Puzzle" author: "Rod Pierce, Math is Fun" - id: winfree-handout-wayback-2001 resource: "https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "Handout for The Art of Scientific Discovery (EEB 479/479H/579), archived" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Telltale Number Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 8. The syllabus sets it for the day of [The Barometer Story](https://tyson-swetnam.github.io/aosd/problems/barometer-story/index.md) and Adams's chapter on emotional blocks, with the note "About assumptions and [Tying Knots](https://tyson-swetnam.github.io/aosd/problems/tying-knots/index.md)".* !!! abstract "The problem" Write down a ten-digit number with this property: its first (leftmost) digit tells how many zeros appear in the whole number, its second digit tells how many ones, its third digit how many twos, and so on, until the tenth digit tells how many nines. In other words, label the ten positions 0, 1, 2, ..., 9 from left to right. The digit sitting in position *k* must equal the number of times the digit *k* occurs anywhere in the number, counting its own position too. There is exactly one such ten-digit number. Find it, and be able to say why there is no other. Before you start filling in digits at random, ask what the wording forces to be true of every such number. A note on the wording: Winfree's archived course handout names the puzzle but never states it, and of the original, in James F. Fixx's *Solve It!* (1978), only the opening clause is readable ("Write a ten-digit number so that the first digit tells ..."). The three paragraphs above are our own reconstruction, checked against the puzzle as it circulates today; the instruction to prove that the answer is unique is our addition. ![Ten empty boxes in a row, labelled beneath from left to right: how many 0s, how many 1s, and so on up to how many 9s, with an arrow showing that each count refers back to the whole row](https://tyson-swetnam.github.io/aosd/assets/images/problems/telltale-number.svg){ width="560" } *The shape of the puzzle: ten boxes, each of which reports on all ten. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 8 is the second meeting of Section 2, "Creative Blocks", and the reading due that day is Chapter 3 of James L. Adams's *Conceptual Blockbusting*, on emotional blocks: fear of making a mistake, an inability to tolerate ambiguity, the urge to judge an idea before it is finished, and the refusal to let a problem incubate. The Telltale Number exercises all four at once. The puzzle is unpleasant at first sight, and that is the point. It looks circular, because every digit depends on all the others, and it looks hopeless, because there are ten billion ten-digit strings to sift. The two natural responses are paralysis and frantic guessing, which are the emotional blocks Adams describes, arriving on cue. Winfree's note for the session, "About assumptions", points at the exit. The assumption worth examining is not about the digits; it is your own assumption that searching is the only way in. One unstated fact does the work. Find it and ten billion cases collapse into a few lines of reasoning; the puzzle's title is itself a hint that a single telltale feature gives the whole thing away. The syllabus describes the purpose of these exercises plainly: "The purpose of the puzzles (many of them silly) is to slow you down for a few minutes so you can examine the working of your own mind." What is worth writing in your GamesWorth book is not the number but the minute in which you stopped guessing. ## Where it comes from The ten-digit self-describing number is a staple of recreational mathematics from the 1970s. James F. Fixx (1932-1984), a Mensa member far better known for *The Complete Book of Running*, published three puzzle collections with Doubleday, and the third, *Solve It!* (1978), carries this one as item 8 under exactly the name Winfree used. Full-text search inside the scanned copies returns the snippet with its running head: "50 SOLVE IT! 8. The Telltale Number Write a ten-digit number so that the first digit tells ...". That is most likely where Winfree found both the puzzle and the name. The book is lending-restricted, so only that opening clause is readable, and Fixx's answer pages are not indexed at all. The answer below therefore rests on the later sources and on exhaustive computation, not on Fixx. Martin Gardner posed the same puzzle in *Mathematical Circus* (Knopf, 1979) as problem 7; Tanya Khovanova reports that in discussing the answer Gardner mentions the name "tally numbers" for such numbers, and she generalized the idea in 2008 to the "biographies" of numbers. MathWorld cites Clifford Pickover's *Keys to Infinity* (1995), Chapter 28, for the same number, and the On-Line Encyclopedia of Integer Sequences carries the whole family as A046043, "autobiographical numbers", each entry written in the base equal to its own number of digits. Who first invented the puzzle is not recorded in any source found. Winfree's own pages add nothing. The archived copy of his course handout, the document transcribed here as the syllabus, carries the session-8 line and no statement, hint or answer for the puzzle; the other item on that line, [Tying Knots](https://tyson-swetnam.github.io/aosd/problems/tying-knots/index.md), is undocumented, so what he did with it in class remains an open question. ??? tip "Hints" - Every digit in the number is a count of something, and the ten things being counted are the ten digits of the number itself. What, then, must the ten digits add up to? - That sum forces most of the digits to be zero. So the first digit, the count of zeros, must be fairly large. What does a large first digit imply about the digit sitting in the position with that same label? - Can any digit larger than 2 appear anywhere except in the first position? Can any digit larger than 2 appear twice? - Work down from the largest possible first digit (9, then 8, then 7 ...) and see how quickly each case contradicts itself. - If ten digits feel like too many, solve the same puzzle for four-digit and five-digit numbers first: a four-digit number whose digits count its zeros, ones, twos and threes. The pattern carries over. ??? success "Resolution" The unique answer is **6 2 1 0 0 0 1 0 0 0**. It contains six zeros, two ones, one two and one six, and no threes, fours, fives, sevens, eights or nines, which is exactly what its digits claim. ![The digits 6, 2, 1, 0, 0, 0, 1, 0, 0, 0 in ten boxes, each labelled with the count it asserts, and the sum six plus two plus one plus one equals ten](https://tyson-swetnam.github.io/aosd/assets/images/problems/telltale-number-solution.svg){ width="560" } *The answer, checked against itself. Drawn for this site (CC BY 4.0).* Why it is the only one. Call the digits a0, a1, ..., a9 from left to right, so that a0 counts the zeros in the number, a1 counts the ones, and in general the digit labelled with a given value counts how many times that value occurs. Since the number has ten digits and each digit is counted exactly once, a0 + a1 + ... + a9 = 10. The number has a0 zeros, so it has 10 - a0 nonzero digits, and a0 itself is one of them (a0 cannot be 0, or the number would contain a zero it fails to count). The other 9 - a0 nonzero digits therefore sum to 10 - a0, which is one more than their count: they are all 1s except for a single 2. So every digit apart from a0 is 0, 1 or 2. That already rules out a0 = 9 (no digit is left to carry the extra 1) and a0 = 8 (the one other nonzero digit would be a 2, so a2 would have to be 1, but no 1 is available). It also rules out a0 = 1 (then a0 is itself a 1 and there are seven more, so a1 would have to be 8, which is not a digit of the number) and a0 = 2 (six 1s, so a1 would have to be 6, likewise absent). For a0 between 3 and 7 the number contains exactly one 2, so a2 = 1, and the 1s are the remaining 8 - a0 of the "other" digits, so a1 = 8 - a0. But a1 is itself one of those other digits, so a1 is 1 or 2. If a1 = 1 then a0 = 7, yet a2 = 1 and a7 = 1 (the digit 7 occurs once, as a0) are already two 1s, a contradiction. If a1 = 2 then a0 = 6, and 6210001000 checks. No other case survives. The same question asked of numbers of other lengths, each read in the base equal to its own number of digits, gives 1210 and 2020 with four digits, 21200 with five, 3211000 with seven, 42101000 with eight, 521001000 with nine and 6210001000 with ten. There is nothing at all with two, three or six digits, and exactly one answer for every length from seven upward (OEIS A046043). ## Sources - **James F. Fixx**, *Solve It! A Perplexing Profusion of Puzzles* (Doubleday, 1978), item 8, p. 50 β€” the source of the puzzle and of its name β€” [Internet Archive](https://archive.org/details/solveitperplexin00fixx){target=_blank} πŸ”“ *(borrow)* - **Internet Archive / Open Library**, full-text search inside *Solve It!* for "Telltale Number Write a ten-digit number" β€” the evidence for the title, the item number and the opening clause β€” [openlibrary.org](https://openlibrary.org/search/inside?q=%22Telltale+Number+Write+a+ten-digit+number%22){target=_blank} πŸ”“ - **Martin Gardner**, *Mathematical Circus* (Knopf, 1979), problem 7, pp. 128 and 135 β€” page numbers cited from OEIS and Khovanova, not read in the book β€” [Internet Archive](https://archive.org/details/mathematicalcirc00gard){target=_blank} πŸ”“ *(borrow)* - **Robert Leduc**, ed. N. J. A. Sloane, "A046043: Autobiographical numbers (or curious numbers)", OEIS β€” [oeis.org](https://oeis.org/A046043){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Self-Descriptive Number", MathWorld β€” [mathworld.wolfram.com](https://mathworld.wolfram.com/Self-DescriptiveNumber.html){target=_blank} πŸ”“ - **Tanya Khovanova**, "Autobiographical Numbers", arXiv:0803.0270 (2008); published as "A Story of Storytelling Numbers", *Math Horizons* 17(1), 14-17 (2009) β€” [arxiv.org](https://arxiv.org/abs/0803.0270){target=_blank} πŸ”“ - **Tanya Khovanova**, "Autobiographical Numbers" (blog post, December 2007) β€” [blog.tanyakhovanova.com](https://blog.tanyakhovanova.com/2007/12/autobiographical-numbers/){target=_blank} πŸ”“ - **Wikipedia contributors**, "Self-descriptive number" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Self-descriptive_number){target=_blank} πŸ”“ - **Wikipedia contributors**, "Jim Fixx" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Jim_Fixx){target=_blank} πŸ”“ - **Rod Pierce**, "10-digit Number Puzzle", Math is Fun β€” [mathsisfun.com](https://www.mathsisfun.com/puzzles/10-digit-number.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, handout for *The Art of Scientific Discovery* (EEB 479/479H/579), 2001 β€” the archived copy of the same document transcribed here as the syllabus β€” [Wayback Machine](https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/barometer-story/ --- title: "The Barometer Story" description: "Calandra's parable of the student who measured a building with a barometer six ways, none of them the one the examiner had in mind: a lesson in hidden assumptions and emotional blocks." type: Activity tags: [course, student-facing, problem, section-2, emotional-blocks, assumptions, physics-teaching, creativity] status: stable problem: section: 2 session: 8 identification: confident kind: case-study generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: calandra-1968-standler resource: "http://www.rbs0.com/baromete.htm" title: "Angels on the Head of a Pin: A Modern Parable (Saturday Review, 21 Dec 1968), with Standler's note on the 1961 and 1964 versions" author: "Alexander Calandra; hosted and annotated by Ronald B. Standler" - id: calandra-1964-panarchy resource: "https://panarchy.org/calandra/barometer.html" title: "The Barometer Story: A problem in teaching critical thinking (Current Science, 1964)" author: "Alexander Calandra; hosted by panarchy.org" - id: calandra-1968-hicks resource: "https://www.stephenhicks.org/2022/07/10/angels-on-the-head-of-a-pin-by-alexander-calandra/" title: "Angels on the Head of a Pin, by Alexander Calandra" author: "Alexander Calandra; posted by Stephen Hicks" - id: calandra-lander-reader resource: "https://philosophy.lander.edu/intro/introbook2.1/x874.html" title: "The Barometer Story, in Reading for Philosophical Inquiry, ch. 3" author: "Alexander Calandra; ed. Lee Archie and John G. Archie" - id: wikipedia-barometer-question resource: "https://en.wikipedia.org/wiki/Barometer_question" title: "Barometer question" author: "Wikipedia contributors" - id: wikipedia-calandra resource: "https://en.wikipedia.org/wiki/Alexander_Calandra" title: "Alexander Calandra" author: "Wikipedia contributors" - id: snopes-barometer resource: "https://www.snopes.com/fact-check/the-barometer-problem/" title: "The Barometer Problem (fact check: was the student Niels Bohr?)" author: "David Mikkelson, Snopes" - id: pride-1959 resource: "https://books.google.com/books?id=DNWgAAAAMAAJ&q=barometer" title: "Pride, volumes 3-4 (American College Public Relations Association, 1959)" author: "American College Public Relations Association" - id: readers-digest-1958 resource: "https://books.google.com/books?id=P7cNAQAAMAAJ&q=aneroid+barometer" title: "Reader's Digest Treasury of Wit & Humor (1958)" author: "Reader's Digest Association" - id: simanek-barometer-fable resource: "https://jcdverha.home.xs4all.nl/scijokes/2_12.html" title: "Science Jokes, section 2.12, including Donald Simanek's The Barometer Fable" author: "Joachim Verhagen (compiler); Donald E. Simanek" - id: hake-2013 resource: "http://betterfilecabinet.com/pipermail/rume_betterfilecabinet.com/2013-January/004364.html" title: "The Old Barometer Story (OBS) Redux" author: "Richard R. Hake" - id: naturelovesmath-2011 resource: "https://www.naturelovesmath.com/en/humor/the-myth-of-niels-bohr-and-the-barometer-question/" title: "The myth of Niels Bohr and the barometer question" author: "Nature Loves Math" - id: adams-conceptual-blockbusting resource: "https://archive.org/details/conceptualblockb00jame" title: "Conceptual Blockbusting: A Guide to Better Ideas (Chapter 3, Emotional Blocks)" author: "James L. Adams" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # The Barometer Story Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 8. The same session discusses [Telltale Number](https://tyson-swetnam.github.io/aosd/problems/telltale-number/index.md) and, under the note "About assumptions", [Tying Knots](https://tyson-swetnam.github.io/aosd/problems/tying-knots/index.md).* !!! abstract "The problem" **Part 1: try it first.** A physics examination asked: *"Show how it is possible to determine the height of a tall building with the aid of a barometer."* (Alexander Calandra, 1959/1964.) Take five minutes and list every way you can think of, without stopping at the first. Then mark which of them a physics instructor would accept for full credit. **Part 2: the story**, paraphrased by the editors; Calandra's text is in copyright. Calandra was asked to referee a grading dispute. The instructor wanted to give a student zero; the student insisted on a perfect score. His answer was, in effect, lower the barometer from the roof on a rope and measure the rope. Calandra thought the answer complete and correct β€” and empty of physics, which is what a physics grade certifies. He gave the student six minutes to try again, with some physics in it. At five minutes the student had written nothing; asked whether he wished to give up, he said no, he had *many* answers and was choosing the best. In the last minute he wrote: drop the barometer from the roof, time the fall, use the free-fall formula. That earned almost full credit. On the way out he listed the rest: compare its shadow with the building's, mark the stairwell off in barometer-lengths, swing it as a pendulum and infer the height from the change in *g*, or β€” probably best, he said β€” trade the barometer to the superintendent for the answer. **For the session.** Which answer would you have graded highest, and what were you feeling as you decided? ![Diagram of a tall building labelled h equals question mark, surrounded by six small panels showing ways to measure it with a barometer: comparing pressure at roof and street, lowering it on a rope, comparing shadows, dropping it and timing the fall, using it as a ruler on the stairs, and trading it to the superintendent for the answer](https://tyson-swetnam.github.io/aosd/assets/images/problems/barometer-story.svg){ width="560" } *Six of the seven methods; the student's pendulum is left out, for the reason given below. Drawn for this site (CC BY 4.0).* ## Why it is in the course The syllabus puts the story at the head of session 8 with "Adams Chapter 3: Emotional blocks", then "About assumptions and Tying Knots". Adams's blocks include fear of taking a risk, no appetite for chaos, and judging ideas instead of generating them. The instructor shows several at once: one answer in mind, instant judgement, no room for the rest. The student shows fluency, holding many answers in suspension through five silent minutes. The story also carries the assumption lesson: "with the aid of a barometer" was silently read as "using the barometer's pressure reading". And it turns the lens on the reader, since most people feel a flash of irritation at one character or the other, and that feeling is the block under discussion. ## Where it comes from Alexander Calandra (1911-2006) taught physics at Washington University in St. Louis from 1948 until his retirement in 1979. His essay "Angels on a Pin" appeared in *Pride* in 1959 and was reprinted in *Current Science* in January 1964 as "The Barometer Story: A problem in teaching critical thinking", whose main title is the one the syllabus uses. The version everyone copies is "Angels on the Head of a Pin", *Saturday Review*, 21 December 1968. The joke is older. The *Reader's Digest Treasury of Wit & Humor* (1958) already has a student lowering an aneroid barometer on a string and measuring the string. Later retellings move the exam to Copenhagen and make the student Niels Bohr; Snopes files the tale under "Legend" and rejects that attribution. ??? tip "Hints" - Inventory the barometer as an object, not an instrument: it has a length, a shadow and a resale value, and it can be tied, dropped, swung and given away. - Ask what the question says and what the grader silently added. - Sort your list by what each method demonstrates: physics, ingenuity, or merely the number. - Check the accuracy. Pressure falls by roughly 1 hPa for every 8 m near sea level; is that measurable on a 30 m building? - Notice your reaction to each character, and name the block behind it. ## What happened **The intended answer.** Read the barometer at street level and on the roof and take the difference: pressure falls with height, near sea level by about 1 hPa per 8 m. This is Pascal's idea, tested in 1648 when his brother-in-law Florin PΓ©rier carried a Torricelli tube up the Puy de DΓ΄me. For one building it is also among the least precise methods on the list. ![Nineteenth-century wood engraving of Florin PΓ©rier and companions in eighteenth-century dress reading a tall Torricelli tube on the summit of the Puy de DΓ΄me](https://tyson-swetnam.github.io/aosd/assets/images/problems/barometer-story-puydedome.jpg){ width="560" } *Florin PΓ©rier measuring the mercury column near the top of the Puy de DΓ΄me, 1648. Wood engraving by Yan Dargent for Louis Figuier, 1867; Library of Congress, LCCN 2006690481. Public domain, via Wikimedia Commons.* **How the six hold up.** The rope is exact, and uses the barometer as a plumb bob. Free fall contains real physics but destroys the instrument, and a hand-timed fall is good to perhaps ten percent. Shadows work by similar triangles; the staircase is exact and tedious; the superintendent has no physics and perfect accuracy. The pendulum is the weak one: the free-air gradient of *g* is about three parts in ten million per metre, so a 100 m building changes *g* by three parts in a hundred thousand, which no stopwatch pendulum can detect. The answer that sounds most sophisticated is the only one that cannot be made to work β€” our objection, not Calandra's or Simanek's. **What it does not settle.** Calandra's 1964 moral warns against teaching "the scientific method" as a recipe divorced from subject matter. Donald Simanek grants that the piece is nicely constructed humour but holds that "as a parable with a moral, it falls flat": Calandra never says plainly what the moral is. And the referee's dilemma is real. The rope answer is correct and shows no physics, so the defective thing was the question, not the student. The lesson is not that the student was right and the instructor wrong, but that the single-answer expectation was an assumption nobody examined. ## Sources - **Alexander Calandra**, "Angels on the Head of a Pin: A Modern Parable", *Saturday Review*, 21 December 1968, p. 60 β€” [full text with Ronald B. Standler's note on the 1961 and 1964 versions](http://www.rbs0.com/baromete.htm){target=_blank} πŸ”“ - **Alexander Calandra**, "The Barometer Story: A problem in teaching critical thinking", *Current Science* (teacher's edition), 1964 β€” [full text, including the seven precautions](https://panarchy.org/calandra/barometer.html){target=_blank} πŸ”“ - **Alexander Calandra**, "Angels on the Head of a Pin" β€” [copy posted by Stephen Hicks](https://www.stephenhicks.org/2022/07/10/angels-on-the-head-of-a-pin-by-alexander-calandra/){target=_blank} πŸ”“ - **Alexander Calandra**, "The Barometer Story", in Lee Archie and John G. Archie (eds), *Reading for Philosophical Inquiry*, ch. 3 β€” [philosophy.lander.edu](https://philosophy.lander.edu/intro/introbook2.1/x874.html){target=_blank} πŸ”“ - **American College Public Relations Association**, *Pride*, vols 3-4 (1959), the essay's first appearance β€” [Google Books](https://books.google.com/books?id=DNWgAAAAMAAJ&q=barometer){target=_blank} πŸ”’ - **Reader's Digest Association**, *Reader's Digest Treasury of Wit & Humor* (1958), p. 303, the earlier one-liner β€” [Google Books](https://books.google.com/books?id=P7cNAQAAMAAJ&q=aneroid+barometer){target=_blank} πŸ”’ - **Joachim Verhagen** (compiler) and **Donald E. Simanek**, *Science Jokes* section 2.12, which reproduces Calandra's text and Simanek's critique "The Barometer Fable" β€” [jcdverha.home.xs4all.nl](https://jcdverha.home.xs4all.nl/scijokes/2_12.html){target=_blank} πŸ”“ - **David Mikkelson**, "The Barometer Problem", *Snopes* (2000, revised since) β€” [snopes.com](https://www.snopes.com/fact-check/the-barometer-problem/){target=_blank} πŸ”“ - **Richard R. Hake**, "The Old Barometer Story (OBS) Redux" (2013) β€” [mailing-list archive](http://betterfilecabinet.com/pipermail/rume_betterfilecabinet.com/2013-January/004364.html){target=_blank} πŸ”“ - **Nature Loves Math**, "The myth of Niels Bohr and the barometer question" (2011) β€” [naturelovesmath.com](https://www.naturelovesmath.com/en/humor/the-myth-of-niels-bohr-and-the-barometer-question/){target=_blank} πŸ”“ - **Wikipedia contributors**, "Barometer question" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Barometer_question){target=_blank} πŸ”“ - **Wikipedia contributors**, "Alexander Calandra" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Alexander_Calandra){target=_blank} πŸ”“ - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas* (1974; 3rd ed. 1986), Chapter 3, Emotional Blocks β€” [Internet Archive](https://archive.org/details/conceptualblockb00jame){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/tying-knots/ --- title: "Tying Knots" description: "Hold one end of a rope in each hand and tie a knot in it without ever letting go: an impossible-looking task that becomes easy the moment you find the rule nobody stated." type: Activity tags: [course, student-facing, problem, section-2, topology, hidden-assumptions, creative-blocks] status: stable problem: section: 2 session: 8 identification: probable kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: ashley-1944 resource: "https://archive.org/details/TheAshleyBookOfKnots" title: "The Ashley Book of Knots, entry #2576 (chapter \"Tricks and Puzzles\")" author: "Clifford W. Ashley" - id: braingle-896 resource: "https://www.braingle.com/brainteasers/896/knot-the-rope.html" title: "Knot the Rope (Brain Teaser #896)" author: "Braingle (submitted by user \"Michelle\")" - id: puzzle-prime-tie-a-knot resource: "https://www.puzzleprime.com/puzzles/brain-teasers/practical/tie-a-knot/" title: "Tie a Knot" author: "Puzzle Prime" - id: mr-magician-handkerchief resource: "https://www.mrmagician.co.uk/knotinhandkerchief.html" title: "Knot in Handkerchief Trick" author: "Chris Welsh (Mr Magician)" - id: gardner-1956 resource: "https://books.google.com/books/about/Mathematics_Magic_and_Mystery.html?id=-kOFBQAAQBAJ" title: "Mathematics, Magic and Mystery" author: "Martin Gardner" - id: adams-blockbusting resource: "https://www.hachette.co.uk/titles/james-l-adams/conceptual-blockbusting-fifth-edition/9781541674042/" title: "Conceptual Blockbusting: A Guide to Better Ideas (5th ed.), Ch. 3 \"Emotional Blocks\"" author: "James L. Adams" - id: wikipedia-overhand-knot resource: "https://en.wikipedia.org/wiki/Overhand_knot" title: "Overhand knot" author: "Wikipedia" - id: wikipedia-ashley-book-of-knots resource: "https://en.wikipedia.org/wiki/The_Ashley_Book_of_Knots" title: "The Ashley Book of Knots" author: "Wikipedia" - id: magicians-own-book-1862 resource: "https://www.gutenberg.org/ebooks/60687" title: "The Magician's Own Book, or The Whole Art of Conjuring" author: "George Arnold and Frank Cahill" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Tying Knots Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 8. The readings due that day are [The Barometer Story](https://tyson-swetnam.github.io/aosd/problems/barometer-story/index.md) and Adams Chapter 3, on emotional blocks; the session also discusses [Telltale Number](https://tyson-swetnam.github.io/aosd/problems/telltale-number/index.md).* !!! abstract "The problem" Winfree's schedule gives four words for this session's exercise: "About assumptions and Tying Knots". Nothing else of his survives for it, so the statement below is a reconstruction: the classic puzzle that fits those words, rebuilt from Clifford Ashley's 1944 version and from the way it circulates today. Lay a piece of rope (a shoelace or a metre of string works) straight on a table in front of you. Pick up one end in your left hand and the other end in your right hand. Now, **without letting go of either end at any moment**, tie an ordinary overhand knot in the middle of the rope. You may move your hands, arms and body however you like, and pass the rope over or under anything; but the rope must never leave your grip, you may not slide your hands along it, and no one may help. Most people conclude, after a few minutes of contorted attempts, that it is impossible. It is not. Before you look for the trick, write down every rule you think you are obeying, and ask which was actually stated. ## Why it is in the course Session 8 sits in the middle of Section 2, on creative blocks, between Adams's chapters on perceptual blocks (session 7) and cultural blocks (session 9). Winfree's wording puts assumptions first and knots second, and that is what the exercise is for. The solver fails not because the task is hard but because of a rule that was never in the problem: that you begin with your arms uncrossed. Every failed attempt demonstrates a constraint the solver supplied. The pairing with Adams Chapter 3, on emotional blocks, matters too. The puzzle is attempted in public; most give up and declare it impossible, and the answer, once seen, looks childishly simple. It is a safe rehearsal of being wrong, and of the reluctance to look foolish that Adams discusses. There is a second lesson for Winfree the topologist. Ordinary attempts fail for a genuine reason, not for want of dexterity: once you are holding both ends, no amount of wriggling can put a knot into the rope. So the exercise also separates what is truly impossible from what merely seems so, in the spirit of the Section 1 sessions on "hidden assumptions" and on "distinguishing things we know vs only imagine". ## Where it comes from The challenge is an old parlour trick. Clifford W. Ashley recorded it in *The Ashley Book of Knots* (1944) as entry #2576, in his chapter of tricks and puzzles: hold the opposite corners of an unknotted handkerchief and, "without once letting go", tie a knot in it. It "has also been called the 'Fourth-Dimensional Knot' because it appears from nowhere in particular", Ashley notes, and he describes the performer folding his arms, toying with the two ends, and producing the knot on unfolding them. The idea of a knot stored in the arms was working knowledge among sailors: Ashley's #2543 forms a clove hitch with the arms "crossed as far as possible" and then rotated, a method he calls "one of the most practical ways to form the knot". *The Magician's Own Book* (1862) is full of neighbouring handkerchief feats but not this one, so Ashley's entry is the earliest printed source verified here. Today the puzzle circulates as a bar bet, on puzzle sites and as the magicians' napkin-knot wager. It belongs to the family of tricks Martin Gardner called "topological tomfoolery", whose secret is that some property cannot change under continuous motion. ??? tip "Hints" - Write down the rules you are obeying. The problem forbids letting go of the ends. It says nothing about the position of your arms and body before you pick the rope up. - Think of your arms, your body and the rope as one closed loop. Once you are holding both ends, can any amount of wriggling change whether that loop is knotted? - If the closed loop cannot become knotted after you grip the ends, then it must already be knotted before you grip them. Where could a knot be hiding if the rope is straight? - Try it in reverse: tie an overhand knot, hold the two ends, and see what posture you end up in if you work the knot out of the rope without ever letting go. ??? success "Resolution" Fold your arms first. Keeping them crossed, bend down and pick up one end of the rope in each hand. Now uncross your arms without letting go. As your arms pass back to the ordinary position, the crossing they carried is transferred into the rope, and an overhand knot appears in its middle. ![Two-panel schematic: a person with arms folded picks up the two ends of a straight rope; after uncrossing the arms an overhand knot appears in the rope](https://tyson-swetnam.github.io/aosd/assets/images/problems/tying-knots.svg){ width="560" } *The knot is stored in your arms before the rope is touched. Drawn for this site (CC BY 4.0).* Why nothing else works: from the moment both ends are held, your arms, your torso and the rope form a single closed loop. Moving about without releasing the rope deforms that loop continuously and can never pass one part of it through another, so its knot type cannot change. An overhand knot in the rope, with the ends joined through your body, is a trefoil, a genuine knot; a straight rope held with uncrossed arms closes up into an unknotted loop. You cannot get from the second to the first, so you must start with the closed loop already a trefoil, which is exactly what folding your arms does. The "impossible" puzzle was impossible only under the unstated assumption about how you begin. ## Sources - **Clifford W. Ashley**, *The Ashley Book of Knots* (Doubleday, 1944), entry #2576 in "Tricks and Puzzles" β€” [archive.org scan](https://archive.org/details/TheAshleyBookOfKnots){target=_blank} πŸ”“ - **Wikipedia**, "The Ashley Book of Knots" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/The_Ashley_Book_of_Knots){target=_blank} πŸ”“ - **Wikipedia**, "Overhand knot" β€” an overhand knot becomes a trefoil knot when its ends are joined β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Overhand_knot){target=_blank} πŸ”“ - **George Arnold and Frank Cahill**, *The Magician's Own Book, or The Whole Art of Conjuring* (Dick & Fitzgerald, 1862) β€” [Project Gutenberg #60687](https://www.gutenberg.org/ebooks/60687){target=_blank} πŸ”“ - **Martin Gardner**, *Mathematics, Magic and Mystery* (Dover, 1956), "Topological Tomfoolery" β€” [Google Books](https://books.google.com/books/about/Mathematics_Magic_and_Mystery.html?id=-kOFBQAAQBAJ){target=_blank} πŸ”’ - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas* (5th ed. 2019; 1st ed. 1974), Chapter 3, "Emotional Blocks" β€” the session's assigned reading β€” [publisher page](https://www.hachette.co.uk/titles/james-l-adams/conceptual-blockbusting-fifth-edition/9781541674042/){target=_blank} πŸ”’ - **Braingle**, "Knot the Rope" (Brain Teaser #896) β€” [braingle.com](https://www.braingle.com/brainteasers/896/knot-the-rope.html){target=_blank} πŸ”“ - **Puzzle Prime**, "Tie a Knot" (2018) β€” [puzzleprime.com](https://www.puzzleprime.com/puzzles/brain-teasers/practical/tie-a-knot/){target=_blank} πŸ”“ - **Chris Welsh (Mr Magician)**, "Knot in Handkerchief Trick" β€” [mrmagician.co.uk](https://www.mrmagician.co.uk/knotinhandkerchief.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001), session 8 β€” the only Winfree record of the exercise β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives four words and nothing more: "About assumptions and Tying Knots". It is the only Winfree document that mentions the exercise; no page of his archived lab site names a knot puzzle. The identification below is therefore an inference from the wording, the session and his tastes, not a documented fact. Candidate readings: - The crossed-arms rope puzzle, Ashley #2576 (high confidence; the reading used above). It is literally about tying a knot, its whole point is an unstated assumption, it takes two minutes with a shoelace, and its explanation is topological, which was Winfree's own research field. - A short talk, demonstrated with string, on why a closed loop cannot be knotted or unknotted without cutting: an "assumption" that is really a theorem (low confidence). The entry reads "About assumptions and ..." where sibling entries say "Discuss ...", which leaves room for a demonstration; but no source attests a lecture, and demonstrating the crossed-arms trick would satisfy this reading too. - Maier's two-string problem, in which two strings hanging from a ceiling must be tied together (low confidence; a standard creativity-block demonstration, but its point is tying strings to each other rather than tying a knot, and it needs a rigged ceiling). The statement above is paraphrased from Ashley #2576 and from the modern puzzle-site versions; only the labelled phrases are quoted from Ashley, whose 1944 book is still in copyright. The topological explanation is stated in the editors' own words. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/mercurys-hidden-hemisphere/ --- title: "Mercury's Mysterious Hidden Hemisphere" description: "For 75 years every textbook said Mercury keeps one face to the Sun and hides a frozen hemisphere no one could ever see; radar in 1965 showed the planet turns three times for every two orbits." type: Activity tags: [course, student-facing, problem, section-2, history-of-science, astronomy, cultural-blocks, observation] status: stable problem: section: 2 session: 9 identification: probable kind: case-study generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: pettengill-dyce-1965 resource: "https://doi.org/10.1038/2061240a0" title: "A Radar Determination of the Rotation of the Planet Mercury" author: "G. H. Pettengill and R. B. Dyce" - id: peale-gold-1965 resource: "https://doi.org/10.1038/2061240b0" title: "Rotation of the Planet Mercury" author: "S. J. Peale and T. Gold" - id: colombo-1965 resource: "https://doi.org/10.1038/208575a0" title: "Rotational Period of the Planet Mercury" author: "G. Colombo" - id: colombo-shapiro-1966 resource: "https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1966ApJ...145..296C&defaultprint=YES&filetype=.pdf" title: "The Rotation of the Planet Mercury" author: "G. Colombo and I. I. Shapiro" - id: goldreich-peale-1966 resource: "https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1966AJ.....71..425G&defaultprint=YES&filetype=.pdf" title: "Spin-orbit coupling in the solar system" author: "P. Goldreich and S. J. Peale" - id: dyce-pettengill-shapiro-1967 resource: "https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1967AJ.....72..351D&defaultprint=YES&filetype=.pdf" title: "Radar determination of the rotations of Venus and Mercury" author: "R. B. Dyce, G. H. Pettengill and I. I. Shapiro" - id: howard-barrett-haddock-1962 resource: "https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1962ApJ...136..995H&defaultprint=YES&filetype=.pdf" title: "Measurement of Microwave Radiation from the Planet Mercury" author: "W. E. Howard, A. H. Barrett and F. T. Haddock" - id: de-piccoli-carpino-2025 resource: "http://www.fedoabooks.unina.it/index.php/fedoapress/catalog/download/669/727/3372?inline=1" title: "Friendly Stilbon, fraudful Hermes. Schiaparelli and the rotation of Mercury" author: "Lorenzo De Piccoli and Mario Carpino" - id: krumenaker-1976 resource: "https://www.hermograph.com/science/cartogrf.htm" title: "The Planet Mercury: Drawing the Right Surface Maps of Mercury" author: "Larry Krumenaker" - id: wikipedia-mercury resource: "https://en.wikipedia.org/wiki/Mercury_(planet)" title: "Mercury (planet)" author: "Wikipedia contributors" - id: nasa-mariner-10-45-years resource: "https://www.nasa.gov/history/45-years-ago-mariner-10-first-to-explore-mercury/" title: "45 Years Ago: Mariner 10 First to Explore Mercury" author: "NASA History Office" - id: wikipedia-mariner-10 resource: "https://en.wikipedia.org/wiki/Mariner_10" title: "Mariner 10" author: "Wikipedia contributors" - id: sciencedaily-messenger-2008 resource: "https://www.sciencedaily.com/releases/2008/10/081030091153.htm" title: "More Hidden Territory On Mercury Revealed By MESSENGER Spacecraft" author: "ScienceDaily (source credited: NASA)" - id: lasp-messenger-2008 resource: "https://lasp.colorado.edu/2008/10/29/messenger-reveals-hidden-territory-on-mercury/" title: "MESSENGER reveals more hidden territory on Mercury" author: "Laboratory for Atmospheric and Space Physics, University of Colorado" - id: solarviews-mercury resource: "https://solarviews.com/eng/mercury.htm" title: "Mercury" author: "Calvin J. Hamilton, Views of the Solar System" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu:80/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery: course handout (EEB 479/479H/579), archived" author: "A. T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Mercury's Mysterious Hidden Hemisphere Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 9. The syllabus pairs the discussion with Adams Chapter 4 on cultural blocks and Platt's Diversity.* !!! abstract "The problem" Reconstructed from the syllabus; Winfree's own wording is not recorded. Mercury never appears more than about 28 degrees from the Sun, so it is seen low in a twilight sky, through turbulent air, its markings at the limit of vision. In the 1880s Giovanni Schiaparelli, among the finest planetary observers alive, concluded from those markings that Mercury turns once per 88-day orbit, keeping one face to the Sun as the Moon keeps one face to us: one hemisphere scorched, the other frozen, a hidden hemisphere no telescope could ever see. Tidal friction explained it, Antoniadi's 1934 map confirmed it, and every textbook repeated it for seventy-five years. It was wrong. Before reading **What happened**: what does a visual observer actually record, and how often? Mercury's *synodic* period, the interval between apparitions of the same kind, is about 116 days β€” is 88 days the only rotation period that would show such observers the same markings every time? And what measurement, not using the eye at all, could settle it? ![Three panels. Left: the old 1:1 picture, four positions on a circular orbit with the same face always turned to the Sun. Right: the true 3:2 resonance, six positions on one eccentric orbit with a marker that turns 90 degrees for every 60 degrees of orbit. Middle: the same marker at three successive perihelion passages, facing the Sun, then away, then toward it again. Bottom: a bar chart to scale of rotation (58.6 d), synodic period (115.9 d) and two rotations (117.3 d), the last two almost equal.](https://tyson-swetnam.github.io/aosd/assets/images/problems/mercurys-hidden-hemisphere.svg){ width="560" } *Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 9 pairs Adams on cultural blocks with Platt's essay on diversity. Mercury is a ready-made specimen: a plausible result from an authoritative observer, which a whole field then believed for seventy-five years. The correction came from outside, from radar astronomers asking a different question with a different instrument β€” Platt's argument for diversity made concrete. Visual astronomy could not correct itself, because its observing schedule was aliased to the very thing it measured. ## Where it comes from Schiaparelli, observing Mercury in daylight from Brera from 1881, found the markings unchanged over hours and announced the 88-day period in 1889. One system of spots had struck him as reappearing in nearly the same place at six eastern elongations in 1882–83. Antoniadi, at Meudon in 1924–29, named Mercury's dark and bright areas and, in Larry Krumenaker's phrase, "firmly established (as he thought)" the period. Features that later vanished or moved were put down to libration, poor seeing or luminescence; the result, Krumenaker writes, "seemed quite secure, both observationally and theoretically". ![Schiaparelli's 1889 map of Mercury: a circular projection ruled with meridians and marked SUD, NORD, EST and OUEST, its dusky streaks lettered, drawn on the assumption that the planet keeps one face to the Sun](https://tyson-swetnam.github.io/aosd/assets/images/problems/mercurys-hidden-hemisphere-schiaparelli.jpg){ width="560" } *Giovanni Schiaparelli, map of Mercury, 1889. Public domain, via Wikimedia Commons (from NASA SP-423).* ??? tip "Hints" - A visual observer gets a few sketches per apparition, at intervals set by the geometry of Earth and Mercury, not by Mercury's rotation. Which rotation periods return the same face to the sketch-pad every time? - Write the rotation period as a fraction of the 116-day synodic period. 88 days is not the only simple answer. - Radar returns a Doppler-shifted echo: the approaching limb shifts it up in frequency, the receding limb down. What does the *width* of that spectrum tell you? ## What happened **The period is 58.65 days, not 88.** In June 1965 Pettengill and Dyce, using the new 1000-foot Arecibo dish, measured the Doppler broadening of radar echoes and got 59 Β± 5 days. In the same issue of *Nature*, Peale and Gold explained why a planet on an eccentric orbit need not lock 1:1; Colombo then pointed out that 58.65 days is exactly two-thirds of the 88-day orbit, and the dynamics of this 3:2 spin-orbit resonance followed. There is no permanently dark hemisphere, and a solar day on Mercury lasts 176 Earth days. **Why the eye was fooled.** Twice the rotation period, 117.3 days, is within a day of the 115.9-day synodic period, so at each apparition of the same kind Mercury turns nearly the same face to Earth β€” and observers worked mainly at the favourable apparitions. The drawings, Dyce, Pettengill and Shapiro noted in 1967, had been made at intervals of very nearly even multiples of Mercury's orbital period, and so could not tell 88 days from 59. **The blocks.** Perceptual: markings at the threshold of vision, read through a published map. Cultural: a great observer's authority, the Moon analogy, seventy-five years of textbooks. Emotional: microwave measurements in 1962, read on the assumption that the night side was cold, implied a temperature under the Sun of about 1100 K where sunlight can supply about 633 K β€” the signature of a night side that is not cold. Howard, Barrett and Haddock reached instead for radioactive heating or an insulating dust layer, and after 1965 others reached for heat-carrying winds. The 88-day period went unquestioned. **Postscript.** Mariner 10's 176-day orbit equalled two Mercury years and three Mercury rotations, so its three flybys of 1974–75 all saw the same sunlit face and mapped only 40–45 percent of the surface. When Winfree taught this course more than half the planet had still never been photographed; MESSENGER's two flybys in 2008 raised coverage to about 95 percent. ## Sources - **G. H. Pettengill and R. B. Dyce**, "A Radar Determination of the Rotation of the Planet Mercury", *Nature* 206, 1240 (19 June 1965) β€” [doi:10.1038/2061240a0](https://doi.org/10.1038/2061240a0){target=_blank} πŸ”’ - **S. J. Peale and T. Gold**, "Rotation of the Planet Mercury", *Nature* 206, 1240–1241 (1965) β€” [doi:10.1038/2061240b0](https://doi.org/10.1038/2061240b0){target=_blank} πŸ”’ - **G. Colombo**, "Rotational Period of the Planet Mercury", *Nature* 208, 575 (6 November 1965) β€” [doi:10.1038/208575a0](https://doi.org/10.1038/208575a0){target=_blank} πŸ”’ - **G. Colombo and I. I. Shapiro**, "The Rotation of the Planet Mercury", *Astrophysical Journal* 145, 296 (1966), doi:10.1086/148762 β€” [free scan, NASA ADS](https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1966ApJ...145..296C&defaultprint=YES&filetype=.pdf){target=_blank} πŸ”“ - **P. Goldreich and S. J. Peale**, "Spin-orbit coupling in the solar system", *Astronomical Journal* 71, 425 (1966), doi:10.1086/109947 β€” [free scan, NASA ADS](https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1966AJ.....71..425G&defaultprint=YES&filetype=.pdf){target=_blank} πŸ”“ - **R. B. Dyce, G. H. Pettengill and I. I. Shapiro**, "Radar determination of the rotations of Venus and Mercury", *Astronomical Journal* 72, 351 (1967), doi:10.1086/110231 β€” [free scan, NASA ADS](https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1967AJ.....72..351D&defaultprint=YES&filetype=.pdf){target=_blank} πŸ”“ - **W. E. Howard, A. H. Barrett and F. T. Haddock**, "Measurement of Microwave Radiation from the Planet Mercury", *Astrophysical Journal* 136, 995 (1962), doi:10.1086/147451 β€” [free scan, NASA ADS](https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1962ApJ...136..995H&defaultprint=YES&filetype=.pdf){target=_blank} πŸ”“ - **Lorenzo De Piccoli and Mario Carpino**, "Friendly Stilbon, fraudful Hermes. Schiaparelli and the rotation of Mercury", *Atti del XLIV Congresso Nazionale SISFA* (2025), 199–206 β€” [open-access PDF](http://www.fedoabooks.unina.it/index.php/fedoapress/catalog/download/669/727/3372?inline=1){target=_blank} πŸ”“ - **Larry Krumenaker**, "The Planet Mercury: Drawing the Right Surface Maps of Mercury", from *A Computer Analysis of Visual Observations of Mercury* (MS thesis, Case Western Reserve University, 1976) β€” [hermograph.com](https://www.hermograph.com/science/cartogrf.htm){target=_blank} πŸ”“ - **Wikipedia contributors**, "Mercury (planet)" (overview) β€” [Wikipedia](https://en.wikipedia.org/wiki/Mercury_%28planet%29){target=_blank} πŸ”“ - **NASA History Office**, "45 Years Ago: Mariner 10 First to Explore Mercury" (2019) β€” [nasa.gov](https://www.nasa.gov/history/45-years-ago-mariner-10-first-to-explore-mercury/){target=_blank} πŸ”“ - **Wikipedia contributors**, "Mariner 10" (overview) β€” [Wikipedia](https://en.wikipedia.org/wiki/Mariner_10){target=_blank} πŸ”“ - **ScienceDaily** (source credited: NASA), "More Hidden Territory On Mercury Revealed By MESSENGER Spacecraft" (31 October 2008) β€” [ScienceDaily](https://www.sciencedaily.com/releases/2008/10/081030091153.htm){target=_blank} πŸ”“ - **Laboratory for Atmospheric and Space Physics, University of Colorado**, "MESSENGER reveals more hidden territory on Mercury" (2008) β€” [LASP](https://lasp.colorado.edu/2008/10/29/messenger-reveals-hidden-territory-on-mercury/){target=_blank} πŸ”“ - **Calvin J. Hamilton**, "Mercury", *Views of the Solar System* β€” [solarviews.com](https://solarviews.com/eng/mercury.htm){target=_blank} πŸ”“ - **A. T. Winfree**, *The Art of Scientific Discovery: course handout (EEB 479/479H/579)*, archived 2002 β€” the session-9 line is the only mention of Mercury in it β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu:80/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives the title, the session and its readings, and nothing more. "Mercury" occurs once in Winfree's archived course handout β€” in that one line β€” and nothing on the planet survives anywhere else in his archived pages and columns. Identification is therefore **probable**. The candidates: - **The 1889–1965 rotation episode** (high confidence): it matches every word of the title, it is a textbook cultural block, and it needs only elementary arithmetic. - **Mariner 10's unimaged hemisphere** (medium): also literally hidden, and still hidden in 2001, but that gap is an engineering consequence rather than a block. - **A geometrical puzzle** about what Earth can see of a synchronously rotating planet (low): nothing supports it over the historical reading. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/ --- title: "Paths Through Mazes" description: "A path-counting puzzle: in how many ways can ABRACADABRA be read down a diamond of letters, and can you check the count a second way?" type: Activity tags: [course, student-facing, problem, section-2, recreational-mathematics, combinatorics, pascal-triangle, problem-solving-strategy] status: stable problem: section: 2 session: 10 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: polya-1962 resource: "https://archive.org/details/mathematicaldisc0000geor" title: "Mathematical Discovery, vol. 1, ch. 3 \"Recursion\", sect. 3.5 \"Abracadabra\" and 3.6 \"The Pascal triangle\"" author: "George Polya" - id: dudeney-1907 resource: "https://www.gutenberg.org/ebooks/27635" title: "The Canterbury Puzzles, and Other Curious Problems, no. 38 \"The Amulet\"" author: "Henry Ernest Dudeney" - id: neuburger-1910 resource: "https://archive.org/details/historyofmedicin01neub" title: "History of Medicine (trans. Ernest Playfair)" author: "Max Neuburger" - id: adams-blockbusting resource: "https://www.hachette.co.uk/titles/james-l-adams/conceptual-blockbusting-fifth-edition/9781541674042/" title: "Conceptual Blockbusting: A Guide to Better Ideas (Ch. 5, \"Intellectual and Expressive Blocks\")" author: "James L. Adams" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Paths Through Mazes Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 10. Discussed together with [Taboo Questions](https://tyson-swetnam.github.io/aosd/problems/taboo-questions/index.md).* !!! abstract "The problem" Reconstruction, based on the syllabus title and a bookmark in Winfree's handout: his own statement is lost (see the note below). This diamond follows George Polya (1962); the array as typed and the wording are the editors'. ``` A B B R R R A A A A C C C C C A A A A A A D D D D D A A A A B B B R R A ``` Start at the top A. Read downwards, each time stepping to one of the two letters just below, left or right (from the widest row down, a letter on the edge has only one), ending at the bottom A. 1. **In how many different ways can the word be read?** 2. Treat the diamond as a map of streets, each letter a corner and each step one block. What does your count say about routes? 3. Find the answer a second, independent way. 4. How many shortest routes cross a grid *m* blocks by *n* blocks, corner to opposite corner? In your GamesWorth book, record your first strategy and why you dropped it. ![A diamond of ABRACADABRA letters joined by faint diagonal lines, with one zigzag reading from top A to bottom A highlighted in red](https://tyson-swetnam.github.io/aosd/assets/images/problems/paths-through-mazes.svg){ width="560" } *One reading of the word. Drawn for this site (CC BY 4.0).* ## Why it is in the course The syllabus sets this beside [Taboo Questions](https://tyson-swetnam.github.io/aosd/problems/taboo-questions/index.md) in session 10, "Adams Chapter 5: Intellectual blocks". Adams's chapter is about tackling a problem in the wrong "language" and sticking with an inadequate strategy. The natural first attack is to trace routes with a pencil. The routes pile up, and the count never settles. The block is treating the puzzle as a picture to trace instead of a structure to count. Polya's way out: restate it as shortest paths through a street grid, solve simpler cases, and find the law that links them. Winfree's instruction applies: "Write down your approaches, your lucky insights, how you got into and out of blind alleys. This is the main thing, not the 'answers'." Why he called it "mazes" is not known. ## Where it comes from ABRACADABRA began as a charm. A Roman medical poem attributed to Quintus Serenus Samonicus (third century AD) prescribes it against fever, written in a shrinking cone of letters (Neuburger, 1910). The puzzle form is older than Polya. H. E. Dudeney's "The Amulet" (*The Canterbury Puzzles*, 1907) asks how many ways the word can be read down an 11-row triangle. Polya's *Mathematical Discovery*, vol. 1 (1962), section 3.5 "Abracadabra", puts the word in a diamond and recasts it as counting shortest zigzag paths through city blocks; section 3.6 names the resulting numbers the Pascal triangle. ??? tip "Hints" - Don't trace every route. How many ways reach one letter a row or two down? - Each letter is entered only from the letters directly above it. How does its count depend on theirs? - Write the counts row by row. Have you seen the top half before? - For a check: every reading is ten steps, each down-left or down-right. How many of each? ??? success "Resolution" **The count.** Label each letter with the number of ways to reach it. From the top down to the widest row, edge letters have only one letter above them, so they count 1. Every other letter gets the sum of the two letters above it: ``` A 1 B 1 1 R 1 2 1 A 1 3 3 1 C 1 4 6 4 1 A 1 5 10 10 5 1 D 6 15 20 15 6 A 21 35 35 21 B 56 70 56 R 126 126 A 252 ``` The word can be read in **252** ways, Polya's answer. **As streets.** Each reading is a shortest route, ten blocks long, between opposite corners of a 5-by-5 grid of blocks stood on one corner, so 5 blocks go down-left and 5 down-right. The addition rule holds because every shortest route to a corner passes through exactly one of the two corners just above it. **Second method.** A route is fixed by choosing which 5 of its 10 steps go down-left: C(10,5) = 10!/(5!Β·5!) = 252. The two methods agree. **Generalization.** For a grid *m* blocks by *n* blocks, the number of shortest corner-to-corner routes is C(m+n, m) = (m+n)!/(m!Β·n!). **The triangle variant.** In Dudeney's 11-row triangle a reading may end on any A in the bottom row. Every letter above the bottom row has two letters below it, so each step doubles the count: 2^10 = **1024**. The bottom row of the Pascal triangle sums to the same total. Same word, same rule, different shape, different answer: the array is part of the problem. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* course handout (archived 2002) β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **George Polya**, *Mathematical Discovery*, vol. 1, sections 3.5-3.6, pp. 68-70 (1962) β€” [Internet Archive](https://archive.org/details/mathematicaldisc0000geor){target=_blank} πŸ”“ *(borrow)* - **Henry Ernest Dudeney**, *The Canterbury Puzzles*, no. 38 "The Amulet" (1907; 1919 edition online) β€” [Project Gutenberg](https://www.gutenberg.org/ebooks/27635){target=_blank} πŸ”“ - **Max Neuburger**, *History of Medicine*, vol. 1, trans. Ernest Playfair (1910) β€” [Internet Archive](https://archive.org/details/historyofmedicin01neub){target=_blank} πŸ”“ - **James L. Adams**, *Conceptual Blockbusting*, Chapter 5 (1974; 5th ed. 2019) β€” [publisher](https://www.hachette.co.uk/titles/james-l-adams/conceptual-blockbusting-fifth-edition/9781541674042/){target=_blank} πŸ”’ - **Arthur T. Winfree**, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the title. In Winfree's archived course handout (2002 capture, listed in Sources), the link on this item points to a bookmark named `abracadabra`. The bookmark's target is not in the capture, and no companion document has been found, so no Winfree wording survives. From the name, the editors infer the ABRACADABRA path-counting puzzle. That fixes the kind of puzzle, not his array or intended answer, so identification is probable. Candidates: - **Polya's diamond**, reconstructed above. It is literally about paths through streets, and Winfree's syllabus puts two other Polya books, *Induction and Analogy in Mathematics* and *Patterns of Plausible Inference*, on reserve. Leading with it is an editorial judgement. - **Dudeney's triangle**, the older version. The bookmark fits it equally well, and it gives a different answer. - **An exercise from Adams's Chapter 5.** A full-text search found no sign of the puzzle there, but the chapter itself could not be read, so this is not ruled out. Before the bookmark was found, the editors read the title literally, as a maze-threading exercise. The bookmark points to counting paths instead, so that reading has been dropped. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/taboo-questions/ --- title: "Taboo Questions" description: "The session-10 discussion of the questions a scientist is not supposed to ask, reconstructed from the syllabus, with James Adams's ping-pong-ball-in-a-pipe exercise as the documented warm-up." type: Activity tags: [course, student-facing, problem, section-2, conceptual-blocks, cultural-blocks, questioning] status: stable problem: section: 2 session: 10 identification: probable kind: discussion generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: adams-blockbusting-outline-2001 resource: "https://web.archive.org/web/20220429044046/https://jamesladams.typepad.com/blog/conceptual-blocks-from-conceptual-blockbusting.html" title: "Conceptual Blockbusting: A Guide to Better Ideas, 4th edition (outline by the author)" author: "James L. Adams" - id: adams-blockbusting-1979 resource: "https://archive.org/details/conceptualblockb00adam_4" title: "Conceptual Blockbusting: A Guide to Better Ideas, 2nd edition" author: "James L. Adams" - id: adams-blockbusting-1986 resource: "https://archive.org/details/conceptualblockb00jame" title: "Conceptual Blockbusting: A Guide to Better Ideas, 3rd edition" author: "James L. Adams" - id: adams-blockbusting-2019 resource: "https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674042/" title: "Conceptual Blockbusting: A Guide to Better Ideas, 5th edition (publisher page)" author: "James L. Adams" - id: halpern-thought-and-knowledge-1984 resource: "https://archive.org/details/thoughtknowledge0000halp" title: "Thought and Knowledge: An Introduction to Critical Thinking" author: "Diane F. Halpern" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Taboo Questions Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 10, with Adams Chapter 5 on intellectual blocks. Paired in the schedule with [Paths Through Mazes](https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/index.md).* !!! abstract "The problem" The syllabus names this item but gives no problem sheet for it, so the page is in two parts. Part 1 reconstructs the discussion. Part 2 is the documented exercise behind it, from the book the class had just been reading. **Part 1. Taboo questions (reconstruction).** A taboo, in Adams's sense, is a prohibition that removes a whole family of solutions from consideration before you have looked at them. The same thing happens to *questions*. In your own field, laboratory or classroom, write down three questions you have wanted to ask but felt you were not supposed to: one that sounds stupid, one that doubts something everyone accepts, one that is simply unfashionable. For each, record who exactly would be displeased, what the prohibition protects, and what you would learn if the usual answer turned out to be wrong. Then pick the one that is most nearly askable and ask it, in class or by email, before the next session. **Part 2. The ball in the pipe (James L. Adams's exercise, not Winfree's).** This is Adams's own exercise from *Conceptual Blockbusting*, Chapter 4, the chapter assigned for the previous session; it is the passage where he introduces taboos. It is paraphrased here, with his list of objects kept exactly. A steel pipe is imbedded in the concrete floor of a bare room. Its inside diameter is only a little larger than the ping-pong ball resting at the bottom. You are one of a group of six people in the room, along with the following objects: - 100 feet of clothesline - a carpenter's hammer - a chisel - a box of Wheaties - a file - a wire coat hanger - a monkey wrench - a light bulb In five minutes, list as many ways as you can think of to get the ball out of the pipe without damaging the ball, tube, or floor. Write them all in your GamesWorth book, including the ones you would rather not say aloud. Then mark which you would actually have proposed to the group and which you would have kept to yourself, and why β€” that is the bridge back to Part 1. ![Cross-section of a steel pipe imbedded in a concrete floor, closed at the bottom, with a ping-pong ball resting at its bottom, beside icons of the eight available objects: clothesline, hammer, chisel, box of Wheaties, file, coat hanger, monkey wrench and light bulb](https://tyson-swetnam.github.io/aosd/assets/images/problems/taboo-questions.svg){ width="560" } *The set-up. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 2 works through creative blocks chapter by chapter with Adams. Session 9 read Chapter 4 on cultural blocks, whose first entry is Taboos. Session 10 reads Chapter 5 on intellectual blocks, and the schedule pairs Taboo Questions with Paths Through Mazes. The two sit well together: a maze's whole difficulty is that some paths are closed, and a taboo closes paths in the mind before you have tried them. Adams's exercise catches a block in the act. Almost everyone finds a mechanical solution, and almost everyone silently discards the one that works best. That is what the syllabus says the puzzles are for: "to slow you down for a few minutes so you can examine the working of your own mind." Widening taboo solutions into taboo questions carries the lesson into research, where the course rehearses "recognizing ignorance, of posing questions". The point is not to break taboos, as Adams is careful to say, but to notice when one has quietly narrowed your alternatives. ## Where it comes from James L. Adams, a Stanford professor of mechanical engineering, first published *Conceptual Blockbusting* in 1974; Winfree's students used the fourth edition (2001). The book sorts the habits that stop people having ideas into perceptual, emotional, cultural and environmental, and intellectual and expressive blocks, and the chapter on cultural blocks opens with Taboos. Adams says he has used the ping-pong ball exercise with many groups, and concludes that "cultural taboos can remove entire families of solutions from the ready grasp of the problem-solver. Taboos therefore are conceptual blocks", adding that this "is not a tirade against taboos". The exercise travelled into later textbooks, among them Diane Halpern's *Thought and Knowledge* (1984). The syllabus returns to the theme two sessions later with Freeman Dyson's *Unfashionable Pursuits* and a one-page biography of Mario Capecchi, whose gene-targeting proposal was initially judged not worth funding. ??? tip "Hints" - A taboo is easier to see from outside a culture. Explain your field's forbidden question to someone in another department and watch whether they find it forbidden too. - A taboo can hide a solution, but it can also be protecting something. Write down what each of your questions might cost as well as what it might gain. - For the pipe: count your solutions after five minutes, then ask which of the eight objects you never touched. That family of ideas may be the one you avoid. - The statement says the pipe is imbedded in a floor and that six people are present. Why six? In a contrived puzzle every detail is there for a reason. - List what a ping-pong ball does that a steel ball of the same size does not. What in the room could supply it? ??? success "Resolution" **Taboo questions.** There is no answer key, and no record of what Winfree's class produced. You are done when you have written down a question you were not supposed to ask, named whose displeasure enforces the prohibition, and asked it anyway. Session 12's readings, Dyson on unfashionable pursuits and Capecchi's unfundable proposal, are the historical cases of such questions paying off. **The ball in the pipe.** The ping-pong ball floats. The solution Adams's exercise is built around is for the six people to urinate into the pipe until the ball rises. Most groups never say it aloud, which is the point; Adams observes that urinating is "somewhat of a closet activity in the U.S." The usual solutions are mechanical: file the wire coat hanger in two, flatten the ends and use the pieces as tongs, or smash the hammer handle and lift the ball with the splinters. Crushing the Wheaties and pouring them in works too; anyone who found a use for the Wheaties, Adams remarks, is "an even more flexible thinker". ## Sources - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, 4th edition (Basic Books, 2001), chapter-by-chapter outline by the author β€” [Wayback Machine](https://web.archive.org/web/20220429044046/https://jamesladams.typepad.com/blog/conceptual-blocks-from-conceptual-blockbusting.html){target=_blank} πŸ”“ - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, 2nd edition (W. W. Norton, 1979) β€” [Internet Archive](https://archive.org/details/conceptualblockb00adam_4){target=_blank} πŸ”“ *(borrow)* - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, 3rd edition (Addison-Wesley, 1986) β€” [Internet Archive](https://archive.org/details/conceptualblockb00jame){target=_blank} πŸ”“ *(borrow)* - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, 5th edition (Basic Books, 2019), publisher page β€” [Hachette Book Group](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674042/){target=_blank} πŸ”’ - **Diane F. Halpern**, *Thought and Knowledge: An Introduction to Critical Thinking* (Lawrence Erlbaum, 1984) β€” [Internet Archive](https://archive.org/details/thoughtknowledge0000halp){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The identification is **probable**. The syllabus gives the name, the session and the neighbour: session 10 reads Adams Chapter 5 on intellectual blocks and lists Taboo Questions beside Paths Through Mazes. No statement, handout or archived page of Winfree's for this item has been found, so Part 1 above is a reconstruction. The argument for it is a word match. In Adams's own outline of the 2001 edition, Taboos is the first cultural block, and the ball-in-the-pipe exercise is its illustration. That chapter was the previous session's reading. Winfree's wording is "questions", not "taboos", so the widening is his, not Adams's. Against the match: Adams's outline lists no taboo anywhere under intellectual and expressive blocks, the chapter actually assigned that day. The candidates: - **Adams's taboos, widened to taboo questions** (medium confidence): the discussion above, taken up a session after the cultural-blocks chapter. - **A Winfree warm-up or GamesWorth prompt** (medium confidence): students list questions they feel forbidden to ask in their own field. Nothing of the sort is archived. - **An item from Adams 5 itself** (low confidence): questions a field cannot ask for want of the right language, strategy or information. This matches the day's reading, but the word "taboo" appears nowhere in that chapter's outline. - **A xeroxed article** (low confidence): the syllabus mentions "a lot of xeroxed handouts, not listed here, many from current periodicals". Nothing identifies which, so this cannot be reconstructed. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/walking-through-walls/ --- title: "Walking Through Walls" description: "Ask for a better door and you get a hinged slab; ask for a better way to get through a wall and the answers change - a drill in restating the problem, from Adams's Conceptual Blockbusting." type: Activity tags: [course, student-facing, problem, section-2, blockbusters, problem-framing, hidden-assumptions, design] status: stable problem: section: 2 session: 11 identification: probable kind: thought-experiment generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: adams-blockbusting-2nd-googlebooks resource: "https://books.google.com/books?id=tYyBl80D5GsC&q=%22through+a+wall%22" title: "Conceptual Blockbusting: A Guide to Better Ideas, 2nd edition (Google Books snippet view; the door/wall passage is at pp. 31-32)" author: "James L. Adams" - id: adams-blockbusting-2nd-archive resource: "https://archive.org/details/conceptualblockb0000adam" title: "Conceptual Blockbusting, 2nd edition (Internet Archive controlled-lending copy)" author: "James L. Adams" - id: adams-blockbusting-archive-2001 resource: "https://archive.org/details/conceptualblockb00adam_2" title: "Conceptual Blockbusting, 4th edition, 2001 Perseus printing (Internet Archive controlled-lending copy)" author: "James L. Adams" - id: adams-blockbusting-5th-googlebooks resource: "https://books.google.com/books?id=DMaCDwAAQBAJ&q=%22through+a+wall%22" title: "Conceptual Blockbusting: A Guide to Better Ideas, 5th edition (Google Books preview)" author: "James L. Adams" - id: adams-blockbusting-publisher resource: "https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674042/" title: "Conceptual Blockbusting (publisher page, Basic Books / Hachette)" author: "James L. Adams" - id: coon-intro-psychology-1998 resource: "https://books.google.com/books?id=Gjrj9k_zzkEC&q=%22better+way+to+get+through+a+wall%22" title: "Introduction to Psychology: Exploration and Application, 8th edition" author: "Dennis Coon" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (EEB 479/479H/579): course handout (the syllabus), Internet Archive capture of 20 April 2002" author: "Arthur T. Winfree" - id: wikipedia-passe-muraille resource: "https://en.wikipedia.org/wiki/Le_Passe-muraille" title: "Le Passe-muraille" author: "Wikipedia contributors" - id: wikipedia-revolving-door resource: "https://en.wikipedia.org/wiki/Revolving_door" title: "Revolving door" author: "Wikipedia contributors" - id: van-kannel-patent-1888 resource: "https://patents.google.com/patent/US387571A/en" title: "US Patent 387,571, Storm-door structure (the revolving door)" author: "Theophilus Van Kannel" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Walking Through Walls Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 11, the only problem the syllabus lists that day. The reading due is Adams's Chapter 7, Blockbusters.* !!! abstract "The problem" Reconstructed from the syllabus: the archived handout carries only the one-line schedule entry for this session, so the rounds below are rebuilt from the passage in Adams's *Conceptual Blockbusting* that the name most probably points to. Finish each round in your notebook before reading the next. **Round 1.** A client says: *design me a better door.* List or sketch as many designs as you can. **Round 2.** Cross out the word *door*. The real request turns out to be: *find me a better way to get through a wall.* What can you propose now that Round 1 did not allow? **Round 3.** Widen it once more: *find a better way to keep two spaces apart - sound, sight, heat, weather, privacy - while still letting people and things pass between them.* **Round 4.** Compare the three lists. What did the word *door* decide about shape, material, hinges and the position of the opening? A request for a better way to get through a wall, Adams writes, "releases one from the preconception of the rectangular slab that swings or slides" (*Conceptual Blockbusting*, 2nd ed., 1980, p. 31). ![Three nested boxes showing the widening problem statements - design a better door, find a better way to get through a wall, find a better way to keep two spaces apart yet let people pass - with pictograms of a hinged slab, a round hole, a curtain, a revolving door and an air curtain](https://tyson-swetnam.github.io/aosd/assets/images/problems/walking-through-walls.svg){ width="560" } *Each statement is a special case of the next one out. Drawn for this site (CC BY 4.0).* ![Patent drawing of a storm-door structure: plan and elevation of three door leaves on a central shaft inside a cylindrical enclosure](https://tyson-swetnam.github.io/aosd/assets/images/problems/walking-through-walls-storm-door.jpg){ width="560" } *One answer that is not a hinged slab. Theophilus Van Kannel, US Patent 387,571 (1888), via Wikimedia Commons, public domain.* ## Why it is in the course Session 11 is the fifth of Section 2's six sessions, and the one where the reading turns from blocks to *blockbusters*. Earlier sessions used Adams's puzzles to expose perceptual, emotional, cultural and intellectual blocks. This one drills a tool for getting past them: a questioning attitude aimed not at the answer but at the problem statement. It follows straight on from session 10, [Taboo Questions](https://tyson-swetnam.github.io/aosd/problems/taboo-questions/index.md) and [Paths Through Mazes](https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/index.md). A maze is only a maze if you accept its walls; a door is only the answer if you accept the word "door". The exercise has no answer. Its product is the difference between your three lists, and your account of what one word had silently decided. That suits a syllabus which says the puzzles exist "to slow you down for a few minutes so you can examine the working of your own mind". ## Where it comes from James L. Adams taught mechanical engineering and design at Stanford and published *Conceptual Blockbusting: A Guide to Better Ideas* in 1974; the fourth edition (2001) was the one in print when the course ran. His chapter on perceptual blocks has a section on the tendency to delimit the problem area too closely, and the door is his example. Ask for a better door and the answers stay inside the word: a hinged slab with a handle. Ask instead for a better way to get through a wall, and his students varied the opening itself - its shape, and whether the thing that closes it swings at all. Widen the request once more, from opening a wall to keeping two spaces separate, and the question becomes what the wall is for rather than what a door looks like. The example became a standard classroom drill, reproduced in Dennis Coon's *Introduction to Psychology*. The title, though, is probably Winfree's own. It echoes Marcel Ayme's 1941 story *Le Passe-muraille*, "The Man Who Walked Through Walls", though nothing links the two. ??? tip "Hints" - First write down every property a "door" has that nobody asked for: shape, material, how it moves, that it is one thing. - In Round 2, change one property of the opening at a time: shape, size, height off the floor, whether it moves or the person does. - In Round 3, ask what the wall is *for*. Cold, noise, sight and strangers are different jobs, and each suggests a different way through. - Count the ideas in each round. That count is the result, not any one design. ??? success "Resolution" There is no single answer. What you can compare your notebook against is what Adams reports. Asked for a better door, people stay inside the word: a hinged slab, better made. Asked for a better way through a wall, his students varied the opening itself - other shapes, curtains and shutters, doors that rotate or fold rather than swing. Asked for a better way to keep two spaces separate, they could reach answers that are not openings at all; Adams's example is the air curtain that keeps warm air inside a shop while people walk straight through it. At that width the statement even invites the question of whether the two spaces need walling apart. His moral is that it is foolish to constrain a problem so closely that the solver's abilities go unused, and that this holds just as much when the person stating the problem and the person solving it are the same. Every statement carries a picture of its answer inside it. ## Sources - **James L. Adams**, *Conceptual Blockbusting: A Guide to Better Ideas*, 2nd ed. (1980), pp. 31-32 - [Google Books search-inside](https://books.google.com/books?id=tYyBl80D5GsC&q=%22through+a+wall%22){target=_blank} πŸ”’ - **James L. Adams**, *Conceptual Blockbusting*, 2nd ed. (1980) - [Internet Archive](https://archive.org/details/conceptualblockb0000adam){target=_blank} πŸ”“ *(borrow)* - **James L. Adams**, *Conceptual Blockbusting*, 4th ed., 2001 Perseus printing - [Internet Archive](https://archive.org/details/conceptualblockb00adam_2){target=_blank} πŸ”“ *(borrow)* - **James L. Adams**, *Conceptual Blockbusting*, 5th ed. (2019) - [Google Books preview](https://books.google.com/books?id=DMaCDwAAQBAJ&q=%22through+a+wall%22){target=_blank} πŸ”’ - **James L. Adams**, *Conceptual Blockbusting* - [publisher page, Basic Books / Hachette](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674042/){target=_blank} πŸ”’ - **Dennis Coon**, *Introduction to Psychology: Exploration and Application*, 8th ed. (1998) - [Google Books](https://books.google.com/books?id=Gjrj9k_zzkEC&q=%22better+way+to+get+through+a+wall%22){target=_blank} πŸ”’ - **Arthur T. Winfree**, course handout (the syllabus) for EEB 479/479H/579 (2001), archived 20 April 2002 - [Internet Archive](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ Β· transcribed at [the schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks) - **Theophilus Van Kannel**, US Patent 387,571, "Storm-door structure" (1888) - [Google Patents](https://patents.google.com/patent/US387571A/en){target=_blank} πŸ”“ - **Wikipedia contributors**, "Revolving door" - [Wikipedia](https://en.wikipedia.org/wiki/Revolving_door){target=_blank} πŸ”“ - **Wikipedia contributors**, "Le Passe-muraille" - [Wikipedia](https://en.wikipedia.org/wiki/Le_Passe-muraille){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only one row for this session: Adams Chapter 7, Blockbusters, as the reading, and Walking Through Walls as the sole problem. Winfree's handout is archived, but it carries that same schedule row and no problem text, so his own wording is lost. The identification is *probable*: it rests on the match between his title and Adams's "a better way to get through a wall", on Adams being the reading assigned in the same row, and on Adams's own report that he set the restated problem to students. Against it: that passage sits in Adams's Chapter 2, read back at session 7, not the Chapter 7 assigned here, and Winfree wrote "walking" where Adams wrote "get through". The live alternative is a joke continuing session 10: having found [Paths Through Mazes](https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/index.md) under the rules, students are invited to walk through the walls instead. Nothing supports that beyond the two names sitting in consecutive sessions, but the teaching point above holds either way. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/ --- title: "Sums of Integers" description: "Find the formula for 1 + 2 + ... + n, then find it again by as many independent routes as you can: an exercise whose point is not the answer but the way separate derivations lock together like jig-saw pieces." type: Activity tags: [course, student-facing, problem, section-2, cross-checking, triangular-numbers, number-theory, recreational-mathematics] status: stable problem: section: 2 session: 12 identification: probable kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-wayback-2001 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu:80/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery, ECOL 479/579 handout and schedule (Session 12: 'Sums of integers: like a jig-saw puzzle of cross-checks')" author: "Arthur T. Winfree" - id: hayes-2006 resource: "https://www.americanscientist.org/article/gausss-day-of-reckoning" title: "Gauss's Day of Reckoning" author: "Brian Hayes" - id: sartorius-1856 resource: "https://archive.org/details/bub_gb_h_Q5AAAAcAAJ" title: "Gauss zum GedΓ€chtniss" author: "Wolfgang Sartorius von Waltershausen" - id: wikipedia-triangular-number resource: "https://en.wikipedia.org/wiki/Triangular_number" title: "Triangular number" author: "Wikipedia contributors" - id: wikipedia-squared-triangular-number resource: "https://en.wikipedia.org/wiki/Squared_triangular_number" title: "Squared triangular number (Nicomachus's theorem)" author: "Wikipedia contributors" - id: wikipedia-faulhabers-formula resource: "https://en.wikipedia.org/wiki/Faulhaber%27s_formula" title: "Faulhaber's formula" author: "Wikipedia contributors" - id: mathworld-power-sum resource: "https://mathworld.wolfram.com/PowerSum.html" title: "Power Sum" author: "Eric W. Weisstein, MathWorld" - id: wikipedia-polite-number resource: "https://en.wikipedia.org/wiki/Polite_number" title: "Polite number" author: "Wikipedia contributors" - id: guy-1982 resource: "https://www.fq.math.ca/Scanned/20-1/guy.pdf" title: "Sums of Consecutive Integers" author: "Robert Guy" - id: leveque-1950 resource: "https://doi.org/10.4153/cjm-1950-036-3" title: "On Representations as a Sum of Consecutive Integers" author: "W. J. LeVeque" - id: sylvester-1882 resource: "https://doi.org/10.2307/2369545" title: "A Constructive Theory of Partitions, Arranged in Three Acts, an Interact and an Exodion" author: "J. J. Sylvester, with insertions by F. Franklin" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Sums of Integers Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md), session 12. The syllabus sets it beside the [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md) lab.* !!! abstract "The problem" Reconstructed from the syllabus: Winfree left only the name of this problem and a gloss, "like a jig-saw puzzle of cross-checks". No handout for it has been found. What follows is the elementary exercise that name most plausibly points to, framed the way the course framed everything else: the point is not the answer but how many independent routes you can find to it, and how those routes check one another. **Part 1.** Find a formula for the sum of the first *n* positive integers, S(*n*) = 1 + 2 + 3 + ... + *n*, and convince yourself it is right. Do not stop at one argument. Find as many genuinely different ways as you can: arithmetic, pictorial, algebraic, inductive, whatever occurs to you. Treat each as a jig-saw piece and check that every piece fits every other. The same formula must come out; it must give the right values for *n* = 1, 2, 3, 4, computed by hand; and it must behave sensibly at *n* = 0. **Part 2.** Do the same for the sums of squares and of cubes, 1² + 2² + ... + *n*² and 1³ + 2³ + ... + *n*³. Before deriving anything, decide what *kind* of formula to expect (a polynomial in *n*? of what degree?) and how you would test a guess. Then look for relations between the three sums that let one formula check another. **Part 3 (extension).** Some numbers are a sum of two or more *consecutive* positive integers (9 = 4 + 5 = 2 + 3 + 4) and some are not (try 8). Which are which, and in how many ways? Make a table, guess a rule, then find at least two independent arguments for it. Keep a record in your GamesWorth book of every route you tried, including the ones that failed, and of every cross-check that caught a slip. ![Left: two identical staircases of 1+2+3+4 squares, one shaded and one pale, fitted together to fill a four-by-five rectangle. Right: the sum 1 to n written above its reversal, with every column adding to n+1](https://tyson-swetnam.github.io/aosd/assets/images/problems/sums-of-integers.svg){ width="560" } *Two of the routes, and the cross-check between them: the picture and the arithmetic must give the same formula. Drawn for this site (CC BY 4.0).* ## Why it is in the course Winfree's gloss names the lesson. A jig-saw piece is confirmed not by staring at it harder but by whether it fits its neighbours, and a derived formula is confirmed the same way: by agreeing with hand-computed small cases, with a second derivation, with a picture, and with an identity that ties it to some other formula. The syllabus asks for that habit everywhere. The final exam rewards "as many cross-checking distinct solutions as you can", and graduate write-ups must show "ways to check and cross-check every part of the solutions". Session 12 closes Section 2, "Creative Blocks", after Adams's chapters on perceptual, emotional, cultural and intellectual blocks and his chapter on blockbusters. A problem whose one-line answer almost everyone half-remembers is a good place to catch the intellectual block of stopping at the first method that works. The readings due that day push from another direction: Dyson on unfashionable pursuits and Narlikar on venture funding are both about the value of a route nobody else is taking. The exercise also fits the course's stated design. Its problems "depend as little as possible on knowledge of any particular subject area", are "mostly made from elementary mathematics so as to require no lab setup", and exist "to slow you down for a few minutes so you can examine the working of your own mind". ## Where it comes from The formula *n*(*n* + 1)/2 is ancient. The triangular numbers 1, 3, 6, 10, ... are usually traced to the Pythagoreans, and a statement of the rule appears in the *Computus* of the Irish monk Dicuil around 816; both attributions come from a secondary overview. Nicomachus of Gerasa (c. 60 – c. 120 CE) noticed that grouping the odd numbers 1, 3 + 5, 7 + 9 + 11, ... produces the cubes, which yields the identity now named after him. Johann Faulhaber (1631) computed formulas for the sums of higher powers one by one, and Jacob Bernoulli systematised them in *Ars Conjectandi* (1713), where the Bernoulli numbers first appear. Faulhaber also claimed, without proof, that such formulas exist for all odd powers. Carl Jacobi proved that claim in 1834. ![Oil portrait of an elderly Carl Friedrich Gauss in a black velvet cap, dark coat and white ruffled collar, painted in 1840](https://tyson-swetnam.github.io/aosd/assets/images/problems/sums-of-integers-gauss.jpg){ width="560" } *Christian Albrecht Jensen, portrait of Carl Friedrich Gauss (1840). Public domain, via Wikimedia Commons.* The story everyone tells about this sum repays the course's own treatment. Brian Hayes went back to the memorial volume *Gauss zum GedΓ€chtniss* (1856), written by Wolfgang Sartorius von Waltershausen the year after Gauss died, and calls it "the key document on which all subsequent accounts seem to depend". In that account the schoolmaster sets the class an arithmetic series to sum, and Gauss throws his slate on the table almost at once with the words "There it lies" (the English is Helen Worthington Gauss's translation of his Braunschweig dialect). What matters about this telling, Hayes says, "is not what's there but what's absent": no numbers 1 to 100, no pairing trick, no formula. Those arrived later, from retellers. Hayes collected "over a hundred exemplars, in eight languages", diverging in nearly every detail. A story we all know, most of which nobody recorded, is exactly the sort of thing this course asks you to notice. The Part 3 extension has its own literature: J. J. Sylvester touched it in 1882, W. J. LeVeque studied it in 1950, and Robert Guy gave a short proof of the counting rule in 1982. ??? tip "Hints" - Write the sum forwards and backwards and add the two rows term by term. Every column then holds the same number. Count the columns. - Draw 1 + 2 + ... + *n* as a staircase of dots. Two identical staircases fit together into a rectangle; what are its sides? For squares, try three copies of a suitable three-dimensional staircase, or Nicomachus's grouping of the odd numbers. - Before deriving, decide what kind of formula to expect. Tabulate the sum for *n* = 1 to 6 and take successive differences; when the differences go constant you know the degree, and a handful of small cases then fixes the polynomial. Any other derivation must reproduce exactly that polynomial. - Hunt for identities that tie the sums together: the sum of the first *n* odd numbers, two consecutive triangular numbers, the relation between the cube sum and the plain sum. Each identity is a cross-check that costs nothing. - For the extension, notice that a sum of *k* consecutive integers is *k* times its average. Ask what that says when *k* is odd, then when *k* is even, then what could go wrong for powers of two. ??? success "Resolution" **Part 1.** S(*n*) = *n*(*n* + 1)/2. Independent routes, each a check on the others: - *Pairing.* Forwards plus backwards gives *n* columns each summing to *n* + 1, so 2S = *n*(*n* + 1). This is the trick traditionally credited to the schoolboy Gauss, though not by any early source. - *Staircase.* Two dot staircases fit into an *n* by (*n* + 1) rectangle. - *Finite differences.* S(*n*) − S(*n* − 1) = *n* is linear, so S is quadratic; three values fix it. - *Telescoping.* (*k* + 1)² − *k*² = 2*k* + 1, summed from 1 to *n*, gives (*n* + 1)² − 1 = 2S + *n*. - *Induction.* Check *n* = 1, assume the formula, add *n* + 1. Checks: S(0) = 0, S(1) = 1, S(4) = 10, and S(*n*) + S(*n* − 1) = *n*², so two consecutive triangular numbers make a square. **Part 2.** The squares sum to *n*(*n* + 1)(2*n* + 1)/6 and the cubes to *n*²(*n* + 1)²/4, which is S(*n*)². - *Degree.* Third differences of the square sum are constant, so it is a cubic; the cube sum is a quartic. Fit with small cases: 1, 5, 14, 30 and 1, 9, 36, 100. - *Telescoping.* (*k* + 1)³ − *k*³ = 3*k*² + 3*k* + 1 summed gives (*n* + 1)³ − 1 = 3(square sum) + 3S + *n*, so one formula delivers the next. Fourth powers give the cube sum the same way. - *Nicomachus.* Odd numbers grouped 1 | 3 + 5 | 7 + 9 + 11 | ... give the cubes, so the cube sum is the sum of the first S(*n*) odd numbers, that is S(*n*)². This is the piece you could not have guessed from degrees alone, and it locks the picture together. - *Pictures.* Three copies of the stack of squares, as unit cubes, rearrange into a box of sides *n*, *n* + 1 and *n* + ½. Checks: the square sums 1, 5, 14 at *n* = 1, 2, 3; the cube sum at *n* = 3 is 36 = 6² = S(3)²; every formula gives 0 at *n* = 0. **Part 3.** A positive integer is a sum of two or more consecutive positive integers exactly when it is not a power of two, and the number of such representations is the number of its odd divisors greater than one. Sketch: a sum of *k* ≥ 2 consecutive integers starting at *a* ≥ 1 equals *k*(2*a* + *k* − 1)/2. The factors *k* and 2*a* + *k* − 1 have opposite parity, so exactly one is odd, and that odd one exceeds 1. Each representation thus gives an odd divisor *d* > 1, and each odd divisor *d* > 1 gives back exactly one representation: write N = *dm*, and take *k* = *d* if *d* < 2*m*, otherwise *k* = 2*m*. Powers of two have no odd divisor above one. Cross-check: 15 = 7 + 8 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 (odd divisors 3, 5, 15); 9 = 4 + 5 = 2 + 3 + 4 (odd divisors 3, 9); 8 has none. Guy counts the one-term representation as well, so his totals include the divisor 1. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery*, ECOL 479/579 handout and schedule (2001), Wayback Machine capture of 20 April 2002 β€” [web.archive.org](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu:80/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Brian Hayes**, "Gauss's Day of Reckoning", *American Scientist* 94(3), 200-205 (2006) β€” [americanscientist.org](https://www.americanscientist.org/article/gausss-day-of-reckoning){target=_blank} πŸ”“ - **Wolfgang Sartorius von Waltershausen**, *Gauss zum GedΓ€chtniss* (Leipzig: S. Hirzel, 1856) β€” [Internet Archive](https://archive.org/details/bub_gb_h_Q5AAAAcAAJ){target=_blank} πŸ”“ - **Wikipedia contributors**, "Triangular number" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Triangular_number){target=_blank} πŸ”“ - **Wikipedia contributors**, "Squared triangular number (Nicomachus's theorem)" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Squared_triangular_number){target=_blank} πŸ”“ - **Wikipedia contributors**, "Faulhaber's formula" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Faulhaber%27s_formula){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Power Sum", MathWorld β€” [mathworld.wolfram.com](https://mathworld.wolfram.com/PowerSum.html){target=_blank} πŸ”“ - **Wikipedia contributors**, "Polite number" β€” [en.wikipedia.org](https://en.wikipedia.org/wiki/Polite_number){target=_blank} πŸ”“ - **Robert Guy**, "Sums of Consecutive Integers", *The Fibonacci Quarterly* 20(1), 36-38 (1982) β€” [fq.math.ca](https://www.fq.math.ca/Scanned/20-1/guy.pdf){target=_blank} πŸ”“ - **W. J. LeVeque**, "On Representations as a Sum of Consecutive Integers", *Canadian Journal of Mathematics* 2, 399-405 (1950) β€” [doi.org](https://doi.org/10.4153/cjm-1950-036-3){target=_blank} πŸ”’ - **J. J. Sylvester**, with insertions by F. Franklin, "A Constructive Theory of Partitions, Arranged in Three Acts, an Interact and an Exodion", *American Journal of Mathematics* 5, 251-330 (1882) β€” [doi.org](https://doi.org/10.2307/2369545){target=_blank} πŸ”’ - **Arthur T. Winfree**, *The Art of Scientific Discovery*, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives one line for session 12: the name of the problem, then "like a jig-saw puzzle of cross-checks". The archived copy of Winfree's own course page carries the same line and nothing more. No handout, no column and no other document describing the exercise has been found, so the page above is a reconstruction from the name, the gloss and the course's stated habits. That is why the identification is *probable* rather than *confident*. One further caution: the same syllabus schedules [Summing a Series](https://tyson-swetnam.github.io/aosd/problems/summing-a-series/index.md) separately at session 26, so Winfree evidently had more than one series exercise, and this one may have been narrower. The candidates the editors weighed: - **Power sums by many routes** (the version written above): formulas for 1 + 2 + ... + *n* and for the sums of squares and cubes, derived several independent ways so that each checks the others. It matches the name, matches the gloss, matches Winfree's repeated demand for "as many cross-checking distinct solutions as you can", and needs only elementary mathematics. - **Sums of consecutive integers**: which numbers can be written as a sum of two or more consecutive positive integers, and in how many ways. Also literally "sums of integers", and also a puzzle whose partial results must fit together, but Winfree wrote "integers", not "consecutive integers". It is offered above as Part 3 so that either reading is served. - **Some undocumented arithmetic puzzle of his own**, built on integer sums with deliberately redundant clues to be checked against each other. Winfree liked puzzles with hidden traps, but no candidate text was found, so this remains speculation. One loose end. Winfree's archived site has a page titled ["How to use this in ASD"](https://web.archive.org/web/20030114050407/http://eebweb.arizona.edu/faculty/winfree/Associativity.htm){target=_blank} πŸ”“, in which ten numbers added forwards and backwards by computer give different sums. Its subject would suit this session or session 06, on ways to check for errors; reading it into either is an inference from the title, not a documented fact. --- *Back to [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-2-creative-blocks)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/ --- title: "Pedestrian Crosswalk Mystery" description: "Winfree's outdoor group lab: watch a real pedestrian crossing, record what happens before explaining it, pool the class's logs, and turn the oddities into questions that can be tested." type: Activity tags: [course, student-facing, problem, section-3, observation, fieldwork, hypothesis-testing] status: stable problem: section: 3 session: 12 identification: unknown kind: lab generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2001 resource: "https://web.archive.org/web/20030111064604/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "ECOL 479/579 The Art of Scientific Discovery: course handout with retrospective syllabus (archived)" author: "A. T. Winfree" - id: fhwa-hawk-2010 resource: "https://www.fhwa.dot.gov/publications/research/safety/10042/10042.pdf" title: "Safety Effectiveness of the HAWK Pedestrian Crossing Treatment (FHWA-HRT-10-042)" author: "Kay Fitzpatrick and Eun Sug Park" - id: nyt-placebo-buttons-2004 resource: "https://web.archive.org/web/20101110072424/http://www.nytimes.com/2004/02/27/nyregion/for-exercise-in-new-york-futility-push-button.html" title: "For Exercise in New York Futility, Push Button (The New York Times, 27 Feb 2004; archived copy)" author: "Michael Luo" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Pedestrian Crosswalk Mystery Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 12, where the lab begins, alongside [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md); the observations are pooled and discussed in session 13.* !!! abstract "The problem" Reconstructed from the syllabus. Winfree's own text is lost; what follows is built from the lab's name and its place in the schedule. Nothing records what the "mystery" was, or even whether the crossing had a signal. Go, as a group, to a pedestrian crosswalk on a busy street. Spend at least half an hour there over the coming week, at more than one time of day, and keep a log in your GamesWorth book. 1. **Describe it before explaining it.** Write down what is actually there: the markings, the signs, the kerbs, whatever machinery there is β€” and the people. Time events to the second, and note what happened just before each change. 2. **Watch the people as closely as the hardware.** Where do they cross relative to the painted lines? Who waits and who steps out? How do two opposing streams of walkers get through each other without colliding? When do drivers stop, and when do they roll through? 3. **Find the mystery.** Somewhere in your log there should be something you cannot yet explain. It may be mechanical, it may be human. Let the log tell you which. 4. **Pool the data.** Next session, combine everyone's timings and counts. Look for regularities no single observer could see, and for disagreements between observers, which are data too. 5. **Question, hypothesize, test.** Write your rules for the crossing as if-then hypotheses, say what would refute each, and go back and look. The product is not "the answer" but a clean record of what happened, a list of the questions it raises, and the explanations you tried to eliminate. ## Why it is in the course The lab straddles a seam in the schedule. It begins in session 12, the last of Section 2 on creative blocks, and is discussed in session 13, the first of Section 3, "Observations and Questions". It is the first exercise whose raw material is not a puzzle on paper but the street outside. The habits it drills are named elsewhere in the syllabus. Session 4 asks for "facts before explanations of facts" and for "distinguishing things we know vs only imagine"; a street corner demands both at once, because almost everyone arrives believing they already know how a crossing works. Pooling matters as much as watching: nobody can watch the traffic, the crossing and the walkers at once, so the logs come back incomplete and contradictory, and Section 3 is where disagreements become evidence. ## Where it comes from This is Winfree's own field exercise, not a puzzle borrowed from anyone. It survives as two lines of the schedule: the lab's name in session 12, and a line in session 13 telling the class to discuss its outdoor observations of the crosswalk. The problem sheet students were handed was never published. The archived course handout that carries the schedule is a plain reading copy with no working links, and Winfree's other surviving pages β€” his lab site and his eighteen "Adventures in Discovery" columns β€” say nothing about a crosswalk. The syllabus marks group labs "on the day they begin in class", which is why this one spans two sessions. ??? tip "Hints" - Time everything. A log without seconds cannot tell an effect from a coincidence. - Include controls. Compare a quiet hour with a busy one; if there is something to press, compare the intervals when someone presses it with those when nobody does. - Count people, not just machinery. Where they walk and who ignores the markings are observations as good as any timing. - Keep three columns: what you saw, what you inferred, and what you assumed because a sign said so. Everyone else acts on the third column too. - Ask what is not there. An event that never happens, a car that never stops, a stretch nobody uses: absences are observations. ## Sources - **A. T. Winfree**, *ECOL 479/579 The Art of Scientific Discovery: course handout with retrospective syllabus* (2001), archived copy β€” [Wayback Machine](https://web.archive.org/web/20030111064604/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Kay Fitzpatrick and Eun Sug Park**, *Safety Effectiveness of the HAWK Pedestrian Crossing Treatment*, FHWA-HRT-10-042 (2010) β€” [PDF](https://www.fhwa.dot.gov/publications/research/safety/10042/10042.pdf){target=_blank} πŸ”“ - **Michael Luo**, "For Exercise in New York Futility, Push Button", *The New York Times*, 27 February 2004 β€” [Wayback Machine](https://web.archive.org/web/20101110072424/http://www.nytimes.com/2004/02/27/nyregion/for-exercise-in-new-york-futility-push-button.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The identification is **unknown**. That the class went outdoors to a pedestrian crossing and pooled the results in the next session is certain from the syllabus. The location is not recorded, the "mystery" is not described, and the syllabus does not even say the crossing was signalized. All four readings below are reconstructions. - **Watching the people rather than the machinery** (medium confidence): the exercise is named for the *pedestrian* crossing, and this fits the course's rule that its exercises "depend as little as possible on knowledge of any particular subject area" and "require no lab setup". - **Inferring an unfamiliar signal's rules** (low-medium confidence): Tucson developed the HAWK β€” High intensity Activated crossWalK β€” beacon in the late 1990s, and its sequence was novel in 2001. But nothing ties Winfree's lab to a beacon, and traffic engineering is the specialist knowledge the course says it avoids. - **Testing whether a push button does anything** (low-medium confidence): a 2004 New York Times report found more than 2,500 of New York City's 3,250 walk buttons deactivated. That reporting postdates the course by three years. - **A rhythm-and-timing observation** (low confidence): two signals drifting in and out of phase, in the spirit of Winfree's work on coupled oscillators. The syllabus speaks of one crossing, and nothing supports this reading. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/ant-walk/ --- title: "Ant Walk" description: "Only the title survives in the syllabus; the editors' reconstruction puts an ant on the wires of a cube and an octahedron, trying to walk every wire exactly once, settled by counting the wires at each corner." type: Activity tags: [course, student-facing, problem, section-3, graph-theory, euler, recreational-mathematics, observation] status: stable problem: section: 3 session: 13 identification: unknown kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2001 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu:80/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery, ECOL 479/579 course handout (archived 20 April 2002)" author: "Arthur T. Winfree" - id: dudeney-1917 resource: "https://www.gutenberg.org/ebooks/16713" title: "Amusements in Mathematics, No. 245 \"The Fly on the Octahedron\" and No. 246 \"The Icosahedron Puzzle\"" author: "Henry Ernest Dudeney" - id: euler-1741 resource: "https://scholarlycommons.pacific.edu/euler-works/53/" title: "Solutio problematis ad geometriam situs pertinentis (E53), Commentarii academiae scientiarum Petropolitanae 8" author: "Leonhard Euler" - id: wikipedia-eulerian-path resource: "https://en.wikipedia.org/wiki/Eulerian_path" title: "Eulerian path" author: "Wikipedia contributors" - id: wikipedia-seven-bridges resource: "https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg" title: "Seven Bridges of KΓΆnigsberg" author: "Wikipedia contributors" - id: wikipedia-ant-rubber-rope resource: "https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope" title: "Ant on a rubber rope" author: "Wikipedia contributors" - id: gardner-1982 resource: "https://archive.org/details/ahagotchaparadox00gard" title: "aha! Gotcha: Paradoxes to Puzzle and Delight (pp. 145-146, the ant on a rubber rope)" author: "Martin Gardner" - id: mccartney-2013 resource: "https://doi.org/10.1080/0020739X.2012.729615" title: "Extending the rubber rope: convergent series, divergent series and the integrating factor" author: "Mark McCartney" - id: escher-mobius-1963 resource: "https://mcescher.com/gallery/mathematical/" title: "MΓΆbius Strip II (Red Ants), woodcut, February 1963 (M. C. Escher gallery: Mathematical)" author: "M. C. Escher" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Ant Walk Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 13. Discussed together with [Seven Bridges](https://tyson-swetnam.github.io/aosd/problems/seven-bridges/index.md), alongside the [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md) and the [chemical pattern-formation lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's own statement of this problem has not survived, and the syllabus gives only the title, paired with Seven Bridges. What follows is the puzzle the editors think was most likely meant; the other readings are listed at the foot of the page. An ant lives on a wire model of a cube: twelve straight wires meeting at eight corners, nothing else, and it walks only along the wires. Starting at any corner it likes, can it travel every one of the twelve wires exactly once? If it can, show a route. If it cannot, say exactly why β€” then find the fewest wires it must walk twice to cover them all in one continuous walk. Now move the ant to a wire octahedron: six corners, twelve wires, four wires at every corner. Ask the same question. Dudeney set this version in 1917, with a fly that "confines its walks entirely to the edges". Before you turn to Seven Bridges, notice what single observation settles both solids. ![Two wire-frame diagrams: a cube drawn as two nested squares with eight corners each meeting three edges, and an octahedron drawn as two nested triangles with six corners each meeting four edges; a red dot marks the ant on one corner of each](https://tyson-swetnam.github.io/aosd/assets/images/problems/ant-walk.svg){ width="560" } *The two wire models, flattened into diagrams. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 13 opens Section 3, Observations and Questions, and the syllabus asks the class to "Discuss Ant Walk and Seven Bridges". Euler's triumph at KΓΆnigsberg was not a calculation. It was an observation β€” count the bridge-ends at each landmass β€” followed by the right question: what property of the map decides whether any route exists at all? An ant confined to the edges of a solid invites the same move. Almost everyone begins by trying routes, gets stuck, and tries another. The discovery is the moment you stop tracing and start counting: look at a corner and ask what a walk must do every time it passes through one. That one question answers the cube, the octahedron and KΓΆnigsberg alike. Note what you threw away to get it β€” everything except which corners join which. ## Where it comes from Euler read his solution of the KΓΆnigsberg bridge problem to the St Petersburg Academy on 26 August 1735 and published it in 1741. Every landmass, he observed, except possibly the two where the walk starts and ends, must have an even number of bridge-ends; KΓΆnigsberg's four are all odd. The same count decides any "walk every edge once" puzzle: such a walk exists exactly when the network has zero or two odd corners. Euler proved the condition necessary, Carl Hierholzer that it is sufficient (1873). Puzzle-makers then moved the idea onto solids. Dudeney's "The Fly on the Octahedron" (*Amusements in Mathematics*, 1917, No. 245) flattens the solid into a diagram with four lines at every point and counts the complete routes from its top corner. The cube, with three wires at each of its eight corners, is the textbook impossible case. Whether Winfree meant this puzzle is not recorded. ??? tip "Hints" - Do not try routes. Count how many wires meet at each corner of the cube, then of the octahedron. - Passing through a corner uses two wires, one to arrive and one to leave. What does that imply for a corner where an odd number meet? - Only the starting and finishing corners can behave differently. How many corners, at most, can be odd? - For the cube: walking a wire twice is like adding an extra wire. How few extra wires leave at most two corners odd? ??? success "Resolution" **Cube.** Three wires meet at each of the eight corners β€” odd. In a walk using every wire once, every corner but the start and the finish is entered and left equally often, so it must be even. At most two may be odd; the cube has eight, so no such walk exists. Covering all twelve wires needs repeats, and each repeated wire flips the count at both of its ends, so it can fix two odd corners at once. Three repeats are enough for a walk that may finish anywhere (fifteen traversals); a walk returning to its start needs four (sixteen). **Octahedron.** Four wires meet at each of the six corners, all even, so a walk over all twelve edges exists from any corner and must end where it began. Dudeney: "if we start at the point A and go over all the lines once, we must always end our route at A." He counts 1,488 such routes from that top point. **Seven Bridges.** Euler's paper makes the identical observation: all four KΓΆnigsberg landmasses are odd, so no walk crosses each bridge once. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery*, ECOL 479/579 course handout β€” the archived web copy of the same syllabus, carrying the same single line β€” [Wayback Machine capture, 2002-04-20](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu:80/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Henry Ernest Dudeney**, *Amusements in Mathematics*, No. 245 "The Fly on the Octahedron" and No. 246 "The Icosahedron Puzzle" (1917) β€” [Project Gutenberg eBook #16713](https://www.gutenberg.org/ebooks/16713){target=_blank} πŸ”“ - **Leonhard Euler**, "Solutio problematis ad geometriam situs pertinentis", *Commentarii academiae scientiarum Petropolitanae* 8, 128-140 (1741) β€” [Euler Archive E53](https://scholarlycommons.pacific.edu/euler-works/53/){target=_blank} πŸ”“ - **Wikipedia contributors**, "Eulerian path" β€” [Wikipedia](https://en.wikipedia.org/wiki/Eulerian_path){target=_blank} πŸ”“ - **Wikipedia contributors**, "Seven Bridges of KΓΆnigsberg" β€” [Wikipedia](https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg){target=_blank} πŸ”“ - **Wikipedia contributors**, "Ant on a rubber rope" β€” [Wikipedia](https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope){target=_blank} πŸ”“ - **Martin Gardner**, *aha! Gotcha: Paradoxes to Puzzle and Delight*, pp. 145-146 (1982) β€” [Internet Archive](https://archive.org/details/ahagotchaparadox00gard){target=_blank} πŸ”“ *(borrow)* - **Mark McCartney**, "Extending the rubber rope: convergent series, divergent series and the integrating factor", *International Journal of Mathematical Education in Science and Technology* 44(4), 554-559 (2013) β€” [DOI](https://doi.org/10.1080/0020739X.2012.729615){target=_blank} πŸ”’ - **M. C. Escher**, "MΓΆbius Strip II (Red Ants)", woodcut, February 1963 β€” [official gallery](https://mcescher.com/gallery/mathematical/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure at all. The syllabus is the only surviving trace, and it gives the title and nothing more; the archived web copy of the same handout carries the same single line, and no other page on Winfree's archived lab site mentions an ant problem. The pairing with Seven Bridges is the strongest clue. The candidates weighed: - **An edge-walk on a solid** β€” the reconstruction above. Both problems in the session are walks over edges settled by the same corner-count. Medium confidence, but inference, not documentation. - **Gardner's ant on a rubber rope**, which crawls 1 cm/s along a 1 km rope stretched by another kilometre each second. It does reach the end, but the puzzle has nothing to do with KΓΆnigsberg. - **An ant on a MΓΆbius band**, as in Escher's 1963 woodcut. KΓΆnigsberg is also the birth of topology, and the syllabus discusses Escher's *Print Gallery* in session 16 β€” but nothing in the archive or the syllabus mentions a MΓΆbius band. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/ --- title: "Chemical Pattern-Formation Lab" description: "A three-session group lab watching chemical waves spread through a petri dish, almost certainly the Belousov-Zhabotinsky reaction that Winfree spent his career on, run as an exercise in recording observations before explaining them." type: Activity tags: [course, student-facing, problem, section-3, chemistry, pattern-formation, excitable-media, observation] status: stable problem: section: 3 session: 13 identification: probable kind: lab generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: scholarpedia-bz-reaction resource: "http://www.scholarpedia.org/article/Belousov-Zhabotinsky_reaction" title: "Belousov-Zhabotinsky reaction" author: "Anatol M. Zhabotinsky" - id: winfree-spiral-waves-1972 resource: "https://doi.org/10.1126/science.175.4022.634" title: "Spiral Waves of Chemical Activity" author: "Arthur T. Winfree" - id: zaikin-zhabotinsky-1970 resource: "https://doi.org/10.1038/225535b0" title: "Concentration Wave Propagation in Two-dimensional Liquid-phase Self-oscillating System" author: "A. N. Zaikin and A. M. Zhabotinsky" - id: winfree-prehistory-1984 resource: "https://doi.org/10.1021/ed061p661" title: "The prehistory of the Belousov-Zhabotinsky oscillator" author: "Arthur T. Winfree" - id: field-lecture-demonstration-1972 resource: "https://doi.org/10.1021/ed049p308" title: "A reaction periodic in time and space. A lecture demonstration" author: "Richard J. Field" - id: winfree-rotating-reactions-1974 resource: "https://www.scientificamerican.com/article/rotating-chemical-reactions/" title: "Rotating Chemical Reactions" author: "Arthur T. Winfree" - id: ruoff-bz-phenomenology resource: "https://www.ux.uis.no/~ruoff/BZ_Phenomenology.html" title: "The Phenomenology of the Belousov-Zhabotinsky Reaction" author: "Peter Ruoff" - id: rsc-classic-demonstrations-1995 resource: "https://web.archive.org/web/20140816215935/http://www.rsc.org/learn-chemistry/content/filerepository/CMP/00/001/001/Classicdemos_full.pdf" title: "Classic Chemistry Demonstrations: 2. An oscillating reaction" author: "Ted Lister / Royal Society of Chemistry" - id: strogatz-winfree-obituary-2003 resource: "https://physicstoday.aip.org/obituaries/arthur-taylor-winfree" title: "Arthur Taylor Winfree (obituary)" author: "Steven Strogatz" - id: glass-winfree-obituary-2003 resource: "https://doi.org/10.1038/421034a" title: "Arthur T. Winfree (1942-2002) (obituary)" author: "Leon Glass" - id: winfree-when-time-breaks-down-1987 resource: "https://archive.org/details/whentimebreaksdo0000winf" title: "When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias" author: "Arthur T. Winfree" - id: winfree-geometry-biological-time resource: "https://archive.org/details/geometryofbiolog0000winf" title: "The Geometry of Biological Time" author: "Arthur T. Winfree" - id: shakhashiri-chemical-demonstrations resource: "https://archive.org/details/chemicaldemonstr0002shak" title: "Chemical Demonstrations: A Handbook for Teachers of Chemistry" author: "Bassam Z. Shakhashiri" - id: wikipedia-bz-reaction resource: "https://en.wikipedia.org/wiki/Belousov%E2%80%93Zhabotinsky_reaction" title: "Belousov-Zhabotinsky reaction (overview and bibliography)" author: "Wikipedia contributors" - id: winfree-lab-home-2002 resource: "https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/" title: "A. T. Winfree in Tucson, AZ (archived lab home page)" author: "Arthur T. Winfree" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (EEB 479/479H/579): course handout (Internet Archive capture)" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Chemical Pattern-Formation Lab Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 13, continuing through sessions 14 and 15. The same session discusses [Ant Walk](https://tyson-swetnam.github.io/aosd/problems/ant-walk/index.md), [Seven Bridges](https://tyson-swetnam.github.io/aosd/problems/seven-bridges/index.md) and the [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: the schedule says only "start lab exercise on chemical pattern-formation", then "more chemical observations", then "last chemical observations". No lab sheet survives; this is the exercise as it was most likely run. **The set-up.** A layer a few millimetres deep of the ferroin-catalysed Belousov-Zhabotinsky reagent, in a covered petri dish about 10 cm across, left unstirred on a white background. The reagent is an acidic mix of bromate, malonic acid and the iron indicator ferroin, red when reduced and blue when oxidised, so the chemistry paints its own picture. It contains sulfuric acid and bromate, so it needs goggles, gloves, supervision, and a published safety-checked recipe for this unstirred dish, not one from this page. Note that the open recipes below (the RSC's, Ruoff's) are for the stirred beaker version instead. **The task.** Over three sessions, in small groups, keep a dated notebook of what you *see*, not of what you think is happening, and sketch the dish at intervals. When do the first blue spots appear? How fast do the rings grow? What happens when two fronts meet, or when one is broken? Then rank the questions your observations raise by how surprising the answer would be, and test one, prediction written down first. ![Photograph of a petri dish of deep red Belousov-Zhabotinsky reagent crossed by pale blue wave fronts, nested as concentric rings around several separate centres, with gas bubbles scattered across the layer](https://tyson-swetnam.github.io/aosd/assets/images/problems/chemical-pattern-formation-lab-spirals.jpg){ width="560" } *Target patterns in a real dish, with the bubbles and stray fronts a diagram leaves out. Photograph by Stephen Morris, chemistry by Michael Rogers, 2004. CC BY 2.0, via Wikimedia Commons.* ![Schematic of a petri dish with concentric target rings spreading from a pacemaker, and a second dish with a pair of counter-rotating spiral waves](https://tyson-swetnam.github.io/aosd/assets/images/problems/chemical-pattern-formation-lab.svg){ width="560" } *The two patterns to watch for. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 3 is "Observations and Questions", and the syllabus labels these three days as observations and nothing else. A dish of this reagent is almost an instrument for the purpose: something visibly happens over minutes, the patterns are strange, and nobody in the room can explain them. That forces session 4's discipline of "facts before explanations of facts", like the outdoor exercise in the same session, where the class watches a [pedestrian crosswalk](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md). The readings press the point: session 14 pairs the lab with Judson's *Chance* and [What Isn't There](https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/index.md), session 15 with Judson's *Evidence* and [Mother Nature as magician](https://tyson-swetnam.github.io/aosd/problems/mother-nature-as-magician/index.md). The syllabus promises puzzles "for pencil and paper and for simple lab manipulation"; this is the lab half, and it asks for no chemistry, only patience and an honest notebook. ## Where it comes from Boris Belousov found the first oscillating reaction in a stirred flask in the early 1950s. Editors rejected it, by Winfree's own account, because a solution changing colour back and forth looked like a violation of thermodynamics; it surfaced only in an obscure 1959 volume. Anatol Zhabotinsky took it up in 1961. In 1970 Zaikin and Zhabotinsky reported in *Nature* that a thin unstirred layer makes expanding rings from point pacemakers. Later that year Winfree, then at Chicago, saw what his own reasoning had ruled out. He had argued on topological grounds that a wave cannot rotate forever in a plain layer: it needs a hole to circulate around. Then came "a bewildering surprise, on October 10, 1970, to behold several perfectly stable spiral waves sedately rotating in a dish of this chemical reagent", in the words of Strogatz's obituary of him. He published the spirals in *Science* in 1972; Field's lecture demonstration the same year helped make the reaction a classroom standard, and such spirals later turned up in fibrillating heart muscle. So the class was watching the phenomenon its instructor's career was built on, and an argument of his own that a dish had overturned. ??? tip "Hints" - Split the notebook page: on the left only what the eye sees, colours, shapes, times, distances; on the right your guesses. The left column fills more slowly than people expect. - Time things. How long from pouring to the first blue spot? How many millimetres a minute does a front travel? Does the gap between rings stay the same? - Watch a collision. When two rings meet, do they cross like ripples on a pond, or do something else? - Watch a free end. A broken front does not simply stop; sketch one end every minute. - Next session change one thing: layer thickness, temperature, a wire drawn through. ??? success "Resolution" There is no answer to grade; the exercise is the notebook. But careful observers reliably find this much. - **Pacemakers and targets.** Blue oxidation waves start at discrete points, a dust speck or a scratch, and spread as concentric rings at a steady few millimetres a minute. Fast pacemakers entrain slow ones, so a few targets come to rule the dish. - **Annihilation, not superposition.** Colliding fronts wipe each other out rather than crossing. Each is a pulse of excitation trailed by a refractory zone that cannot fire again yet: the signature of an *excitable medium*, and what links the dish to nerve and heart tissue. - **Spirals.** A broken front curls around its own refractory tail, usually giving a counter-rotating pair. Spirals turn faster than pacemakers and outlive them, so an old dish fills with them. Their tips are phase singularities: the "hole" Winfree's argument demanded, shrunk to a point. ## Sources - **A. N. Zaikin and A. M. Zhabotinsky**, "Concentration Wave Propagation in Two-dimensional Liquid-phase Self-oscillating System", *Nature* 225, 535–537 (1970) β€” [doi:10.1038/225535b0](https://doi.org/10.1038/225535b0){target=_blank} πŸ”’ - **Arthur T. Winfree**, "Spiral Waves of Chemical Activity", *Science* 175(4022), 634–636 (1972) β€” [doi:10.1126/science.175.4022.634](https://doi.org/10.1126/science.175.4022.634){target=_blank} πŸ”’ - **Arthur T. Winfree**, "Rotating Chemical Reactions", *Scientific American* 230(6), 82–95 (June 1974) β€” [scientificamerican.com](https://www.scientificamerican.com/article/rotating-chemical-reactions/){target=_blank} πŸ”’ - **Arthur T. Winfree**, "The prehistory of the Belousov-Zhabotinsky oscillator", *Journal of Chemical Education* 61(8), 661 (1984) β€” [doi:10.1021/ed061p661](https://doi.org/10.1021/ed061p661){target=_blank} πŸ”’ - **Richard J. Field**, "A reaction periodic in time and space. A lecture demonstration", *Journal of Chemical Education* 49(5), 308 (1972) β€” [doi:10.1021/ed049p308](https://doi.org/10.1021/ed049p308){target=_blank} πŸ”’ - **Anatol M. Zhabotinsky**, "Belousov-Zhabotinsky reaction", *Scholarpedia* 2(9), 1435 (2007) β€” [scholarpedia.org](http://www.scholarpedia.org/article/Belousov-Zhabotinsky_reaction){target=_blank} πŸ”“ - **Peter Ruoff**, "The Phenomenology of the Belousov-Zhabotinsky Reaction" (University of Stavanger; authorship inferred from the site) β€” [ux.uis.no](https://www.ux.uis.no/~ruoff/BZ_Phenomenology.html){target=_blank} πŸ”“ - **Ted Lister / Royal Society of Chemistry**, "Classic Chemistry Demonstrations: 2. An oscillating reaction" (1995; the stirred version, not the petri-dish one) β€” [Internet Archive PDF](https://web.archive.org/web/20140816215935/http://www.rsc.org/learn-chemistry/content/filerepository/CMP/00/001/001/Classicdemos_full.pdf){target=_blank} πŸ”“ - **Bassam Z. Shakhashiri**, *Chemical Demonstrations: A Handbook for Teachers of Chemistry*, vol. 2 (University of Wisconsin Press) β€” [archive.org](https://archive.org/details/chemicaldemonstr0002shak){target=_blank} πŸ”“ *(borrow)* - **Steven Strogatz**, "Arthur Taylor Winfree" (obituary), *Physics Today* 56(6), 74–75 (2003) β€” [physicstoday.aip.org](https://physicstoday.aip.org/obituaries/arthur-taylor-winfree){target=_blank} πŸ”“ - **Leon Glass**, "Arthur T. Winfree (1942-2002)" (obituary), *Nature* 421, 34 (2003) β€” [doi:10.1038/421034a](https://doi.org/10.1038/421034a){target=_blank} πŸ”’ - **Arthur T. Winfree**, *When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias* (1987) β€” [archive.org](https://archive.org/details/whentimebreaksdo0000winf){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Geometry of Biological Time* (1980; 2nd ed. 2001) β€” [archive.org](https://archive.org/details/geometryofbiolog0000winf){target=_blank} πŸ”“ *(borrow)* - **Wikipedia contributors**, "Belousov-Zhabotinsky reaction" β€” [overview and bibliography](https://en.wikipedia.org/wiki/Belousov%E2%80%93Zhabotinsky_reaction){target=_blank} πŸ”“ - **Arthur T. Winfree**, "A. T. Winfree in Tucson, AZ" (archived lab home page, 2002) β€” [Internet Archive](https://web.archive.org/web/20021225142609/http://eebweb.arizona.edu/faculty/winfree/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery* course handout (archived 2002) β€” [Internet Archive](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Fairly, but not certainly. The syllabus gives three lines, and no lab sheet survives in the Internet Archive. - **The Belousov-Zhabotinsky reaction in petri dishes**, as above. Winfree's 2002 home page says his funding covered "chemical, cardiac, and neural excitable media", so the reagents were down the hall, and no other bench-top chemical pattern unfolds over hours with no instruments. High confidence. - **Liesegang rings**, periodic precipitation bands in a gel: as slow and as equipment-light, but nothing ties Winfree to them. Low confidence. - **A menu of systems.** The syllabus names no reaction. Low confidence. The protocol above is ordinary practice for this demonstration, not Winfree's instructions. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/seven-bridges/ --- title: "Seven Bridges of KΓΆnigsberg" description: "Can a walker cross all seven bridges of KΓΆnigsberg exactly once? Euler's answer threw away the map and kept only the connections β€” the founding move of graph theory." type: Activity tags: [course, student-facing, problem, section-3, graph-theory, topology, euler, asking-the-right-question] status: stable problem: section: 3 session: 13 identification: confident kind: puzzle generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: euler-1741 resource: "https://scholarlycommons.pacific.edu/euler-works/53/" title: "Solutio problematis ad geometriam situs pertinentis (E53)" author: "Leonhard Euler" - id: euler-latin-text resource: "https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler.html" title: "Solutio problematis ad geometriam situs pertinentis: Latin text with figures (republication)" author: "Leonhard Euler; republished by Michael Behrend" - id: hopkins-wilson-2004 resource: "https://doi.org/10.2307/4146895" title: "The Truth about KΓΆnigsberg" author: "Brian Hopkins and Robin J. Wilson" - id: wikipedia-seven-bridges resource: "https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg" title: "Seven Bridges of KΓΆnigsberg" author: "Wikipedia contributors" - id: wikipedia-ehler resource: "https://en.wikipedia.org/wiki/Carl_Gottlieb_Ehler" title: "Carl Gottlieb Ehler" author: "Wikipedia contributors" - id: wikipedia-eulerian-path resource: "https://en.wikipedia.org/wiki/Eulerian_path" title: "Eulerian path" author: "Wikipedia contributors" - id: mactutor-topology resource: "https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/" title: "A history of Topology (MacTutor History of Mathematics)" author: "J. J. O'Connor and E. F. Robertson" - id: mactutor-konigsberg resource: "https://mathshistory.st-andrews.ac.uk/Extras/Konigsberg/" title: "KΓΆnigsberg bridges (MacTutor Extras: historic pictures)" author: "MacTutor History of Mathematics" - id: shields-2012 resource: "https://doi.org/10.1177/0263276412451161" title: "Cultural Topology: The Seven Bridges of KΓΆnigsburg, 1736" author: "Rob Shields" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Seven Bridges of KΓΆnigsberg Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 13. Discussed together with [Ant Walk](https://tyson-swetnam.github.io/aosd/problems/ant-walk/index.md); the same session takes up outdoor observations on the [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md) and starts the [chemical pattern-formation lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md).* !!! abstract "The problem" *The setting below is the editors' summary of Euler's own description of the city and its bridges. Nothing in it is a quotation.* The Prussian city of KΓΆnigsberg (today Kaliningrad, Russia) stood on the river Pregel. In the middle of the city the river runs in two branches around the island of Kneiphof, and a second island, the Lomse, lies between the same two branches. Seven bridges joined these four pieces of land: the island Kneiphof (call it A), the mainland bank on one side (B), the mainland bank on the other (C) and the second island (D). Two bridges (a and b) joined A to B, two (c and d) joined A to C, one (e) joined A to D, one (f) joined B to D and one (g) joined C to D. Euler opens his 1741 paper by reporting the puzzle as it reached him: could anyone plan a walk that crosses every one of the seven bridges exactly once? He adds that he was told some people denied it was possible and others doubted it, but that nobody asserted it. **Your task.** Find such a walk, or explain convincingly why none exists. You may start and finish anywhere, and need not return to your starting point. **Then the wider question**, also Euler's: whatever the shape of the river and however many bridges there are, state a rule that decides the matter for any map, without listing all the possible walks. ![Schematic map of KΓΆnigsberg with island A, banks B and C, land D between the river branches, and seven bridges a to g; beside it the same layout drawn as a graph with four dots and seven lines, each dot marked with its number of bridges](https://tyson-swetnam.github.io/aosd/assets/images/problems/seven-bridges.svg){ width="560" } *Drawn for this site (CC BY 4.0). Schematic: it keeps the connections of Euler's own figure, not the shapes or distances of the real city, and like his figure it draws D as an open wedge of land between the two branches.* ## Why it is in the course Section 3 is "Observations and Questions", and session 13 pairs Seven Bridges with [Ant Walk](https://tyson-swetnam.github.io/aosd/problems/ant-walk/index.md). KΓΆnigsberg is a clean case of observations piling up while the right question goes unasked. The townspeople had plenty of data β€” many attempted walks, none successful β€” and Euler records where that left them: some denied a walk was possible, others doubted it, nobody asserted it. That is not yet knowledge. Failed attempts are facts, but "nobody has found a walk" is not the same claim as "no walk exists". The syllabus makes that distinction the work of session 04, where [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md) is about "distinguishing things we know vs only imagine". Euler refused the brute search. In Β§3 he considers listing every possible route and rejects it: too laborious, hopeless for larger maps, and it turns up a great deal nobody asked for. He asked instead what property of the map decides the matter, and threw away everything that could not matter β€” distances, the shapes of the banks, the streets β€” until only the four lands and the number of bridges at each one remained. That stripping down, and the small counting argument it makes possible, is the point of the session. A good question turns a heap of observations into a rule you can check, one that settles every similar map at once. ## Where it comes from KΓΆnigsberg's townspeople had argued over the question for some time before it reached a mathematician. It came to Euler, then at the St Petersburg Academy, through Carl Gottlieb Ehler, an astronomer who later became mayor of Danzig. In a letter of March 1736 Ehler asked Euler for his solution to the problem of the seven KΓΆnigsberg bridges, with a proof, presenting it as a fine specimen of the calculus of position. Euler's reply played the problem down: its solution, he said, had little to do with mathematics and rested on reason alone, and he admitted he did not know what Leibniz and Wolff had meant by a geometry of position. (The letters are published in translation by Hopkins and Wilson, p. 202; the editors report them here at second hand and do not quote them.) He had in fact already dealt with it. "Solutio problematis ad geometriam situs pertinentis" β€” "solution of a problem belonging to the geometry of position" β€” was presented to the Academy on 26 August 1735 and printed in 1741 in the *Commentarii academiae scientiarum Petropolitanae*, volume 8, pages 128–140. It is now read as the first paper in graph theory and one of the first in topology, because nothing in it depends on distance or shape, only on what is connected to what. ![Euler's Figure 1 from the 1741 paper: a schematic of the river with island A, regions B, C and D, and the bridges lettered a to g](https://tyson-swetnam.github.io/aosd/assets/images/problems/seven-bridges-euler-fig1.png){ width="560" } *Leonhard Euler, Fig. 1 of "Solutio problematis ad geometriam situs pertinentis" (1741). Public domain, via Wikimedia Commons.* ??? tip "Hints" - Before searching for a walk, say what a walk really is. Does the route through the streets matter, or only the order in which you visit the four pieces of land? Try writing a walk as a string of letters A, B, C, D. - If a walk crosses seven bridges, how many letters does its string have? Each crossing adds one letter. - Count the bridges touching each piece of land. Every time you pass through a land you use two of its bridges, one in and one out. What does an odd count force about starting or finishing there? - How many lands can be the start or the finish of a single walk? Compare that with how many lands at KΓΆnigsberg have an odd number of bridges. - Test your rule on a map you invent: add one bridge to KΓΆnigsberg and ask whether the walk becomes possible, and where it must then begin. ## What happened No such walk exists. Describe a walk by the sequence of lands it visits: one letter for the land you start in, one more for each bridge crossed, so a walk over the seven bridges is written with eight letters. Each time the walker passes through a land, arriving and leaving, two of that land's bridges are used β€” so a land reached by an odd number of bridges must be either the start or the end of the walk. A walk has one start and one end, so at most two lands may have an odd count. At KΓΆnigsberg the island A has five bridges and B, C and D have three each: four odd lands, and therefore no walk that crosses every bridge exactly once. Euler's own bookkeeping (Β§9) makes the clash visible. A land reached by five bridges forces its letter to appear three times in the record of the walk, and a land reached by three bridges forces two appearances: 3 + 2 + 2 + 2 = 9 letters, in a word with only eight places. The word cannot be built, so the walk cannot be taken. Euler's general rule (Β§20, in paraphrase): if more than two regions have an odd number of bridges, no such crossing exists; if exactly two do, the crossing can be made, provided the walk begins in one of those two; if none do, it can be made beginning anywhere. Euler proved the impossibility half and asserted the rest; Carl Hierholzer gave the first full proof of the other cases, published after his death in 1873. In today's language, a connected graph has an Eulerian trail exactly when it has zero or two vertices of odd degree. Only the pattern of connections carried the answer. Euler called that new kind of geometry the geometry of position; we call it topology and graph theory. A postscript. Two of the seven bridges did not survive the bombing of KΓΆnigsberg in the Second World War, and two more were later demolished for a highway, leaving five at the old sites. With that layout only two lands have an odd count, so the walk is now possible β€” but it must begin on one island and end on the other. (This present-day account comes from the Wikipedia overview listed below, consulted in September 2026 and not checked on the ground.) ## Sources - **Leonhard Euler**, "Solutio problematis ad geometriam situs pertinentis", *Commentarii academiae scientiarum Petropolitanae* 8, 128–140 (1741; presented 1735) β€” [Euler Archive E53](https://scholarlycommons.pacific.edu/euler-works/53/){target=_blank} πŸ”“ - **Leonhard Euler**, the full Latin text with Euler's three figures, republished by Michael Behrend β€” [cantab.net](https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler.html){target=_blank} πŸ”“ (the §§1–4, 9 and 20 used above were read there, in the Latin and in the companion English translation; that translation names no translator and carries no licence, so this page paraphrases it and quotes nothing from it) - **Brian Hopkins and Robin J. Wilson**, "The Truth about KΓΆnigsberg", *The College Mathematics Journal* 35(3), 198–207 (2004) β€” [doi:10.2307/4146895](https://doi.org/10.2307/4146895){target=_blank} πŸ”’ (the scholarly source for the Ehler–Euler letters; the editors could not read it directly) - **Wikipedia contributors**, "Carl Gottlieb Ehler" β€” [Wikipedia](https://en.wikipedia.org/wiki/Carl_Gottlieb_Ehler){target=_blank} πŸ”“ (renders the 1736 letters from Hopkins and Wilson, p. 202; the second-hand basis for the account above) - **Wikipedia contributors**, "Seven Bridges of KΓΆnigsberg" β€” [Wikipedia](https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg){target=_blank} πŸ”“ (dates, the Kneiphof and Lomse, and the fate of the bridges after 1945) - **Wikipedia contributors**, "Eulerian path" β€” [Wikipedia](https://en.wikipedia.org/wiki/Eulerian_path){target=_blank} πŸ”“ (Hierholzer's 1873 proof; the modern statement of the rule) - **J. J. O'Connor and E. F. Robertson**, "A history of Topology", MacTutor History of Mathematics (1996) β€” [MacTutor](https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/){target=_blank} πŸ”“ - **MacTutor History of Mathematics**, "KΓΆnigsberg bridges": historic pictures of the city and its bridges (2000) β€” [MacTutor Extras](https://mathshistory.st-andrews.ac.uk/Extras/Konigsberg/){target=_blank} πŸ”“ - **Rob Shields**, "Cultural Topology: The Seven Bridges of KΓΆnigsburg [sic], 1736", *Theory, Culture & Society* 29(4–5), 43–57 (2012) β€” [doi:10.1177/0263276412451161](https://doi.org/10.1177/0263276412451161){target=_blank} πŸ”’ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/ --- title: "What Isn't There (Surprisingly Hard)" description: "List the things that never happen in the world you observe, then ask why they were so hard to notice: an exercise in turning absences into expectations you can test." type: Activity tags: [course, student-facing, problem, section-3, observation, negative-evidence, perception] status: stable problem: section: 3 session: 14 identification: probable kind: discussion generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: winfree-sas08-2002 resource: "https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html" title: "Adventure of the Rainbow Moon (Adventures in Discovery column, SAS E-Bulletin, 11 January 2002)" author: "Arthur T. Winfree" - id: winfree-sas09-2002 resource: "https://web.archive.org/web/20030114041305/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS09/SAS09.html" title: "Adventure of the Rainbow Moon: crucial experiment this afternoon in Tucson (SAS E-Bulletin, 25 January 2002)" author: "Arthur T. Winfree" - id: judson-1980 resource: "https://archive.org/details/searchforsolutio00juds" title: "The Search for Solutions (chapter 4, 'Chance')" author: "Horace Freeland Judson" - id: anderson-1990 resource: "https://doi.org/10.1063/1.2810433" title: "Some Thoughtful Words (Not Mine) on Research Strategy for Theorists" author: "Philip W. Anderson" - id: doyle-silver-blaze-1892 resource: "https://www.gutenberg.org/files/834/834-h/834-h.htm" title: "The Memoirs of Sherlock Holmes: 'Silver Blaze' (Project Gutenberg eBook #834)" author: "Arthur Conan Doyle" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # What Isn't There (Surprisingly Hard) Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 14. The same session continues the [chemical pattern-formation lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md); the next session discusses [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: the original text is lost, so this follows the item's name, its place in the schedule, and Winfree's framing of the same idea in a later column. Some absences are easy to notice. As Winfree put it, "I bet you never saw a stone fall upwards, and you would have noticed." Others are not. He passed on two he had only been told: that no fish blinks both eyes at once (though sharks do), and that certain animals, sharks among them, never get any kind of cancer. 1. Over a week, list things that **never happen**, or are **never there**, in the world you observe: in the sky, on the walk to class, in your chemical observations. Aim for absences you had never registered until you went looking. 2. For each, ask: how much can you work out about why you never saw it, and what deliberate observations could settle the questions that raises? 3. Sort the list. Which cannot happen? Which happen where you were never placed to see them? Which has nobody checked? Which were you merely told? Then the question that gives the exercise its name: why is this so surprisingly hard? ## Why it is in the course Section 3 is "Observations and Questions". Session 14 pairs "More chemical observations" with this item, one session before the Rainbow Moon, with Judson's *Chance* and Anderson on research strategy as readings. The syllabus gives only a name and a warning: surprisingly hard. In Winfree's words from a later column, "Other things that never happen are hard to notice." Working out why is the point of the exercise. It also returns to session 04's "distinguishing things we know vs only imagine": "I never saw it", "it never happens" and "I was told it never happens" are three different claims. ## Where it comes from The editors found no published puzzle with this name. The archived course handout lists it for session 14, but its problem descriptions lived in a companion document that was never archived. Winfree's clearest statement of the idea opens his column "Adventure of the Rainbow Moon" (Society for Amateur Scientists E-Bulletin, 11 January 2002), written about three months after the session. It shows how he framed the theme, not the exercise as set. He contrasts easy absences with hard ones, then poses a hard one: nobody remembers seeing the Moon behind the colour band of a rainbow. The follow-up (25 January 2002) describes engaging "the absence of an experience that 'ought' to be remembered, but isn't". Winfree had seen a Rainbow Moon himself in Tucson on 6 September 2001, a month before the class discussed it. The idea is old. Holmes makes the point in Conan Doyle's "Silver Blaze" (1892), where the curious incident is the dog that did nothing in the night-time. Nothing in Winfree's materials mentions the story; it is an aside here, not the source. ??? tip "Hints" - Start with the easy absences (stones never fall up) and ask why those were easy. What did you already know that let you notice them? - Separate "I have never seen it", "nobody has ever seen it" and "it cannot happen". Which kind is each entry, and what observation would move it from one kind to another? - For absences that survive, ask what enforces them: geometry, timing, physics, selection. - To catch something that does not happen, first name a specific thing that *could* happen, then look for it. ??? success "Resolution" There is no single answer; the resolution is seeing *why* it is hard. An absence is not an experience. You can notice that something is missing only once you hold a definite expectation that it should be there. The remedy is to manufacture expectations and test them: predict when and where the missing thing should appear, then go and look. Winfree did that with his own example. Predicting when the Moon should stand in a rainbow led him to a month of sextant measurements and several small discoveries about the Moon's motion. The geometry belongs on the [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md) page. The other half of the lesson is Winfree's hedge: he introduced his fish and cancer examples with "I have been told" and "I have also heard". An absence reported at second hand is a claim to check, not a fact. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, "Adventure of the Rainbow Moon", Adventures in Discovery column, SAS E-Bulletin (11 January 2002) β€” [Wayback Machine](https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, "Adventure of the Rainbow Moon: crucial experiment this afternoon in Tucson", SAS E-Bulletin (25 January 2002) β€” [Wayback Machine](https://web.archive.org/web/20030114041305/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS09/SAS09.html){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions* (1980), chapter 4, "Chance" β€” [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* - **Philip W. Anderson**, "Some Thoughtful Words (Not Mine) on Research Strategy for Theorists", *Physics Today* 43(2), 9 (1990) β€” [doi:10.1063/1.2810433](https://doi.org/10.1063/1.2810433){target=_blank} πŸ”’ (very likely the session's "Anderson on research strategy"; not read by the editors) - **Arthur Conan Doyle**, "Silver Blaze", in *The Memoirs of Sherlock Holmes* (1892) β€” [Project Gutenberg](https://www.gutenberg.org/files/834/834-h/834-h.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the name and its place in session 14, just before the Rainbow Moon. No text of the problem survives, so the reading is **probable**, not confident. - **Likeliest: an exercise in noticing things that never happen.** It matches the "hard to notice" framing of Winfree's Rainbow Moon column, and the schedule runs straight from this item to that case. Against it: the column dates from January 2002, about three months after the session, so it shows Winfree's framing of the theme, not the exercise as set. - **Negative evidence and the dog in the night-time.** This site's earlier guess. The classic illustration, but no Winfree source mentions it, so it is rejected as the identification. - **A perceptual absence such as the eye's blind spot**, leading into session 15's "Mother Nature as magician; hallucinations". It fits the theme, but nothing in the archived material supports it. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/mother-nature-as-magician/ --- title: "Mother Nature as Magician; Hallucinations" description: "A reconstructed session-15 discussion on evidence from the senses: illusions in which honest observation leads to a false inference, and hallucinations, where the percept has no outside cause at all." type: Activity tags: [course, student-facing, problem, section-3, perception, illusions, hallucinations, evidence] status: stable problem: section: 3 session: 15 identification: probable kind: discussion generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-sas15-2002 resource: "https://web.archive.org/web/20030114043354/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS15/SAS15.html" title: "Adventures in Discovery, column 15: Vibrating the Brain (19 April 2002)" author: "Arthur T. Winfree" - id: winfree-sas01-2001 resource: "https://web.archive.org/web/20030114041445/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS01/SAS01.html" title: "Adventures in Discovery, column 1: Are straight lines curved in the sky? (9 November 2001)" author: "Arthur T. Winfree" - id: winfree-sas05-2001 resource: "https://web.archive.org/web/20030114051703/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS05/SAS05.html" title: "Adventures in Discovery, column 5: A Personal Encounter with Non-Euclidean Geometry (7 December 2001)" author: "Arthur T. Winfree" - id: winfree-sas11-2002 resource: "https://web.archive.org/web/20030114041649/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS11/SAS11.html" title: "Adventures in Discovery, column 11: Moon and Sun Violate Reason? Nope: \"Reason\" Takes a Lesson From Them (22 February 2002)" author: "Arthur T. Winfree" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: judson-1980 resource: "https://archive.org/details/searchforsolutio0000juds_r0s0" title: "The Search for Solutions" author: "Horace Freeland Judson" - id: feynman-1974 resource: "https://calteches.library.caltech.edu/51/2/CargoCult.htm" title: "Cargo Cult Science" author: "Richard P. Feynman" - id: ermentrout-cowan-1979 resource: "https://doi.org/10.1007/BF00336965" title: "A mathematical theory of visual hallucination patterns" author: "G. B. Ermentrout and J. D. Cowan" - id: bressloff-2001 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC1088430/" title: "Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex" author: "P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. C. Wiener" - id: rule-2011 resource: "https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002158" title: "A Model for the Origin and Properties of Flicker-Induced Geometric Phosphenes" author: "M. Rule, M. Stoffregen and B. Ermentrout" - id: winfree-1972 resource: "https://doi.org/10.1126/science.175.4022.634" title: "Spiral Waves of Chemical Activity" author: "Arthur T. Winfree" - id: wikipedia-moon-illusion resource: "https://en.wikipedia.org/wiki/Moon_illusion" title: "Moon illusion" author: "Wikipedia contributors" - id: wikipedia-fraser-spiral resource: "https://en.wikipedia.org/wiki/Fraser_spiral_illusion" title: "Fraser spiral illusion" author: "Wikipedia contributors" - id: wikipedia-form-constant resource: "https://en.wikipedia.org/wiki/Form_constant" title: "Form constant" author: "Wikipedia contributors" - id: wikipedia-hallucination resource: "https://en.wikipedia.org/wiki/Hallucination" title: "Hallucination" author: "Wikipedia contributors" - id: commons-fraser-spiral resource: "https://commons.wikimedia.org/wiki/File:Fraser_spiral.svg" title: "Fraser spiral illusion (vector redrawing)" author: "Mysid, after James Fraser (1908)" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Mother Nature as Magician; Hallucinations Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 15, with Judson Chapter 8: Evidence. The same session takes the last observations in the [chemical pattern-formation lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md) and discusses [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md).* !!! abstract "The problem" *A reconstruction.* The syllabus gives only the title and the reading. The examples below are standard illusions and observations from Winfree's own writing, most of it from after the course (dates under "Where it comes from"). Work them before reading "What happened". **The magician.** A stage magician breaks no law of nature: everything the audience sees is real, but the obvious inference is false. Nature plays the same trick on careful observers. Check each case yourself: 1. **The Moon at the horizon** looks far bigger than the Moon high in the sky. Hold a coin at arm's length so it just covers the high Moon, then try it at moonrise. 2. **A taut shoestring**, held level at arm's length along the horizon, looks straight. Raise it, still level, about 30 degrees higher: its middle now seems to bow upward. 3. **The Fraser spiral** in the figure below. Trace one of its "spiral" arms with a finger. For each, write down what you observed, what you inferred, where in the chain *object, light, eye, brain* the trick happens, and what test would show it. **The hallucination.** An outdoor fluorescent lamp in a white globe is switched on by a light sensor. At dusk it flickers, because its own light keeps turning it off. The globe seems covered with shifting purple blotches and squirming yellow worms. Are they on the lamp, or in you? You cannot photograph them. Plan observations that would (a) locate the patterns, (b) measure something about them, and (c) let someone else check your claim. ![Two panels. Left: a string held level above the horizon appears bowed upward, while a dotted straight line shows its true shape. Right: the chain object, light, eye, brain, running from observation to inference, with a check named under each link: ruler, light meter, camera, another observer](https://tyson-swetnam.github.io/aosd/assets/images/problems/mother-nature-as-magician.svg){ width="560" } *Drawn for this site (CC BY 4.0).* ![The Fraser spiral illusion: twisted black-and-white cords on a checkered ground that appear to form a spiral](https://tyson-swetnam.github.io/aosd/assets/images/problems/mother-nature-as-magician-fraser.svg){ width="560" } *Fraser spiral illusion, redrawn by Mysid after James Fraser (1908). Wikimedia Commons, public domain.* ## Why it is in the course Section 3 is "Observations and Questions", and the reading for session 15 is Judson's chapter "Evidence". Earlier sessions set the ground rules: "facts before explanations of facts" with the [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md), and "distinguishing things we know vs only imagine" with [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md). [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md) showed a whole laboratory fooling itself. This session moves the deception into perception itself: the audience sees truly and concludes falsely. The exercise is to split observation from inference and find where they came apart. A hallucination makes the evidence problem sharpest: when the thing exists only in the observer, what takes the place of a photograph? ## Where it comes from The idea is old. Feynman's "Cargo Cult Science", read in session 2, says: "The first principle is that you must not fool yourself, and you are the easiest person to fool." The Moon illusion is mentioned by Aristotle, and Purkinje described flicker patterns in 1819. Ermentrout and Cowan (1979) explained the geometric forms of drug-induced hallucinations as patterns that appear when activity in the visual cortex becomes unstable. Only one piece of Winfree's own material predates the session: the lamp observation, which he dates to January 2001. In a column of 19 April 2002 he posed the question it raised: "How do you investigate, how do you communicate to others, and how do you describe without pictures, something that has no 'objective' existence?" The rest of his material is later. The shoestring observation is dated 23 October 2001, and his explanation of it appeared on 7 December 2001. His tests on the lamp are undated and were published only in April 2002. In a column of 22 February 2002 he took up another case of Nature seeming to cheat: the Moon plainly circles the sky once a month, yet the "phase angle" in every ephemeris never reaches 180 degrees. None of these columns records what the class discussed. ??? tip "Hints" - Write the observation and the inference on separate lines. The trick lives in the gap between them. - Check each link with something that cannot be fooled the same way: a ruler on the object, a light meter on the light, a camera in place of your eye, another person in place of your brain. - For a percept with no outside cause, first measure the stimulus with an instrument that cannot hallucinate. - "Objective" is a claim about other observers. Bring one who does not know what to expect, and vary the stimulus until the effect comes and goes. ## What happened Nothing records what the class concluded; the examples resolve like this. - **Moon illusion.** The coin covers the Moon equally well at the horizon and overhead: about half a degree. The enlargement is made in the head, by a mechanism still debated. - **Shoestring.** Winfree argued (7 December 2001) that we build the visual world as a sphere of directions, on which a straight line becomes a great-circle arc: straight in space, curved in the view. - **Fraser spiral.** The finger comes back to where it started: the figure is concentric circles. The tilted strands mislead local orientation, and the brain joins them into a spiral. - **The flickering lamp.** Winfree's answer (April 2002): an oscilloscope showed the flicker had period-doubled to 30 Hz; a strobe on a white wall made the same blotches between about 30 and 18 Hz only; an uninformed graduate student saw them too. Made by eye and brain, yet measured and confirmed. Such patterns are now modelled as pattern formation in the visual cortex (Rule, Stoffregen and Ermentrout, 2011). ## Sources - **Arthur T. Winfree**, "Vibrating the Brain", *Adventures in Discovery* column, Society for Amateur Scientists E-Bulletin (19 April 2002) β€” [Wayback Machine](https://web.archive.org/web/20030114043354/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS15/SAS15.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, "Are straight lines curved in the sky?", *Adventures in Discovery* column (9 November 2001) β€” [Wayback Machine](https://web.archive.org/web/20030114041445/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS01/SAS01.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, "A Personal Encounter with Non-Euclidean Geometry", *Adventures in Discovery* column (7 December 2001) β€” [Wayback Machine](https://web.archive.org/web/20030114051703/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS05/SAS05.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, "Moon and Sun Violate Reason? Nope: 'Reason' Takes a Lesson From Them", *Adventures in Discovery* column (22 February 2002) β€” [Wayback Machine](https://web.archive.org/web/20030114041649/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS11/SAS11.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579), course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions* (Holt, Rinehart and Winston, 1980), Chapter 8: Evidence β€” [Internet Archive](https://archive.org/details/searchforsolutio0000juds_r0s0){target=_blank} πŸ”“ *(borrow)* - **Richard P. Feynman**, "Cargo Cult Science", Caltech commencement address (1974) β€” [Caltech](https://calteches.library.caltech.edu/51/2/CargoCult.htm){target=_blank} πŸ”“ - **G. B. Ermentrout and J. D. Cowan**, "A mathematical theory of visual hallucination patterns", *Biological Cybernetics* 34, 137–150 (1979) β€” [doi:10.1007/BF00336965](https://doi.org/10.1007/BF00336965){target=_blank} πŸ”’ - **P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. C. Wiener**, "Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex", *Phil. Trans. R. Soc. Lond. B* 356, 299–330 (2001) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC1088430/){target=_blank} πŸ”“ - **M. Rule, M. Stoffregen and B. Ermentrout**, "A Model for the Origin and Properties of Flicker-Induced Geometric Phosphenes", *PLoS Computational Biology* 7, e1002158 (2011) β€” [PLoS](https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002158){target=_blank} πŸ”“ - **Arthur T. Winfree**, "Spiral Waves of Chemical Activity", *Science* 175, 634–636 (1972) β€” [doi:10.1126/science.175.4022.634](https://doi.org/10.1126/science.175.4022.634){target=_blank} πŸ”’ - **Wikipedia contributors**, "Moon illusion", "Fraser spiral illusion", "Form constant" and "Hallucination" β€” [Moon illusion](https://en.wikipedia.org/wiki/Moon_illusion){target=_blank}, [Fraser spiral](https://en.wikipedia.org/wiki/Fraser_spiral_illusion){target=_blank}, [Form constant](https://en.wikipedia.org/wiki/Form_constant){target=_blank}, [Hallucination](https://en.wikipedia.org/wiki/Hallucination){target=_blank} πŸ”“ - **Mysid, after James Fraser (1908)**, "Fraser spiral illusion", public domain β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Fraser_spiral.svg){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The identification is **probable**. The syllabus gives the title, the session and the reading. The phrase appears nowhere except Winfree's handout, no document records what was said in class, and his archived site holds related material only, so the examples above are a reconstruction. The candidates: - **Perceptual evidence** (medium confidence): Nature as a conjuror who misleads honest observers, with hallucination as the limiting case. Fits the title, the reading, the section and Winfree's January 2001 lamp; used on this page. - **Chemical spirals to hallucinated spirals** (low confidence): hallucinated spiral forms (Bressloff and colleagues, 2001) resemble the chemical spiral waves Winfree studied (1972). Nothing on his archived pages makes this link. - **Pattern-seeking and coincidence** (low confidence): apophenia and confirmation bias; no Winfree document supports it. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/ --- title: "Rainbow Moon" description: "Winfree's own puzzle: why you have never seen the Moon sitting inside the colour band of a rainbow, when it can happen, and what a failed prediction says about your assumptions." type: Activity tags: [course, student-facing, problem, section-3, rainbow, moon, observation, evidence] status: stable problem: section: 3 session: 15 identification: confident kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-sas08-2002 resource: "https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html" title: "Adventure of the Rainbow Moon (Adventures in Discovery column SAS08, 11 January 2002)" author: "Arthur T. Winfree" - id: winfree-sas09-2002 resource: "https://web.archive.org/web/20030114041305/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS09/SAS09.html" title: "Adventure of the Rainbow Moon: crucial experiment this afternoon in Tucson (SAS09, 25 January 2002)" author: "Arthur T. Winfree" - id: winfree-sas10-2002 resource: "https://web.archive.org/web/20040106042325/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS10/SAS10.html" title: "Moon Violates Reason (SAS10, 8 February 2002)" author: "Arthur T. Winfree" - id: winfree-asd-intro resource: "https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html" title: "About This Column (Adventures in Discovery introduction)" author: "Arthur T. Winfree" - id: sky-telescope-index-2001 resource: "https://archive.org/details/sim_sky-and-telescope_2001_101_index" title: "The Near Sky: A Circumzenithal-Arc-Tinted Moon, Sky & Telescope 101(1): 112 (January 2001), via the 2001 volume index" author: "Fred Schaaf" - id: wikipedia-rainbow resource: "https://en.wikipedia.org/wiki/Rainbow" title: "Rainbow" author: "Wikipedia contributors" - id: wikipedia-lunar-theory resource: "https://en.wikipedia.org/wiki/Lunar_theory" title: "Lunar theory" author: "Wikipedia contributors" - id: wikipedia-moonbow resource: "https://en.wikipedia.org/wiki/Moonbow" title: "Moonbow" author: "Wikipedia contributors" - id: jpl-horizons resource: "https://ssd.jpl.nasa.gov/horizons/app.html" title: "JPL Horizons System" author: "NASA Jet Propulsion Laboratory, Solar System Dynamics" - id: usno-moon-phases resource: "https://aa.usno.navy.mil/data/MoonPhases" title: "Phases of the Moon" author: "US Naval Observatory, Astronomical Applications Department" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Rainbow Moon Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 15. The same session takes up [Mother Nature as magician; hallucinations](https://tyson-swetnam.github.io/aosd/problems/mother-nature-as-magician/index.md) and the last observations for the [chemical pattern-formation lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md).* !!! abstract "The problem" Winfree opened this puzzle with a bet: *"I bet you never saw the Moon in the sky behind the color band of an earthly rainbow. Why not? or have you?"* (A. T. Winfree, "Adventure of the Rainbow Moon", 11 January 2002.) The rest of this box is the editors' paraphrase. 1. **Could it happen?** A rainbow is part of a circle centred on the point opposite the Sun, marked by the shadow of your head. The primary bow lies about 41 degrees from that point, with a colour band about 2 degrees wide; the Moon is half a degree across. Where is the Moon at full moon, and at the quarters? Must it cross the ring? 2. **When?** The Moon drifts about half a degree per hour against the stars. Predict how many hours before or after full moon it sits in the band. The Sun must be up to make the bow, the Moon up to be seen, and there must be rain or spray. What fraction of all time is left? 3. **Go and look**, with a table of full-moon times and a spray bottle, both when a Rainbow Moon is predicted and when it is not. If the prediction fails, ask which quiet assumption is wrong, and design an observation that tests it. ![Diagram. Top: the primary rainbow as a 41-degree ring around the antisolar point, with the Moon inside its colour band. Bottom: the Moon's elongation from the Sun over a month, crossing the band at 139 and 221 degrees, about 81 hours either side of full moon](https://tyson-swetnam.github.io/aosd/assets/images/problems/rainbow-moon.svg){ width="560" } *Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 3 is "Observations and Questions". Session 15 assigns Judson's chapter on *Evidence* beside "Mother Nature as magician; hallucinations", one session after [What Isn't There](https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/index.md). The Rainbow Moon starts from an absence: something you have never seen and would never notice missing. It needs nothing but a spray bottle, yet it goes straight to evidence. A prediction that keeps failing is data about your assumptions, not only about the sky. The schedule dates the discussion to 9 October 2001, though Winfree calls it "a retrospective syllabus of Spring 2001". On either reading the class met the puzzle while it was still open: Winfree's first sighting came by accident on 6 September 2001, his month of measurements began on 18 September, and he did not write any of it up until January 2002. ## Where it comes from The puzzle is Winfree's own. He worked on it through 2001 as one of his daily "Gamesworth" exercises. He then published it in his column "Adventures in Discovery" for the Society for Amateur Scientists: SAS08 and SAS09 in January 2002, with a coda in SAS10 (February 2002). He credits Fred Schaaf in *Sky & Telescope* (January 2001, p. 112) for drawing his attention to it. The magazine's index shows that Schaaf's item was about a circumzenithal arc, which ice crystals make, not a raindrop bow. It is not the moonbow, a rainbow made by moonlight. An earlier draft of this site guessed that. ![A nearly complete ring of rainbow seen from an aircraft above shallow sea, cut off only by the bottom of the frame](https://tyson-swetnam.github.io/aosd/assets/images/problems/rainbow-moon-fullcircle.jpg){ width="560" } *A rainbow is really a circle; from the air you can see almost all of it. Photo by Jakob Owens (Unsplash), March 2017, CC0, via [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Full_Rainbow_%28Unsplash%29.jpg){target=_blank}.* ??? tip "Hints" - Pin down the rainbow first: its centre, its radius, and its band width compared with the Moon. A fingertip at arm's length spans about two degrees. - The full Moon sits at the antisolar point; the quarter Moon is 90 degrees from it. Convert 41 degrees at half a degree per hour into hours. - If your predicted times come up empty, list the assumptions behind them (uniform speed, a circular orbit, a fixed observer) and find a simple daily measurement that tests each one. ## What happened It can happen. The Moon crosses the ring twice a month, roughly 81 hours before and after full moon, and stays in the band about four hours. Both the Sun and the Moon must be up, and with 139 degrees between them one of them sets within about three hours. Winfree put the chance at about one thirty-second of one percent of all time, before even asking for rain. From January to August 2001 he watched at his predicted times and never saw one. Then at 07:33 on 6 September 2001, in Tucson, he spotted one by accident, 89 hours after full moon: eight hours late. He had been trying to confirm predictions, he concluded, when he should have used observation to test the assumptions behind them. From 18 September he measured the Moon's angle from the Sun, the antisolar point or a star on every clear day for a month. He used a plastic sextant and a cross-staff he had built himself. He found: 1. The Moon's speed is not uniform: it ran up to 7-8 degrees ahead of or behind schedule, enough to shift a Rainbow Moon by up to 15 hours. His 8-hour miss in September was well within that. 2. Its apparent diameter changed by about one part in eight, so the orbit is not a circle centred on Earth. Angular speed divided by diameter squared stayed roughly constant, as Kepler's second law predicts; the JPL Horizons ephemeris confirmed it. 3. A daily wobble of about one degree is parallax from standing on a rotating Earth. Behind all this lies the Sun's uneven pull on Earth and Moon, the three-body problem that troubled Newton. His calculations also cleared up an older sighting. At 4 PM on 6 March 2001 he thought he had seen a Rainbow Moon, but the Moon was then only 39 degrees from the antisolar point, too close for a water bow. That day his spray bottle held mostly alcohol. Alcohol bends light more strongly than water, so, he reasoned, the bow shrank enough to reach the Moon. Using Horizons, he listed eight dates in 2002 when Tucson could see a Rainbow Moon. He tried the first on 25 January 2002, in the Catalina Foothills. The Moon cleared a ridge at 15:12. He made a bow with a garden hose, checked with the sextant that it stood 41 degrees from his head's shadow, and photographed the scene. At 15:46 the Moon was about 39 degrees from the antisolar point (Horizons gave 39.6): about four Moon diameters inside the main band. The timing was marginal; two hours earlier would have been perfect, but the Moon had not yet risen. In the spray, a fainter bow just inside the main one crossed the Moon. He also caught a software error: his planetarium program ignored parallax, worth up to two hours here. ## Sources - **Arthur T. Winfree**, "Adventure of the Rainbow Moon", *Adventures in Discovery* column SAS08, Society for Amateur Scientists E-Bulletin (11 January 2002) β€” [archived copy](https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html){target=_blank} πŸ”“ (copyright Winfree; paraphrased here) - **Arthur T. Winfree**, "Adventure of the Rainbow Moon: crucial experiment this afternoon in Tucson", SAS09 (25 January 2002) β€” [archived copy](https://web.archive.org/web/20030114041305/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS09/SAS09.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, "Moon Violates Reason", SAS10 (8 February 2002) β€” [archived copy](https://web.archive.org/web/20040106042325/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS10/SAS10.html){target=_blank} πŸ”“ (the 25 January 2002 observation) - **Arthur T. Winfree**, "About This Column", introduction to *Adventures in Discovery* β€” [archived copy](https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html){target=_blank} πŸ”“ - **Fred Schaaf**, "The Near Sky: A Circumzenithal-Arc-Tinted Moon", *Sky & Telescope* 101(1): 112 (January 2001), as listed in the 2001 volume index β€” [Internet Archive](https://archive.org/details/sim_sky-and-telescope_2001_101_index){target=_blank} πŸ”“ (the article itself was not read) - **Wikipedia contributors**, "Rainbow" β€” [Wikipedia](https://en.wikipedia.org/wiki/Rainbow){target=_blank} πŸ”“ - **Wikipedia contributors**, "Lunar theory" β€” [Wikipedia](https://en.wikipedia.org/wiki/Lunar_theory){target=_blank} πŸ”“ - **Wikipedia contributors**, "Moonbow" β€” [Wikipedia](https://en.wikipedia.org/wiki/Moonbow){target=_blank} πŸ”“ (for contrast) - **NASA Jet Propulsion Laboratory**, Horizons System β€” [ssd.jpl.nasa.gov](https://ssd.jpl.nasa.gov/horizons/app.html){target=_blank} πŸ”“ (phase angle for your own site) - **US Naval Observatory**, Phases of the Moon β€” [aa.usno.navy.mil](https://aa.usno.navy.mil/data/MoonPhases){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/escher-print-gallery/ --- title: "Escher Print Gallery" description: "Loops that pass inspection piece by piece: Escher's Print Gallery, which closes on itself around a blank centre, and the Penrose impossible triangle and staircase, whose error lives in no single corner." type: Activity tags: [course, student-facing, problem, section-3, escher, impossible-figures, visual-perception, self-reference] status: stable problem: section: 3 session: 16 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: escher-foundation-print-gallery resource: "https://mcescher.com/gallery/recognition-success/" title: "Print Gallery, May 1956, lithograph (Recognition and Success gallery)" author: "M.C. Escher Foundation / The M.C. Escher Company" - id: escher-foundation-impossible resource: "https://mcescher.com/gallery/impossible-constructions/" title: "Impossible constructions (gallery): Relativity, Belvedere, Ascending and Descending, Waterfall" author: "M.C. Escher Foundation / The M.C. Escher Company" - id: de-smit-lenstra-2003 resource: "https://web.archive.org/web/20030412011949/http://www.ams.org:80/notices/200304/fea-escher.pdf" title: "Artful Mathematics: The Heritage of M. C. Escher (includes \"The Mathematical Structure of Escher's Print Gallery\")" author: "B. de Smit and H. W. Lenstra Jr. (and others)" - id: de-smit-2005 resource: "https://pub.math.leidenuniv.nl/~smitbde/papers/bridges-2005-desmit.pdf" title: "The Droste-effect and the exponential transform" author: "Bart de Smit" - id: ernst-1976 resource: "https://archive.org/details/magicmirrorofmce0000erns" title: "The Magic Mirror of M. C. Escher" author: "Bruno Ernst (trans. John E. Brigham)" - id: schattschneider-2010 resource: "https://web.archive.org/web/20210415033924/http://www.ams.org/notices/201006/rtx100600706p.pdf" title: "The Mathematical Side of M. C. Escher" author: "Doris Schattschneider" - id: penrose-penrose-1958 resource: "https://doi.org/10.1111/j.2044-8295.1958.tb00634.x" title: "Impossible objects: a special type of visual illusion" author: "L. S. Penrose and R. Penrose" - id: penrose-1993-visual-mind resource: "https://archive.org/details/visualmindartmat0000unse" title: "The Visual Mind: Art and Mathematics (chapter 5, \"On the Cohomology of Impossible Figures\")" author: "Michele Emmer (ed.); chapter by Roger Penrose" - id: wikipedia-penrose-stairs resource: "https://en.wikipedia.org/wiki/Penrose_stairs" title: "Penrose stairs" author: "Wikipedia contributors" - id: commons-impossible-staircase resource: "https://commons.wikimedia.org/wiki/File:Impossible_staircase.svg" title: "Impossible staircase (SVG, public domain)" author: "Sakurambo" - id: penrose-1979 resource: "https://doi.org/10.1111/j.1469-1809.1979.tb00677.x" title: "The topology of ridge systems" author: "R. Penrose" - id: strogatz-2003 resource: "https://archive.org/details/syncemergingscie00stro" title: "Sync: The Emerging Science of Spontaneous Order" author: "Steven Strogatz" - id: winfree-1987-when-time-breaks-down resource: "https://archive.org/details/whentimebreaksdo0000winf" title: "When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Escher Print Gallery Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 16. Discussed together with [Hairy People](https://tyson-swetnam.github.io/aosd/problems/hairy-people/index.md) and [Green Stars](https://tyson-swetnam.github.io/aosd/problems/green-stars/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's write-up is lost, so these questions are the editors', not his. Part 1 follows the title. Part 2 follows a clue that points to Roger Penrose (see the box at the end of the page). **Part 1. Print Gallery.** Find M. C. Escher's lithograph *Print Gallery* (1956) on the [Escher Foundation's gallery page](https://mcescher.com/gallery/recognition-success/){target=_blank} (in copyright, so not reproduced here). 1. A young man at the lower left looks at a print of a harbour town. Follow the town's buildings round the picture. Where do you end up? 2. Do things get bigger or smaller as you go round? How can the picture still return to the same gallery? 3. The middle is a blank disc carrying only Escher's monogram and signature. Is it unfinished, or could it not be drawn? What would have to be shown just outside it? **Part 2. Impossible figures.** The first figure below shows three bars of cubes, each meeting the next at a right angle. 1. Cover all but one corner. Is anything wrong? Try the other corners. 2. Could you build it from wood? If not, where is the impossibility: in a corner, in a bar, or somewhere else? 3. The second figure is a staircase in four flights. Walk round it one way: every step goes down, yet you come back to where you began. Is it the same trick as the triangle? Is Print Gallery's loop impossible in the same sense? ![An isometric drawing of an impossible triangle made of twelve grey cubes: a vertical column on the left, a bar sloping down to the right from its top, and a bar sloping up from its foot to meet it, with corners labelled A, B and C](https://tyson-swetnam.github.io/aosd/assets/images/problems/escher-print-gallery.svg){ width="560" } *Every corner is an ordinary right-angled joint. Drawn for this site (CC BY 4.0).* ![The Penrose stairs: four flights of steps running round a rectangular block and joining up with themselves](https://tyson-swetnam.github.io/aosd/assets/images/problems/escher-print-gallery-staircase.svg){ width="560" } *An impossible staircase. Sakurambo (2005), public domain, via [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Impossible_staircase.svg){target=_blank}; white background added.* ## Why it is in the course Section 3 is "Observations and Questions". Session 16 reads "Discuss Hairy People, Green Stars, and Escher Print Gallery"; two sessions earlier the class had to "Discuss [What Isn't There](https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/index.md) (surprisingly hard)". The rest is the editors' reading. Print Gallery fits that theme closely: its one blank patch is where the whole picture's logic comes to a point. The impossible figures test looking in another way. Every local piece passes inspection, and the trouble shows only when you follow the whole loop. Science has the same trap. Checking each step of an argument is not checking that the steps close up consistently. Escher also comes up around Winfree's research, though nothing links these passages to the course item. In *When Time Breaks Down* (1987) he compared a scene of rotating waves to an Escher drawing of a waterfall. Steven Strogatz recalls in *Sync* (2003) that when Winfree sketched a twisted scroll ring by hand, he "accidentally produced a nonsense picture in the style of Escher". ## Where it comes from **Print Gallery** (*Prentententoonstelling*) is a lithograph of May 1956. Escher, quoted by de Smit and Lenstra from Bruno Ernst, wanted a "cyclic expansion...without beginning or end". His four straight preparatory sketches together make a picture that contains itself at 1/256 of the size. He then switched to a curved grid that closes on itself, expanded 256 times, as you go once clockwise round the centre (de Smit and Lenstra, 2003; de Smit, 2005). De Smit and Lenstra published the exact mathematics of the print in 2003, after Winfree's course. **The Penrose figures.** Roger Penrose saw Escher's prints at an exhibition held for the 1954 International Congress of Mathematicians in Amsterdam (Schattschneider, 2010). He then drew the tribar, three perpendicular bars that seem to form a triangle, and his father L. S. Penrose devised an endless staircase; they published both in 1958. Schattschneider says Penrose sent Escher the sketches, while Escher's 1960 letter to the Penroses says a friend sent him a photocopy of their article. Escher used the figures in *Ascending and Descending* (1960) and *Waterfall* (1961). ??? tip "Hints" - Start from the young man, not the hole. Follow his print until you are standing *in* it. - If each trip round enlarges things by the same factor, what happens running backwards, inwards? - For the triangle, cover all but one corner, then slide the cover on. - For each bar, note which end is nearer to you. Compare your first answer with your last. For the staircase, track height. ??? success "Resolution" **Print Gallery.** The buildings grow until one of them is the young man's gallery: the loop closes, by distortion rather than contradiction. Run backwards, the scene repeats ever smaller inwards: de Smit and Lenstra showed that the idealised picture contains a copy of itself turned clockwise about 157.6 degrees and shrunk about 22.58 times, and a copy of that copy, and so on towards a single point at the centre. Infinitely many copies would have to fit there, so that point cannot be drawn. Escher's blank disc is larger than the point: he did not carry his grid in towards the middle. **The triangle.** A drawing never says how far away each part is, so the eye decides at each joint. Here B is nearer than A, C nearer than B, and A nearer than C: A is nearer than itself. No corner is wrong; the impossibility belongs to the loop. (Penrose later made this precise with cohomology.) **The staircase.** Round a closed path the height changes must total zero, yet walked one way every flight goes down. Triangle and staircase both break a quantity that must return to its start. Print Gallery lets size change steadily instead, and pays with a centre that cannot be drawn. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579) course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **M.C. Escher Foundation**, *Print Gallery*, May 1956, lithograph (Recognition and Success gallery) β€” [mcescher.com](https://mcescher.com/gallery/recognition-success/){target=_blank} πŸ”“ - **M.C. Escher Foundation**, Impossible constructions gallery: *Relativity*, *Ascending and Descending*, *Waterfall* β€” [mcescher.com](https://mcescher.com/gallery/impossible-constructions/){target=_blank} πŸ”“ - **B. de Smit and H. W. Lenstra Jr.**, "The Mathematical Structure of Escher's Print Gallery", in "Artful Mathematics: The Heritage of M. C. Escher", *Notices of the AMS* 50(4), 446–451 (2003) β€” [PDF, Wayback Machine](https://web.archive.org/web/20030412011949/http://www.ams.org:80/notices/200304/fea-escher.pdf){target=_blank} πŸ”“ - **Bart de Smit**, "The Droste-effect and the exponential transform", *Bridges Proceedings* 2005, 169–178 β€” [PDF](https://pub.math.leidenuniv.nl/~smitbde/papers/bridges-2005-desmit.pdf){target=_blank} πŸ”“ - **Bruno Ernst**, *The Magic Mirror of M. C. Escher* (1976) β€” [Internet Archive](https://archive.org/details/magicmirrorofmce0000erns){target=_blank} πŸ”“ *(borrow)* (Escher's words quoted above were read as quoted by de Smit and Lenstra) - **Doris Schattschneider**, "The Mathematical Side of M. C. Escher", *Notices of the AMS* 57(6), 706–718 (2010) β€” [PDF, Wayback Machine](https://web.archive.org/web/20210415033924/http://www.ams.org/notices/201006/rtx100600706p.pdf){target=_blank} πŸ”“ - **L. S. Penrose and R. Penrose**, "Impossible objects: a special type of visual illusion", *British Journal of Psychology* 49(1), 31–33 (1958) β€” [doi:10.1111/j.2044-8295.1958.tb00634.x](https://doi.org/10.1111/j.2044-8295.1958.tb00634.x){target=_blank} πŸ”’ (not read by the editors; its contents are reported from Schattschneider) - **Roger Penrose**, "On the Cohomology of Impossible Figures", in Michele Emmer (ed.), *The Visual Mind: Art and Mathematics* (MIT Press, 1993) β€” [Internet Archive](https://archive.org/details/visualmindartmat0000unse){target=_blank} πŸ”“ *(borrow)* - **Wikipedia contributors**, "Penrose stairs" β€” [Wikipedia](https://en.wikipedia.org/wiki/Penrose_stairs){target=_blank} πŸ”“ (Escher's 1960 letter to the Penroses) - **Sakurambo**, "Impossible staircase" (2005), public domain β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Impossible_staircase.svg){target=_blank} πŸ”“ - **R. Penrose**, "The topology of ridge systems", *Annals of Human Genetics* 42, 435–444 (1979) β€” [doi:10.1111/j.1469-1809.1979.tb00677.x](https://doi.org/10.1111/j.1469-1809.1979.tb00677.x){target=_blank} πŸ”’ (not read by the editors; listed for the rival reading) - **Steven Strogatz**, *Sync: The Emerging Science of Spontaneous Order* (2003) β€” [Internet Archive](https://archive.org/details/syncemergingscie00stro){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *When Time Breaks Down* (1987) β€” [Internet Archive](https://archive.org/details/whentimebreaksdo0000winf){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the title. In Winfree's handout, the link on this item points to a bookmark named `Escher_Penrose`; its target was never archived. The link text names the print and the bookmark adds Penrose, so the reconstruction pairs them. A bookmark name shows a pairing of topics, not the exercise, so identification is probable. Candidates: - **Print Gallery as an observation puzzle** (strongest): fits the title and session 14, but not the word Penrose. - **Penrose's impossible figures**, likely alongside Print Gallery: the only reading that explains the bookmark. - **Penrose's 1962 tiling puzzle for Escher**: unlinked to Print Gallery. - **Penrose's fingerprint-ridge topology** (1979), cited by Winfree: unlinked to Escher. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/green-stars/ --- title: "Green Stars" description: "Stars look red, orange, yellow, white or blue, and the Sun's light peaks in the green, yet nobody sees a green star: a question about noticing an absence and about whether the answer lies in the stars or in the eye." type: Activity tags: [course, student-facing, problem, section-3, astronomy, colour-vision, perception, black-body-radiation] status: stable problem: section: 3 session: 16 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: astronomy-2007 resource: "https://www.astronomy.com/science/why-are-there-no-green-stars/" title: "Why are there no green stars?" author: "Astronomy magazine staff" - id: plait-cull-2008 resource: "https://www.discovermagazine.com/the-sciences/why-are-there-no-green-stars" title: "Why Are There No Green Stars?" author: "Phil Plait and Monica Cull" - id: wikipedia-planckian-locus resource: "https://en.wikipedia.org/wiki/Planckian_locus" title: "Planckian locus" author: "Wikipedia contributors" - id: wikipedia-stellar-classification resource: "https://en.wikipedia.org/wiki/Stellar_classification" title: "Stellar classification" author: "Wikipedia contributors" - id: wikipedia-beta-librae resource: "https://en.wikipedia.org/wiki/Beta_Librae" title: "Beta Librae" author: "Wikipedia contributors" - id: minnaert-1954 resource: "https://archive.org/details/bwb_W9-CRB-649" title: "The Nature of Light and Colour in the Open Air" author: "M. Minnaert" - id: winfree-sas02-2001 resource: "https://web.archive.org/web/20030114043916/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS02/SAS02.html" title: "Magnitudes of bright lights in the sky, Part 1 (Adventures in Discovery, 16 November 2001)" author: "Arthur T. Winfree" - id: winfree-sas03-2001 resource: "https://web.archive.org/web/20030214222857/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS03/SAS03.html" title: "Little Discoveries Using Stellar Magnitudes, Part 2 (Adventures in Discovery, 23 November 2001)" author: "Arthur T. Winfree" - id: winfree-sas08-2002 resource: "https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html" title: "Adventure of the Rainbow Moon (Adventures in Discovery, 11 January 2002)" author: "Arthur T. Winfree" - id: commons-planckian-locus resource: "https://commons.wikimedia.org/wiki/File:PlanckianLocus.png" title: "PlanckianLocus.png (CIE 1931 chromaticity diagram with the Planckian locus)" author: "en:User:PAR" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Green Stars Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 16. Discussed together with [Hairy People](https://tyson-swetnam.github.io/aosd/problems/hairy-people/index.md) and [Escher Print Gallery](https://tyson-swetnam.github.io/aosd/problems/escher-print-gallery/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: only the title survives, so this is the standard question the title most likely refers to. On a clear night the brightest stars show colours: Betelgeuse reddish-orange, Arcturus orange, Capella yellow-white, Vega white, Rigel bluish. A star's colour is set by its surface temperature, from about 3,000 K (red) through the Sun's 5,800 K to 30,000 K and more (blue), and the peak of its broad spectrum slides from the infrared through red, yellow and green toward the ultraviolet as it gets hotter. Now the observation: **nobody ever sees a green star.** Yet the Sun's own spectrum peaks in the green, near 500 nm. (That is per unit wavelength; plotted per unit frequency, the same spectrum peaks in the infrared. Worth a minute's thought.) 1. Why does no star look green? Is the reason in the stars, or in us? 2. Would a star whose spectrum peaked exactly in the green look green through a telescope? Through a green filter? To a camera? 3. Sky guides sometimes call Beta Librae "greenish". Is that colour real? How would you test it? Answer from what you already know about light and colour before looking anything up. You may have looked at the night sky for years without asking why one colour is missing from it. ![Three black-body spectra from 300 to 1100 nm with the visible band from 400 to 700 nm shaded: the 3000 K curve rises steeply toward the red, the 12000 K curve falls toward the red, and the Sun's 5800 K curve peaks at 500 nm but stays between about 0.8 and 1 across the whole visible band](https://tyson-swetnam.github.io/aosd/assets/images/problems/green-stars.svg){ width="560" } *Drawn for this site (CC BY 4.0). Planck's law per unit wavelength, each curve scaled to its own peak.* ## Why it is in the course Section 3 is "Observations and Questions". On the editors' reading, all three problems of session 16 ask about something that is not there, continuing session 14's discussion of what isn't there ("surprisingly hard"). In a later column Winfree noted that some things that never happen are easy to notice and others are not ([SAS08](https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html){target=_blank}). A colour missing from the night sky is the hard kind. The question needs only the night sky and ordinary experience of colour. It has a tempting wrong answer ("no star has that temperature") and a right one that means questioning the observer as well as the observed. And it keeps a fact apart from its explanation, the habit session 04 calls "facts before explanations of facts". ## Where it comes from Star colours have been noticed since antiquity, and nineteenth-century double-star observers recorded some companions as green. The physics came with Planck's law of black-body radiation (1900) and the Harvard sequence of spectral classes, O B A F G K M, given its modern form by Annie Jump Cannon and Edward Pickering from 1901. "Why are there no green stars?" became a standard teaching question, answered in *Astronomy* magazine (2007) and on Phil Plait's Bad Astronomy blog (2008). The syllabus schedule is the Spring 2001 course with its dates moved to Fall 2001, where session 16 falls on Thursday 11 October. About a month later, in November 2001, Winfree's columns worked out stellar magnitudes from everyday observations, cited Minnaert's book on light and colour, and picked out Sun-like stars by their colour. They show his interest in naked-eye astronomy at the time; they are not material the class read. ??? tip "Hints" - Split the question in two: what a star emits, and what an eye does with it. Which half is the puzzle really about? - A star is not a one-colour lamp. How much of a "green-peaked" star's light is red and blue? - What colour does a roughly equal mix of red, green and blue light look like? What colour is the Sun at noon? - Heat a body from a dull red glow to a welding arc. Does its colour ever pass through green on the way from red to blue? - For stars reported as green: what colour usually sits beside them in the eyepiece, and how does the eye judge colour in dim light? ??? success "Resolution" A star radiates roughly as a black body: a smooth, very broad spectrum. Even with its peak at 500 nm, the Sun gives out nearly as much red and blue light as green; across the visible band its spectrum varies by only about a fifth. The eye's three kinds of colour receptor respond to the whole mixture, not to the peak, and a balanced mixture looks white. Cooler stars send out more red and look orange or red; hotter stars send out more blue and look blue-white. No temperature tips the balance toward green alone: on a chromaticity diagram, the curve of black-body colours (the Planckian locus) runs from red through white to pale blue and never enters the green region. So green-peaked stars are common; the Sun is one. The missing colour is a fact about broad spectra and human vision, not about the stars. Only a narrow spectrum, such as a green laser or a filtered beam, can look green. That answers part 2. A telescope gathers more light but does not change the mixture, so a green-peaked star still looks white. A green filter throws away the red and blue, so the star then looks green, as any white light would. A colour camera, like the eye, records the mixture, not the peak. ![CIE 1931 chromaticity diagram: the black curve of black-body colours runs from red through orange and white to pale blue, well below the green region at the top](https://tyson-swetnam.github.io/aosd/assets/images/problems/green-stars-locus.png){ width="560" } *CIE 1931 chromaticity diagram with the Planckian locus. PlanckianLocus.png by en:User:PAR, Wikimedia Commons, public domain.* The "green" stars of the sky guides are perceptual. Beta Librae is a B-type star near 11,900 K that most observers today call white or blue-white; green companions of double stars are usually white or blue stars beside a bright orange primary. Why some observers still see Beta Librae as green has never been settled, so part 3 stays open. ## Sources - **Astronomy magazine staff**, "Why are there no green stars?", *Astronomy* (1 April 2007) β€” [astronomy.com](https://www.astronomy.com/science/why-are-there-no-green-stars/){target=_blank} πŸ”“ - **Phil Plait and Monica Cull**, "Why Are There No Green Stars?", Bad Astronomy (2008, updated 2023) β€” [Discover](https://www.discovermagazine.com/the-sciences/why-are-there-no-green-stars){target=_blank} πŸ”“ - **Wikipedia contributors**, "Stellar classification" β€” [Wikipedia](https://en.wikipedia.org/wiki/Stellar_classification){target=_blank} πŸ”“ (the spectral sequence, Cannon and Pickering, and the absence of green stars in typical viewing) - **Wikipedia contributors**, "Planckian locus" β€” [Wikipedia](https://en.wikipedia.org/wiki/Planckian_locus){target=_blank} πŸ”“ (the path of black-body colours; its figure is reproduced above) - **Wikipedia contributors**, "Beta Librae" β€” [Wikipedia](https://en.wikipedia.org/wiki/Beta_Librae){target=_blank} πŸ”“ (temperature and the green reports) - **M. Minnaert**, *The Nature of Light and Colour in the Open Air* (Dover, 1954) β€” [Internet Archive](https://archive.org/details/bwb_W9-CRB-649){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, "Magnitudes of bright lights in the sky, Part 1", Adventures in Discovery (16 November 2001) β€” [Wayback Machine](https://web.archive.org/web/20030114043916/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS02/SAS02.html){target=_blank} πŸ”“ (related material; does not state this problem) - **Arthur T. Winfree**, "Little Discoveries Using Stellar Magnitudes, Part 2", Adventures in Discovery (23 November 2001) β€” [Wayback Machine](https://web.archive.org/web/20030214222857/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS03/SAS03.html){target=_blank} πŸ”“ (related material; does not state this problem) - **Arthur T. Winfree**, "Adventure of the Rainbow Moon", Adventures in Discovery (11 January 2002) β€” [Wayback Machine](https://web.archive.org/web/20030114040501/http://eebweb.arizona.edu/faculty/winfree/SAS/SAS08/SAS08.html){target=_blank} πŸ”“ - **en:User:PAR**, "PlanckianLocus.png" β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:PlanckianLocus.png){target=_blank} πŸ”“ (public domain) - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Probable, not certain. The syllabus gives only the title. The archived handout links it to a companion problem document that was never captured, so no Winfree text of the problem survives. The identification rests on the title, which matches a standard astronomy question, on Winfree's documented naked-eye astronomy, on the session's place in "Observations and Questions", and on the theme of absence that the editors (not Winfree) see in all three session-16 problems. The candidates: - **Why no star looks green** (most likely): the question above. - **The stars reported as green**, such as Beta Librae: probably part of the same discussion rather than a separate problem. - **An unrelated puzzle called "Green Stars"**: the editors found none, though no exhaustive search was possible. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/hairy-people/ --- title: "Hairy People" description: "Must two people in Tucson have exactly the same number of hairs on their heads? A four-century-old wager that proves a fact nobody could ever observe, and shows why the whole proof rests on an honest upper bound." type: Activity tags: [course, student-facing, problem, section-3, pigeonhole-principle, counting, estimation, proof] status: stable problem: section: 3 session: 16 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: rittaud-heeffer-2014 resource: "https://biblio.ugent.be/publication/4115264/file/4115265.pdf" title: "The Pigeonhole Principle, Two Centuries Before Dirichlet" author: "Benoit Rittaud and Albrecht Heeffer" - id: rittaud-heeffer-2014-doi resource: "https://doi.org/10.1007/s00283-013-9389-1" title: "The Pigeonhole Principle, Two Centuries Before Dirichlet (publisher version)" author: "Benoit Rittaud and Albrecht Heeffer" - id: recreation-mathematique-1624 resource: "https://archive.org/details/recreationsmathe00unse" title: "Recreations mathematiques, composees de plusieurs problemes plaisans & facetieux (1629 Rouen printing)" author: "Published under the name H. van Etten; long attributed to Jean Leurechon, and by Heeffer to Jean Appier Hanzelet" - id: leurechon-selectae-1622 resource: "https://archive.org/details/bub_gb_RGwTAAAAQAAJ" title: "Selectae propositiones in tota sparsim mathematica pulcherrimae (1629 printing)" author: "Jean Leurechon" - id: wikipedia-pigeonhole resource: "https://en.wikipedia.org/wiki/Pigeonhole_principle" title: "Pigeonhole principle" author: "Wikipedia contributors" - id: wikipedia-leurechon resource: "https://en.wikipedia.org/wiki/Jean_Leurechon" title: "Jean Leurechon" author: "Wikipedia contributors" - id: wikipedia-hair resource: "https://en.wikipedia.org/wiki/Hair" title: "Hair" author: "Wikipedia contributors" - id: wikipedia-tucson resource: "https://en.wikipedia.org/wiki/Tucson,_Arizona" title: "Tucson, Arizona" author: "Wikipedia contributors" - id: wikipedia-hairy-ball resource: "https://en.wikipedia.org/wiki/Hairy_ball_theorem" title: "Hairy ball theorem" author: "Wikipedia contributors" - id: mathworld-hairy-ball resource: "https://mathworld.wolfram.com/HairyBallTheorem.html" title: "Hairy Ball Theorem" author: "Eric W. Weisstein, MathWorld" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Hairy People Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 16. Discussed together with [Green Stars](https://tyson-swetnam.github.io/aosd/problems/green-stars/index.md) and [Escher Print Gallery](https://tyson-swetnam.github.io/aosd/problems/escher-print-gallery/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree left only the name, so the wording, and the choice of Tucson, are the editors'. **The wager.** Someone bets you that, right now, at least two people in Tucson have *exactly* the same number of hairs on their heads. Nobody will count, and you may not pick the two. Should you take the bet? Can you prove or refute the claim from your armchair? 1. What is the most hairs any head could carry? Estimate it, add a safety margin, and write the number down. 2. Look up Tucson's population: the city, or the county? Does your margin still leave you an argument? 3. Try other crowds: the world, a class of twenty. How many people must share one hair count? 4. Hunt for booby traps: bald people, falling hairs, and not knowing *which* two match. ![Seven people above a row of six boxes labelled 0 hairs, 1 hair, 2 hairs, a dashed box standing for the omitted boxes, M minus 1 hairs and M hairs; one arrow from each person to a box, and two arrows land in the box for 2 hairs](https://tyson-swetnam.github.io/aosd/assets/images/problems/hairy-people.svg){ width="560" } *Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 3 is "Observations and Questions". The hair wager states a fact no observation could settle, since nobody can count every head in a city, yet a short argument makes it certain. That sharpens the session 04 theme of [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md), "distinguishing things we know vs only imagine". In Winfree's words, "The purpose of the puzzles (many of them silly) is to slow you down for a few minutes so you can examine the working of your own mind." It is also a lesson about evidence. Session 15 assigned Judson's chapter on *Evidence*. Here the only evidence is your own estimate of a head of hair, and any argument you build is only as good as that estimate. ## Where it comes from The earliest known printed form is one Latin sentence in the Jesuit Jean Leurechon's *Selectae Propositiones* (1622). A French book of recreations of 1624, published under the name H. van Etten, gave a full proof headed, in Rittaud and Heeffer's translation, "That it is absolutely necessary that two men have as many hairs or pistoles as the other." Long credited to Leurechon, the book was written, Heeffer argues, by Jean Appier Hanzelet. In 1737 Castel de Saint-Pierre printed Pierre Nicole's story of offering the wager about Paris to Madame de Longueville. In Rittaud and Heeffer's translation Nicole recalls: "She told me, I could never be sure of it until I had counted the hairs of these two men." The principle's usual name, the drawer principle, comes from Dirichlet, two centuries later. ![Title page of RΓ©crΓ©ations mathΓ©matiques, Lyon, Claude Prost, 1642](https://tyson-swetnam.github.io/aosd/assets/images/problems/hairy-people-title-page.jpg){ width="560" } *Title page of RΓ©crΓ©ations mathΓ©matiques (Lyon, 1642), a later edition of the 1624 book, credited on Commons to Jean Leurechon. Scan by the Biblioteca Europea di Informazione e Cultura (BEIC), via Wikimedia Commons. Public domain.* ??? tip "Hints" - Write the argument with a letter M for the most hairs any head could have. How large may M be before the argument fails for your city? - Count the possible answers to "how many hairs are on your head?": 0, 1, 2, ... up to M. Compare with the number of people answering. - Picture one box per hair count and drop each person in. What if there are more people than boxes? - You never find the matching pair. Does that make the claim less certain, or only less observable? ??? success "Resolution" **The argument.** If no head has more than M hairs, the possible counts are 0, 1, ..., M: M + 1 boxes. With more than M + 1 people, two share a box. This is the pigeonhole principle. **Choosing M is the whole problem.** Quoted counts for a full scalp run from about 90,000 hairs (redheads) to 150,000 (blondes), so M = 300,000 is defensible with room to spare. **Tucson.** At the 2000 census the city had 486,699 people and Pima County 843,746. Against 300,001 boxes the city wins by a factor of about 1.6, so the wager is safe. But the textbook London version uses one million as a "safe" bound. Adopt it here and the argument collapses: 1,000,001 boxes can hold every Tucsonan one apiece, and even the county cannot fill them. A bound must be both true and small. **Other crowds.** For a class of twenty no honest bound works. If P people fill M + 1 boxes, some box holds at least P / (M + 1), rounded up: with about 6.1 billion people (2001) and M = 300,000, at least 20,334 people on Earth share one hair count. **Booby traps.** Bald heads help: they crowd box 0. Falling hairs change *which* people match, never whether someone does. And the argument never names the pair: certainty and observability are different things. ## Sources - **Benoit Rittaud and Albrecht Heeffer**, "The Pigeonhole Principle, Two Centuries Before Dirichlet", *The Mathematical Intelligencer* 36(2), 27–29 (2014) β€” [authors' version, Ghent University](https://biblio.ugent.be/publication/4115264/file/4115265.pdf){target=_blank} πŸ”“; [publisher version](https://doi.org/10.1007/s00283-013-9389-1){target=_blank} πŸ”’ (source of the translations quoted above and of the Nicole anecdote) - **H. van Etten** (attributed to Jean Leurechon; by Heeffer to Jean Appier Hanzelet), *Recreations mathematiques* (1624; 1629 Rouen printing) β€” [Internet Archive](https://archive.org/details/recreationsmathe00unse){target=_blank} πŸ”“ - **Jean Leurechon**, *Selectae propositiones in tota sparsim mathematica pulcherrimae* (1622; 1629 printing) β€” [Internet Archive](https://archive.org/details/bub_gb_RGwTAAAAQAAJ){target=_blank} πŸ”“ (the hairs sentence could not be found in this scan's poor OCR; it is reported from Rittaud and Heeffer) - **Wikipedia contributors**, "Pigeonhole principle" β€” [Wikipedia](https://en.wikipedia.org/wiki/Pigeonhole_principle){target=_blank} πŸ”“ (the London version with its million-hair bound) - **Wikipedia contributors**, "Jean Leurechon" β€” [Wikipedia](https://en.wikipedia.org/wiki/Jean_Leurechon){target=_blank} πŸ”“ - **Wikipedia contributors**, "Hair" β€” [Wikipedia](https://en.wikipedia.org/wiki/Hair){target=_blank} πŸ”“ (hairs per head by hair colour) - **Wikipedia contributors**, "Tucson, Arizona" β€” [Wikipedia](https://en.wikipedia.org/wiki/Tucson,_Arizona){target=_blank} πŸ”“ (2000 census figures) - **Wikipedia contributors**, "Hairy ball theorem" β€” [Wikipedia](https://en.wikipedia.org/wiki/Hairy_ball_theorem){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Hairy Ball Theorem", MathWorld β€” [MathWorld](https://mathworld.wolfram.com/HairyBallTheorem.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery* course handout (archived 20 April 2002) β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus, like the archived 2002 course handout, gives only the name and the session: "Discuss Hairy People, Green Stars, and Escher Print Gallery". The name appears on no other archived Winfree page. The identification is therefore **probable**, resting on the name and on the syllabus's note that the exercises are "mostly made from elementary mathematics so as to require no lab setup". Candidates: - **The hair-counting wager** (strongest): fits the name exactly, and the syllabus later schedules a "resolution of wagers" (session 26). - **The hairy ball theorem applied to scalps** (weaker): singularities were Winfree's research subject, and Escher's Print Gallery also turns on one. But the name points at people, not a theorem, and a scalp is not a whole sphere, so the theorem does not force a whorl there. - **A Fermi estimate** of hairs on a head: the wager's first step. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/martian-honeycombs/ --- title: "Martian HoneyCombs" description: "Winfree's statement is lost; this reconstruction asks which features of a bee's honeycomb a Martian comb would have to share, and so which of our honeycomb 'facts' rest on mathematics, physics, Earth biology, or nothing measured at all." type: Activity tags: [course, student-facing, problem, section-3, honeycomb, geometry, observation, thought-experiment] status: stable problem: section: 3 session: 17 identification: unknown kind: thought-experiment generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: thompson-1917 resource: "https://archive.org/details/ongrowthform1917thom" title: "On Growth and Form (1st ed.), ch. VII, section on the bee's cell" author: "D'Arcy Wentworth Thompson" - id: darwin-1859 resource: "https://www.gutenberg.org/ebooks/1228" title: "On the Origin of Species (1st ed.), chapter on Instinct: cell-making instinct of the hive-bee" author: "Charles Darwin" - id: hales-2001 resource: "https://arxiv.org/abs/math/9906042" title: "The Honeycomb Conjecture" author: "Thomas C. Hales" - id: fejes-toth-1964 resource: "https://doi.org/10.1090/S0002-9904-1964-11155-1" title: "What the bees know and what they do not know" author: "Laszlo Fejes Toth" - id: wikipedia-honeycomb resource: "https://en.wikipedia.org/wiki/Honeycomb" title: "Honeycomb" author: "Wikipedia contributors" - id: msss-polygons-2002 resource: "https://web.archive.org/web/20161027001241/http://www.msss.com/mars_images/moc/polygons_5_02/" title: "Southern Hemisphere Polygonal Patterned Ground (MGS MOC Release MOC2-315)" author: "Malin Space Science Systems / NASA JPL" - id: ehrlich-2001 resource: "https://archive.org/details/ninecrazyideasin00ehrl" title: "Nine Crazy Ideas in Science: A Few Might Even Be True" author: "Robert Ehrlich" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Martian HoneyCombs Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 17. Discussed together with [Zygotes](https://tyson-swetnam.github.io/aosd/problems/zygotes/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's own statement is lost. This exercise is the editors' substitute, built on the name and the section theme. It is not his text. **A. Observe.** Get a piece of real honeycomb, or a good close-up photograph. Before reading anything, write down what you can actually *see*: the openings, the closed ends, the tilt, the cell sizes, where the pattern breaks. In a second list, write what you "know" about honeycombs but did *not* see. **B. The Martian comb.** A probe brings back a comb built on Mars by a social creature that, like a bee, stores food in a mass of cells made of a soft material. For each feature on your lists, would you bet on finding it in the Martian comb? Say what each bet rests on: - a *mathematical* fact that holds on any planet; - a *physical* fact that holds if the material and building process are similar (Mars has about 0.38 of Earth's gravity); - a *biological* accident of Earth bees that a Martian need not share. **C. The questions.** Which of your "facts" has anyone measured, and how precisely could a wax cell be measured? Which are theorems, and about what idealised object? What would you ask before saying what a Martian comb *must* look like? ![Close-up of empty honeycomb cells with rounded openings and faceted floors](https://tyson-swetnam.github.io/aosd/assets/images/problems/martian-honeycombs-comb.jpg){ width="560" } *Real comb for Part A: look at the openings and the cell floors. Photo by Doug Bowman, via Flickr and Wikimedia Commons, CC BY 2.0.* ![Left: a patch of hexagonal cells, one highlighted. Right: one cell as a hexagonal prism closed by three rhombi meeting at a tip](https://tyson-swetnam.github.io/aosd/assets/images/problems/martian-honeycombs.svg){ width="560" } *Hexagonal cells, and one cell closed by three rhombi (the facets of a rhombic dodecahedron). Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 3 is "Observations and Questions". The honeycomb is a classic case of an observation buried under its explanations. The Martian framing forces the separation the syllabus asks for earlier: "facts before explanations of facts" and "distinguishing things we know vs only imagine" (session 04). Sorting what you believe about a familiar object into mathematics, physics and Earth biology, and seeing how little rests on your own observation, is the exercise. The reading due that day is Ehrlich, Chapter 3. ## Where it comes from The syllabus gives only the name. Winfree's problem write-ups were kept in a companion document that was never archived, so his statement is lost. The honeycomb itself has a long history. Pappus of Alexandria (about 300 CE) wrote that bees had "wisely selected" the hexagon because it holds more honey. In 1712 Giacomo Maraldi measured the rhombi that close the end of a cell as about 110 and 70 degrees. D'Arcy Thompson (1917) remarked that "nobody appears to have thought of the impossibility of measuring such a thing as the end of a bee's cell to the nearest minute". Darwin (1859), Thompson (1917) and the mathematicians Fejes Toth (1964) and Thomas Hales (1999) all took up the question of why the comb has the shape it has. Their answers are in the resolution below. ??? tip "Hints" - Separate what you saw from what you read. Did you look at the closed ends, or where small cells meet large ones? - For the hexagons, ask what is being optimised and by whom: a geometer bee, soft walls pressing together, or a theorem. Only one of these is planet-independent. - Angles quoted to the minute of arc should make you suspicious. How would you measure a wax facet thinner than a millimetre? - Gravity is the one Martian variable you can reason about. Which features might depend on it? ??? success "Resolution" No answer key survives. For this reconstruction only: **Mathematics (holds anywhere).** Hales proved (1999, published 2001) that no way of dividing the plane into equal-area regions has less total boundary than the regular hexagons. A Martian that tiles a flat comb with equal cells and saves wall material is pushed to hexagons. The theorem says nothing about why cells should be equal or flat. **Physics (holds if the process is similar).** Darwin, with the geometer W. H. Miller, showed that equal spheres in two staggered layers, walled off where they overlap, give hexagonal prisms with three-rhomb bottoms. Thompson added that soft wax settles like soap films, so the rhombi come out at about 109.5 and 70.5 degrees without any bee "knowing" them. A Martian packing cells of a soft material closely would be expected to get the same shapes. Earth cells tilt slightly upward, usually explained by gravity keeping liquid honey in; at 0.38 of Earth's gravity that tilt could differ. **Biology (Earth only).** Cell size, worker and drone cells, and the double-sided hanging comb belong to honey bees, not to combs in general. **The caution.** The famous 109 degrees 28 minutes was theory, not a measurement. And in 1964 Fejes Toth showed that the three-dimensional cell the bees build does not use the least possible wall area. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579), course handout, session 17 β€” [Wayback Machine capture, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **D'Arcy Wentworth Thompson**, *On Growth and Form*, chapter VII, the bee's cell (1917) β€” [Internet Archive](https://archive.org/details/ongrowthform1917thom){target=_blank} πŸ”“ - **Charles Darwin**, *On the Origin of Species*, 1st ed., chapter on Instinct (1859) β€” [Project Gutenberg](https://www.gutenberg.org/ebooks/1228){target=_blank} πŸ”“ - **Thomas C. Hales**, "The Honeycomb Conjecture", *Discrete & Computational Geometry* 25, 1-22 (2001) β€” [arXiv](https://arxiv.org/abs/math/9906042){target=_blank} πŸ”“ - **Laszlo Fejes Toth**, "What the bees know and what they do not know", *Bulletin of the AMS* 70, 468-481 (1964) β€” [DOI](https://doi.org/10.1090/S0002-9904-1964-11155-1){target=_blank} πŸ”“ - **Wikipedia contributors**, "Honeycomb" β€” [Wikipedia](https://en.wikipedia.org/wiki/Honeycomb){target=_blank} πŸ”“ - **Malin Space Science Systems / NASA JPL**, "Southern Hemisphere Polygonal Patterned Ground" (2002) β€” [Wayback Machine](https://web.archive.org/web/20161027001241/http://www.msss.com/mars_images/moc/polygons_5_02/){target=_blank} πŸ”“ - **Robert Ehrlich**, *Nine Crazy Ideas in Science* (2001) β€” [Internet Archive](https://archive.org/details/ninecrazyideasin00ehrl){target=_blank} πŸ”’ *(print-disabled readers only)* - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure. The syllabus gives the name, the session (17, with Zygotes) and the section theme. The phrase appears nowhere else the editors searched, and Winfree's write-up was never archived. The readings weighed: - **A thought experiment on honeycomb geometry** (medium confidence): the reconstruction above, with "Martian" stripping away Earth-bound assumptions. - **The classical bee's-cell story as a case study** (medium): angles "known" to the minute that nobody measured, and a design admired as optimal without checking. - **Literal Martian "honeycombs"** (low): polygonal patterned ground photographed by Mars Global Surveyor from 2000 onward. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/zygotes/ --- title: "Zygotes" description: "Winfree's statement is lost; an editors' reconstruction asks how many twin pairs came from one zygote when all you can see is whether the twins share a sex." type: Activity tags: [course, student-facing, problem, section-3, twins, conditional-probability, hidden-assumptions, genetics] status: stable problem: section: 3 session: 17 identification: unknown kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: weinberg-1901 resource: "https://doi.org/10.1007/BF01657695" title: "BeitrΓ€ge zur Physiologie und Pathologie der Mehrlingsgeburten beim Menschen" author: "Wilhelm Weinberg" - id: wikipedia-wilhelm-weinberg resource: "https://en.wikipedia.org/wiki/Wilhelm_Weinberg" title: "Wilhelm Weinberg" author: "Wikipedia contributors" - id: marshall-knox-1980 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC1052031/" title: "Disease concordance and sex similarity in twins" author: "T. Marshall and E. G. Knox" - id: fellman-eriksson-2006 resource: "https://doi.org/10.1353/hub.2006.0044" title: "Weinberg's Differential Rule Reconsidered" author: "Johan Fellman and Aldur W. Eriksson" - id: kanazawa-segal-de-meza-2018 resource: "https://doi.org/10.1093/humrep/dey046" title: "Why are there more same-sex than opposite-sex dizygotic twins?" author: "Satoshi Kanazawa, Nancy L. Segal and David de Meza" - id: talwalkar-2017 resource: "https://mindyourdecisions.com/blog/2017/07/16/can-you-solve-the-identical-twins-puzzle/" title: "Can You Solve The Identical Twins Puzzle?" author: "Presh Talwalkar" - id: bianconi-2013 resource: "https://doi.org/10.3109/03014460.2013.807878" title: "An estimation of the number of cells in the human body" author: "Eva Bianconi et al." - id: ehrlich-2001 resource: "https://archive.org/details/ninecrazyideasin00ehrl" title: "Nine Crazy Ideas in Science: A Few Might Even Be True" author: "Robert Ehrlich" - id: nhgri-identical-twins resource: "https://commons.wikimedia.org/wiki/File:Identical_twins_lg.jpg" title: "Identical twins (Talking Glossary of Genetic Terms illustration)" author: "National Human Genome Research Institute" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Zygotes Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 17. Discussed together with [Martian Honeycombs](https://tyson-swetnam.github.io/aosd/problems/martian-honeycombs/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's text is lost; this is the editors' best guess from the name. The counts are invented, not data from any real registry. *Identical* twins come from one fertilized egg (one zygote) that splits in two, so they are always the same sex. *Fraternal* twins come from two zygotes, and each is a boy or a girl independently of the other. Without genetic tests, nobody can tell whether a same-sex pair came from one zygote or two. **Part 1.** A birth registry records 1,000 twin pairs: 700 same-sex and 300 opposite-sex. Nothing else is known. How many pairs are identical? State every assumption you need, and how you would check it. **Part 2.** A friend from that registry says she is a twin and her twin is a sister. What is the probability they are identical? Is it simply the overall fraction of identical pairs? **Part 3.** A large study reports that 57 to 60 per cent of *fraternal* pairs are same-sex, not 50. What does that do to your answers, and what would you ask about how the twins were classified? ![Fraternal twins: two eggs, each fertilized by its own sperm, grow into two different babies. Identical twins: one fertilized egg divides and grows into two babies](https://tyson-swetnam.github.io/aosd/assets/images/problems/zygotes-identical-twins.jpg){ width="560" } *One zygote or two. National Human Genome Research Institute, Talking Glossary of Genetic Terms. Public domain (US government work), via Wikimedia Commons.* ## Why it is in the course Section 3 is "Observations and Questions". If the reconstruction is right, the lesson is Wilhelm Weinberg's: one thing you can observe, whether twins share a sex, lets you count something you cannot see, how many zygotes each pair came from. The count is only as good as its assumptions, so the real work is naming them, as in Section 1's "hidden assumptions". The syllabus says the purpose of its puzzles, "many of them silly", "is to slow you down for a few minutes so you can examine the working of your own mind". Part 2 is built for that: the first answer most people give is wrong, and noticing why is the exercise. The session's reading is Chapter 3 of Robert Ehrlich's *Nine Crazy Ideas in Science*; its subject could not be confirmed, so no link is claimed. ## Where it comes from The syllabus gives only the name. The note at the end of this page weighs what else survives. Wilhelm Weinberg (1862-1937), a Stuttgart obstetrician better known for the Hardy-Weinberg principle, showed in 1901 how to estimate identical and fraternal twin numbers from same-sex and opposite-sex counts. His "differential rule" served twin research for most of the twentieth century. Its assumptions (an even sex ratio, independent sexes in fraternal pairs, equal survival) were questioned repeatedly, and a 2018 study found fraternal pairs same-sex more often than chance predicts. The reverse question circulates as a probability puzzle. Whether Winfree used either form is not documented. ??? tip "Hints" - Biology forbids an identical boy-girl pair. Which of the two counts can contain only fraternal pairs? - If fraternal sexes are independent and even, what fraction of fraternal pairs is opposite-sex? Estimate all the fraternal pairs, then subtract. - For Part 2, tabulate the 1,000 pairs by type and sex, and count only the rows your friend's statement allows. - List every assumption, including how anyone decided which pairs were "identical". Each is a question to put to the data. ??? success "Resolution" This resolves the reconstruction only. **Part 1.** Only fraternal pairs can be opposite-sex, and half of them should be. So fraternal pairs are about 2 Γ— 300 = 600, and identical pairs about 700 βˆ’ 300 = 400. Marshall and Knox write Weinberg's rule with L same-sex and U opposite-sex pairs: identical (L βˆ’ U)/(L + U), fraternal 2U/(L + U). It assumes an even sex ratio, independent sexes in fraternal pairs, and no loss or selection that depends on type. ![Bar of 1,000 twin pairs split into identical (400), fraternal same-sex (300) and fraternal opposite-sex (300), with brackets showing the observed 700 same-sex and 300 opposite-sex pairs and the inferred one-zygote and two-zygote totals](https://tyson-swetnam.github.io/aosd/assets/images/problems/zygotes.svg){ width="560" } *Illustrative figures. Only the split into 700 same-sex and 300 opposite-sex pairs is observed; the dashed line is Weinberg's assumption. Drawn for this site (CC BY 4.0).* **Part 2.** 400/700 = 4/7, not 400/1,000 = 2/5. Learning the pair is same-sex rules out the 300 opposite-sex pairs. In Talwalkar's popular version, with 1 in 10 pairs identical, two brothers are identical with probability 2/11, not 1/10. **Part 3.** If fraternal pairs are same-sex more than half the time, the rule undercounts fraternal pairs and overcounts identical ones. At 60 per cent same-sex, 300 opposite-sex pairs imply 750 fraternal pairs and only 250 identical, and Part 2's answer drops to 250/700 = 5/14. Kanazawa, Segal and de Meza found 60.4 and 57.4 per cent in British and American cohorts, but mothers, not DNA tests, classified the twins. The honest answer to Part 1 is a range plus a list of things to measure. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579), course handout; the session-17 line and its bookmark β€” [Wayback Machine capture, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Wilhelm Weinberg**, "BeitrΓ€ge zur Physiologie und Pathologie der Mehrlingsgeburten beim Menschen", *Archiv fΓΌr die gesammte Physiologie des Menschen und der Thiere* 88, 346-430 (1901) β€” [DOI](https://doi.org/10.1007/BF01657695){target=_blank} πŸ”’ - **Wikipedia contributors**, "Wilhelm Weinberg" β€” [Wikipedia](https://en.wikipedia.org/wiki/Wilhelm_Weinberg){target=_blank} πŸ”“ - **T. Marshall and E. G. Knox**, "Disease concordance and sex similarity in twins", *Journal of Epidemiology and Community Health* 34(1), 1-8 (1980) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC1052031/){target=_blank} πŸ”“ - **Johan Fellman and Aldur W. Eriksson**, "Weinberg's Differential Rule Reconsidered", *Human Biology* 78(3), 253-275 (2006) β€” [DOI](https://doi.org/10.1353/hub.2006.0044){target=_blank} πŸ”’ - **Satoshi Kanazawa, Nancy L. Segal and David de Meza**, "Why are there more same-sex than opposite-sex dizygotic twins?", *Human Reproduction* 33(5), 930-934 (2018) β€” [DOI](https://doi.org/10.1093/humrep/dey046){target=_blank} πŸ”“ - **Presh Talwalkar**, "Can You Solve The Identical Twins Puzzle?", *Mind Your Decisions* (2017) β€” [blog post](https://mindyourdecisions.com/blog/2017/07/16/can-you-solve-the-identical-twins-puzzle/){target=_blank} πŸ”“ - **Eva Bianconi et al.**, "An estimation of the number of cells in the human body", *Annals of Human Biology* 40(6), 463-471 (2013) β€” [DOI](https://doi.org/10.3109/03014460.2013.807878){target=_blank} πŸ”’ - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True* (2001) β€” [Internet Archive](https://archive.org/details/ninecrazyideasin00ehrl){target=_blank} πŸ”’ *(print-disabled readers only)* - **National Human Genome Research Institute**, identical and fraternal twins illustration, Talking Glossary of Genetic Terms β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Identical_twins_lg.jpg){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure. The syllabus gives one word. In Winfree's handout, the link on this item points to a bookmark named `Zygote_zygote`, inside a companion problem document that was not archived, so no description survives. The readings weighed: - **Twin zygosity** (medium confidence): the reconstruction above, the best-known elementary puzzle about zygotes. The editors' guess, not Winfree's text: the doubled word in the bookmark may mean one zygote against two, or may be an accident of the word processor. - **Cell doublings** (low): how many doublings turn one zygote into a body of about 3.7 Γ— 10ΒΉΒ³ cells (Bianconi and colleagues' estimate), and why that naive count misleads. - **Cleavage geometry** (low): how the early embryo packs its cells. In this schedule, problems sharing a session are usually unrelated, so the pairing with Martian Honeycombs is no evidence. A former student's memory of the handout would outweigh all of this. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/cevians/ --- title: "Cevians" description: "Join each corner of a triangle to the one-third point of the opposite side: what fraction of the area is the small triangle in the middle? A puzzle about trusting a measurement over a confident first guess." type: Activity tags: [course, student-facing, problem, section-3, geometry, triangles, area, routh-theorem] status: stable problem: section: 3 session: 18 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: potts-1859 resource: "https://archive.org/details/euclidselements00unkngoog" title: "Euclid's Elements of Geometry, fifth school edition (Problems 59 and 100, pp. 78 and 80)" author: "Robert Potts" - id: glaisher-1879 resource: "https://archive.org/details/solutionsofcambr00glaiuoft" title: "Solutions of the Cambridge Senate-House Problems and Riders for the Year 1878 (rider vii, p. 33)" author: "J. W. L. Glaisher, G. H. Prior and N. M. Ferrers" - id: routh-1896 resource: "https://archive.org/details/cihm_33231" title: "A Treatise on Analytical Statics, with Numerous Examples, vol. I, 2nd ed. (Chapter IV, footnote, p. 82)" author: "Edward John Routh" - id: steinhaus-1960 resource: "https://archive.org/details/mathematicalsnap0000stei" title: "Mathematical Snapshots (revised English edition)" author: "Hugo Steinhaus" - id: coxeter-1969 resource: "https://archive.org/details/introductiontoge0000coxe" title: "Introduction to Geometry, 2nd ed." author: "H. S. M. Coxeter" - id: klamkin-liu-1981 resource: "https://cms.math.ca/wp-content/uploads/crux-pdfs/Crux_v7n07_Aug.pdf" title: "Three more proofs of Routh's theorem, Crux Mathematicorum 7 (1981), 199-203" author: "M. S. Klamkin and A. Liu" - id: randi-2001 resource: "https://web.archive.org/web/20060427055758/http://www.randi.org/jr/02-09-2001.html" title: "Commentary, February 9, 2001" author: "James Randi" - id: cook-wood-2004 resource: "https://doi.org/10.1017/S002555720017514X" title: "88.46 Feynman's triangle" author: "R. J. Cook and G. V. Wood" - id: wikipedia-one-seventh-area-triangle resource: "https://en.wikipedia.org/wiki/One-seventh_area_triangle" title: "One-seventh area triangle" author: "Wikipedia contributors" - id: wikipedia-routh-theorem resource: "https://en.wikipedia.org/wiki/Routh%27s_theorem" title: "Routh's theorem" author: "Wikipedia contributors" - id: mathworld-routh-theorem resource: "https://mathworld.wolfram.com/RouthsTheorem.html" title: "Routh's Theorem" author: "Eric W. Weisstein" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Cevians Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 18. Discussed together with [Superposed Filters](https://tyson-swetnam.github.io/aosd/problems/superposed-filters/index.md).* !!! abstract "The problem" *The syllabus gives only the name. This is the editors' statement of the puzzle it most probably means; see the note at the end.* A **cevian** is a straight segment from a corner of a triangle to a point on the opposite side (a median is one example). Draw a large triangle *ABC*. Mark *D* one-third of the way from *B* to *C*, *E* one-third of the way from *C* to *A*, and *F* one-third of the way from *A* to *B*. Draw the cevians *AD*, *BE* and *CF*. They enclose a small triangle in the middle. 1. Before measuring, write down a guess: what fraction of the area of *ABC* is the central triangle? 2. Measure it and compare. 3. Explain the number, then find a second, independent explanation. 4. Does the answer depend on the shape of the triangle? 5. What if the marks sit at one-quarter of each side, or at different fractions on different sides? ![Triangle ABC with each side divided into thirds by tick marks; segments from A to D, B to E and C to F cross to enclose a small shaded triangle in the middle](https://tyson-swetnam.github.io/aosd/assets/images/problems/cevians.svg){ width="560" } *Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 3 is "Observations and Questions", and here a measurement overrules a confident first impression. The figure is built entirely from thirds. When James Randi set the puzzle in early 2001, he reported that an instinctive answer "might be one-ninth, which many readers gave us." A careful drawing disagrees, and the honest next step is a question: why that number? The syllabus says the puzzles exist "to slow you down for a few minutes so you can examine the working of your own mind". Writing the guess down first is how you catch your mind at work. As with session 12's [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md), "like a jig-saw puzzle of cross-checks", the answer can be reached several ways that check one another. ## Where it comes from Robert Potts's 1859 school edition of Euclid poses the figure for an equilateral triangle (Problem 59). A later exercise (Problem 100) asks the reader to show what share of the area each of its two triangles has: one joining the marks, and one enclosed by the cevians. (His numbers are in the Resolution.) The general formula, for cevians cutting the sides in any ratios, was set in the Cambridge Tripos of January 1878 and solved in Glaisher's 1879 volume of solutions. Edward John Routh, himself a Tripos coach then, printed it in a footnote to his *Treatise on Analytical Statics* (first edition 1891; p. 82 of the 1896 second edition), noting that "the author has not met with these expressions". It is called Routh's theorem all the same. The puzzle became famous through Hugo Steinhaus's *Mathematical Snapshots*, which gives a proof with no algebra. Martin Gardner sent that proof to Randi's column in 2001. A 2004 *Mathematical Gazette* note, "Feynman's triangle", reportedly tells how Richard Feynman was set the puzzle in a dinner conversation and worked it out himself. The editors have not read it. ??? tip "Hints" - Draw it big on graph paper and count squares before reasoning. - Start with a convenient triangle: (0,0), (3,0), (0,3). Then ask whether an affine map, which takes any triangle to any other and scales every area by the same factor, could change the fraction. - Find how each cevian is cut by the other two. The ratio is simple. - No algebra: through each corner of the central triangle, draw a line parallel to the side opposite that corner. Add parallels in the same three directions through *A*, *B* and *C*. Compare the pieces outside *ABC* with the gaps inside. ??? success "Resolution" The central triangle is exactly **one-seventh** of *ABC*, for any triangle. **Coordinates.** With *A* = (0,0), *B* = (3,0), *C* = (0,3) (area 9/2), *D* = (2,1), *E* = (0,2), *F* = (1,0). The cevians cross at *P* = (6/7, 3/7) (*AD* and *CF*), *Q* = (12/7, 6/7) (*AD* and *BE*) and *R* = (3/7, 12/7) (*BE* and *CF*). The shoelace formula gives triangle *PQR* an area of 9/14, and (9/14)/(9/2) = 1/7. Affine maps preserve area ratios, so this holds for every triangle. **Ratios.** Each cevian is cut 3 : 3 : 1 counting from its corner: *P* and *Q* sit 3/7 and 6/7 of the way from *A* to *D*. Triangle *AFC* is one-third of *ABC*, because *AF* is one-third of *AB*. Triangle *AFP* has the same apex *A* and a base *FP* one-seventh as long as *FC*, so it is 1/21 of *ABC*. The same argument makes *BDQ* and *CER* 1/21 each. Triangle *ABD* is also one-third (7/21) of *ABC*, and it holds *AFP*, *BDQ* and one quadrilateral, so each quadrilateral is 5/21. The centre gets what is left: 21 βˆ’ 3 βˆ’ 15 = 3 twenty-firsts, or 1/7. **Steinhaus.** Through each corner of the central triangle, draw a line parallel to the opposite side. Each passes through one of the unused one-third marks. Parallels in the same directions through *A*, *B* and *C* complete six copies of the central triangle around it. The parts of the copies that stick out beyond the sides of *ABC* match the gaps left inside it, so *ABC* has the area of seven central triangles. **Routh's theorem.** If *CD*/*BD* = *x*, *AE*/*CE* = *y*, *BF*/*AF* = *z*, the central fraction is (*xyz* βˆ’ 1)Β² / [(*xy* + *y* + 1)(*yz* + *z* + 1)(*zx* + *x* + 1)]. With all ratios equal to *n* it is (*n* βˆ’ 1)Β² / (*n*Β² + *n* + 1): 0 for medians, 1/7 for thirds, 4/13 for quarter points. Potts's Problem 100 gives "one-third and one-seventh" for his two triangles. For more proofs see Coxeter's *Introduction to Geometry* and Klamkin and Liu (1981). ## Sources - **Robert Potts**, *Euclid's Elements of Geometry*, fifth school edition, Problems 59 and 100, pp. 78 and 80 (1859) β€” [Internet Archive](https://archive.org/details/euclidselements00unkngoog){target=_blank} πŸ”“ - **J. W. L. Glaisher, G. H. Prior and N. M. Ferrers**, *Solutions of the Cambridge Senate-House Problems and Riders for the Year 1878*, rider (vii), p. 33 (1879) β€” [Internet Archive](https://archive.org/details/solutionsofcambr00glaiuoft){target=_blank} πŸ”“ - **Edward John Routh**, *A Treatise on Analytical Statics, with Numerous Examples*, vol. I, 2nd ed., footnote on p. 82 (1896) β€” [Internet Archive](https://archive.org/details/cihm_33231){target=_blank} πŸ”“ - **Hugo Steinhaus**, *Mathematical Snapshots*, revised English edition (1960) β€” [Internet Archive](https://archive.org/details/mathematicalsnap0000stei){target=_blank} πŸ”“ *(borrow)* - **H. S. M. Coxeter**, *Introduction to Geometry*, 2nd ed. (1969) β€” [Internet Archive](https://archive.org/details/introductiontoge0000coxe){target=_blank} πŸ”“ *(borrow)* - **M. S. Klamkin and A. Liu**, "Three more proofs of Routh's theorem", *Crux Mathematicorum* 7, 199–203 (1981) β€” [CMS scan of the issue](https://cms.math.ca/wp-content/uploads/crux-pdfs/Crux_v7n07_Aug.pdf){target=_blank} πŸ”“ - **James Randi**, Commentary of 9 February 2001 (the puzzle's answer and the Steinhaus proof sent by Martin Gardner) β€” [Wayback Machine](https://web.archive.org/web/20060427055758/http://www.randi.org/jr/02-09-2001.html){target=_blank} πŸ”“ - **R. J. Cook and G. V. Wood**, "88.46 Feynman's triangle", *The Mathematical Gazette* 88(512), 299–302 (2004) β€” [doi:10.1017/S002555720017514X](https://doi.org/10.1017/S002555720017514X){target=_blank} πŸ”’ (not read by the editors) - **Wikipedia contributors**, "One-seventh area triangle" β€” [Wikipedia](https://en.wikipedia.org/wiki/One-seventh_area_triangle){target=_blank} πŸ”“ - **Wikipedia contributors**, "Routh's theorem" β€” [Wikipedia](https://en.wikipedia.org/wiki/Routh%27s_theorem){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Routh's Theorem", MathWorld β€” [MathWorld](https://mathworld.wolfram.com/RouthsTheorem.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Probable, not certain. The syllabus lists only "Cevians" in session 18, beside Superposed Filters, and no surviving Winfree document describes it. This puzzle is the standard one told in terms of cevians, needs no apparatus, has a surprising measurable answer and cross-checking proofs, and was in Randi's column in 2001 (the syllabus says many handouts came "from current periodicals"). Two weaker readings remain: - **Counting regions** made by several cevians from each corner, a pattern hunt; but the disk-slicing problem, [n dots on a circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md), is listed separately in session 19. - **Ceva's theorem**: discover when three cevians meet in one point, as medians do. The name comes from it, but it lacks a one-number surprise. --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/superposed-filters/ --- title: "Superposed Filters" description: "Stack polarizing filters, write down what you expect, then look: what happens when a third absorbing filter is slid between two crossed ones, and does the order of the filters matter?" type: Activity tags: [course, student-facing, problem, section-3, polarization, optics, malus-law, prediction] status: stable problem: section: 3 session: 18 identification: confident kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: hyperphysics-crossed-polarizers resource: "https://hyperphysics.gsu.edu/hbase/phyopt/polcross.html" title: "Crossed Polarizers / Polarizer Puzzle / Law of Malus (HyperPhysics)" author: "Rod Nave, Georgia State University" - id: malus-arcueil-1809 resource: "https://archive.org/details/bub_gb_Nl87AAAAcAAJ_2" title: "MΓ©moires de Physique et de Chimie de la SociΓ©tΓ© d'Arcueil, tome second (1809), containing Malus, 'Sur une propriΓ©tΓ© de la lumiΓ¨re rΓ©flΓ©chie'" author: "SociΓ©tΓ© d'Arcueil; memoir by Γ‰tienne-Louis Malus" - id: wikipedia-malus resource: "https://en.wikipedia.org/wiki/%C3%89tienne-Louis_Malus" title: "Γ‰tienne-Louis Malus" author: "Wikipedia contributors" - id: wikipedia-polarizer resource: "https://en.wikipedia.org/wiki/Polarizer" title: "Polarizer" author: "Wikipedia contributors" - id: wikipedia-polaroid-polarizer resource: "https://en.wikipedia.org/wiki/Polaroid_%28polarizer%29" title: "Polaroid (polarizer)" author: "Wikipedia contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Superposed Filters Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md), session 18. Discussed in the same session as [Cevians](https://tyson-swetnam.github.io/aosd/problems/cevians/index.md).* !!! abstract "The problem" *Reconstructed from the syllabus and Winfree's handout. His own wording of the exercise is lost; the steps below are the editors' version.* You need two pieces of linear polarizing filter (the lenses of two pairs of polarized sunglasses, two photographic polarizers, or two squares of polarizing sheet) and, for the second half, a third piece. Before each step, write down your prediction. 1. Look through one filter at the sky, a puddle's reflection or a laptop screen, and turn it. What changes? 2. Lay the second filter on the first and slowly turn the top one through a full turn. How many times does the pair go dark, and why that number? 3. Estimate the light getting through at 0, 30, 45, 60 and 90 degrees between the two axes. Propose a formula. What must it give at 0 and at 90 degrees? 4. Set the pair at its darkest. Predict what happens if you slide a **third** filter between them, its axis at 45 degrees to both. Then do it. 5. Put the third filter on top of the pair instead, or underneath. Does the order matter? 6. Going further: stack many filters, each turned a little from the one below, so the first and last are at 90 degrees. What happens as the number grows? Keep the record of what you predicted, what you saw, and how far apart the two were. ## Why it is in the course Section 3 is "Observations and Questions", and session 18 pairs this exercise with [Cevians](https://tyson-swetnam.github.io/aosd/problems/cevians/index.md): one problem you reason out on paper, one you have to look at. The syllabus says most exercises are made from elementary mathematics, then adds: "There are some hands-on lab-type exercises too." The filters test a hidden assumption: that a filter only takes light away, so more filters can only mean less light. Step 4 is built to check that picture. The habit is the one the syllabus names in session 04: "distinguishing things we know vs only imagine", and "facts before explanations of facts". Write the prediction, then look, then measure the gap. Step 2 adds a quieter lesson. The quantity that matters is an angle, a variable that goes round a circle, the kind of cyclic variable Winfree spent his career on. ## Where it comes from In Winfree's handout, the link on "Superposed filters" points to a bookmark named `Polaroids`. Several of his bookmarks name what a problem is about rather than repeating its label (`Tictactoe_LoShu` for LoShu, `Sliced_Disk` for "slice the disk"), so this one points to polarizing (Polaroid) sheets rather than colour filters. The bookmark led into a companion document that was never archived, so Winfree's own statement of the problem is lost. Γ‰tienne-Louis Malus (1775–1812) discovered that reflected light is polarized and published the finding in 1809 in the memoirs of the SociΓ©tΓ© d'Arcueil. The rule that a second polarizer passes a fraction cosΒ²ΞΈ of already-polarized light carries his name. Through the nineteenth century the effect needed crystals such as Iceland spar. Polarizing sheet (first patented in 1929, developed by Edwin Land from 1932, with his H-sheet following in 1938) put it in anyone's pocket. HyperPhysics poses the three-filter version as the "Polarizer Puzzle". ??? tip "Hints" - Count the dark positions in one full turn. Can the pair tell which face of a filter is up, or only the angle between the axes? How often must the pattern repeat? - Fix the formula's endpoints first: the same at 0 and 180 degrees, zero at 90. Then test candidates against your estimates. - A filter does more than remove light. What property does the light that got through now have? Is the second filter meeting the same light the first one met? - If moving the third filter to a different place in the stack changes the result, "each filter removes a fixed share" cannot be the whole story. ??? success "Resolution" ![Three stacks of polarizing filters. Two filters with parallel axes look bright; the same two crossed at right angles look black; inserting a third filter at 45 degrees between the crossed pair makes the stack dim but not black, passing about an eighth of the light. Below, a graph shows the light passed by two filters following cos squared of the angle between them, dark twice per full turn.](https://tyson-swetnam.github.io/aosd/assets/images/problems/superposed-filters.svg){ width="560" } *Drawn for this site (CC BY 4.0). Ideal filters; each is drawn face-on, with light passing up through the stack. Real sheet polarizers leak a little when crossed.* A sheet polarizer passes the part of the light's electric field along its axis and absorbs the rest, so unpolarized light loses about half its intensity at the first filter and leaves polarized. **Two filters.** The second filter meets polarized light and passes cosΒ²ΞΈ of it (Malus's law), where ΞΈ is the angle between the axes: all of it at 0 degrees, none at 90. Because cosΒ²ΞΈ repeats every 180 degrees, a full turn gives two bright and two dark positions. **Three filters.** Insert a filter at 45 degrees between a crossed pair. It passes cosΒ²45Β° = 1/2 of what reaches it *and re-polarizes it at 45 degrees*, so the last filter sees light 45 degrees from its axis instead of 90 and passes another half. That is 25% of the light leaving the first filter, about 12.5% of the original beam. On top of the pair or underneath it, the same filter leaves the stack black: order matters, because each filter both removes light and rewrites what is left. **The stack.** N filters above the first, each turned 90/N degrees, pass [cosΒ²(90/N)]^N of the polarized light: 25% for N = 2, 78% for N = 10, 97% for N = 90. Many small turns cost almost nothing. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery (ECOL 479/579): course handout*, archived 20 April 2002; the session-18 link to the `Polaroids` bookmark β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Rod Nave**, "Crossed Polarizers" and "Polarizer Puzzle", *HyperPhysics*, Georgia State University β€” [hyperphysics.gsu.edu](https://hyperphysics.gsu.edu/hbase/phyopt/polcross.html){target=_blank} πŸ”“ - **Γ‰tienne-Louis Malus**, "Sur une propriΓ©tΓ© de la lumiΓ¨re rΓ©flΓ©chie", *MΓ©moires de Physique et de Chimie de la SociΓ©tΓ© d'Arcueil*, tome 2 (1809) β€” [Internet Archive](https://archive.org/details/bub_gb_Nl87AAAAcAAJ_2){target=_blank} πŸ”“ - **Wikipedia contributors**, "Γ‰tienne-Louis Malus" β€” [Wikipedia](https://en.wikipedia.org/wiki/%C3%89tienne-Louis_Malus){target=_blank} πŸ”“ - **Wikipedia contributors**, "Polarizer" β€” [Wikipedia](https://en.wikipedia.org/wiki/Polarizer){target=_blank} πŸ”“ - **Wikipedia contributors**, "Polaroid (polarizer)" β€” [Wikipedia](https://en.wikipedia.org/wiki/Polaroid_%28polarizer%29){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-3-observations-and-questions)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/ --- title: "N Dots on the Rim of a Circle" description: "Join every pair of n dots on a circle by chords and count the pieces of the disk: a lesson in the difference between a pattern observed and a pattern explained." type: Activity tags: [course, student-facing, problem, section-4, combinatorics, induction, patterns, geometry] status: stable problem: section: 4 session: 19 identification: confident kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: wikipedia-moser-circle resource: "https://en.wikipedia.org/wiki/Moser%27s_circle_problem" title: "Moser's circle problem" author: "Wikipedia contributors" - id: oeis-a000127 resource: "https://oeis.org/A000127" title: "A000127: Maximal number of regions obtained by joining n points around a circle by straight lines" author: "N. J. A. Sloane, ed. (OEIS Foundation)" - id: oeis-a006533 resource: "https://oeis.org/A006533" title: "A006533: Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) regions" author: "N. J. A. Sloane, ed. (OEIS Foundation)" - id: mathworld-circle-division resource: "https://mathworld.wolfram.com/CircleDivisionbyChords.html" title: "Circle Division by Chords" author: "Eric W. Weisstein (MathWorld)" - id: moser-ross-1949 resource: "https://www.jstor.org/stable/3219224" title: "Mathematical Miscellany (On the Danger of Induction)" author: "Leo Moser and W. Bruce Ross" - id: guy-1988 resource: "https://doi.org/10.1080/00029890.1988.11972074" title: "The Strong Law of Small Numbers" author: "Richard K. Guy" - id: noy-1996 resource: "https://doi.org/10.1080/0025570X.1996.11996383" title: "A Short Solution of a Problem in Combinatorial Geometry" author: "Marc Noy" - id: conway-guy-1996 resource: "https://link.springer.com/book/10.1007/978-1-4612-4072-3" title: "The Book of Numbers (ch. 'How Many Regions', pp. 76-79)" author: "John H. Conway and Richard K. Guy" - id: honsberger-1973 resource: "https://archive.org/details/mathematicalgems0001hons_m1e8" title: "Mathematical Gems I, ch. 9 'A Problem in Combinatorics', pp. 99-107" author: "Ross Honsberger" - id: gardner-1979 resource: "https://archive.org/details/mathematicalcirc00gard" title: "Mathematical Circus" author: "Martin Gardner" - id: judson-1980 resource: "https://archive.org/details/searchforsolutio00juds" title: "The Search for Solutions (ch. 2, 'Pattern')" author: "Horace Freeland Judson" - id: 3blue1brown-2023 resource: "https://www.youtube.com/watch?v=YtkIWDE36qU" title: "This pattern breaks, but for a good reason | Moser's circle problem" author: "Grant Sanderson (3Blue1Brown)" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # N Dots on the Rim of a Circle Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 19. Discussed together with [Presidents and States](https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/index.md).* !!! abstract "The problem" *The editors' statement of the standard puzzle, in the terms the syllabus uses. Winfree's own wording is not recorded.* Mark **n** dots on the rim of a circle. Draw every chord joining one dot to another, so that each pair of dots is connected by a straight line. The chords cut the disk into separate pieces. **How many pieces do you get?** Draw and count n = 1 to 5 by hand, with a big circle and a sharp pencil. Then predict n = 6 **before** you draw or count it, write the prediction down, and check it. (The figure below shows the chords but not the counts.) Does the count depend on where the dots sit, or only on how many there are? Find a rule for any n, and give a reason it must be true, not just that it fits your drawings. ![Six circles with one, two, three, four, five and six red dots on the rim; in each circle every pair of dots is joined by a straight chord, with no region counts shown](https://tyson-swetnam.github.io/aosd/assets/images/problems/n-dots-on-circle.svg){ width="560" } *The construction for n = 1 to 6, counts left for you. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 19 opens Section 4, "Patterns, Empirical Generalizations", with Judson's chapter "Pattern". The syllabus pairs this puzzle with [Presidents and States](https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/index.md), another pattern that looks like a law. The puzzle is about the gap between a pattern you have *observed* and one you can *explain*. The first few counts suggest a rule so strongly that most people announce it; whether it survives the next case is the exercise. The syllabus says its exercises are "contrived much as the organizers of an Easter Egg Hunt do in the hour before little kids arrive with their baskets", and this one has more than one egg hidden in it. It also carries forward a Section 1 phrase, "distinguishing things we know vs only imagine": a rule that fits five drawings is imagined until you can say why it holds. Committing to a prediction before drawing n = 6 is a good moment to record in the GamesWorth notebook. ## Where it comes from The puzzle is usually called **Moser's circle problem**, after the Canadian mathematician Leo Moser, who published it with W. Bruce Ross in the "Mathematical Miscellany" column of *Mathematics Magazine* in 1949. The OEIS bibliography gives that column the subtitle "On the Danger of Induction". The same numbers had turned up earlier for a different question: the most pieces that flat cuts can divide four-dimensional space into. Moser's contribution was a construction anyone can draw. It became a standard cautionary tale. Richard Guy placed it among the opening examples of "The Strong Law of Small Numbers" (1988). Books by Honsberger, Gardner, and Conway with Guy treat it or a close variant; Marc Noy published a short proof in 1996. The syllabus line for session 19 reads "discuss n dots on rim of circle, connected to slice the disk". In Winfree's [course handout](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} (archived 2002), the link on "slice the disk" points to a bookmark named `Sliced_Disk`, which fits this reading. The document that bookmark pointed into was not archived, so how Winfree posed the problem is not recorded. ??? tip "Hints" - Draw big and number each region as you count it: the thin slivers next to the rim are the ones people miss. - Try n = 6 twice: once with the six dots evenly spaced, once with them scattered unevenly. If the two counts differ, find what is special about the symmetric picture, and decide which drawing deserves to be called the answer. - Count things other than regions. How many chords are there? How many crossing points inside the disk, if no three chords meet at one point? Both are simple combinations of n. - Add the chords one at a time. A new chord that crosses k existing chords inside the disk splits how many regions? ??? success "Resolution" **The pattern breaks at n = 6.** The counts are 1, 2, 4, 8, 16, **31**, 57, 99, 163, 256, ... and not 1, 2, 4, 8, 16, 32 (OEIS A000127). The tempting rule, `2^(n-1)`, is wrong. The maximum number of pieces for n dots is `C(n,4) + C(n,2) + 1`, where `C(n,k)` is the number of ways to choose k things from n. **Why.** Two counts do the work. - Every set of 4 dots gives exactly one pair of crossing chords, so there are `C(n,4)` crossing points inside the disk, provided no three chords meet at one point. - Every set of 2 dots gives one chord, so there are `C(n,2)` chords. Add the chords one at a time. A chord that crosses k chords already drawn is cut into k + 1 segments, and each segment splits a region in two. Over all chords, the regions added total (chords) + (crossings). Starting from one region, the disk ends with `1 + C(n,2) + C(n,4)`. **Why it looks like doubling.** The formula equals `C(n-1,0) + C(n-1,1) + C(n-1,2) + C(n-1,3) + C(n-1,4)`, the first five terms of the binomial sum for `2^(n-1)`. For n up to 5 those are all the terms; from n = 6 on, doubling has extra terms and overshoots. **The symmetry trap.** Six evenly spaced dots give only **30** regions, because the three long diagonals of a regular hexagon all pass through the centre. Nudge one dot and a tiny triangle opens up there: 31. Eight evenly spaced dots give 88 instead of 99 (OEIS A006533). **The moral.** Agreement over the first few cases is weak evidence. A pattern is not a law until you can say why it must hold. ## Sources - **Leo Moser and W. Bruce Ross**, "Mathematical Miscellany" ("On the Danger of Induction"), *Mathematics Magazine* 23(2), 109–114 (1949) β€” [JSTOR](https://www.jstor.org/stable/3219224){target=_blank} πŸ”’ (the editors could not read it directly; the citation is confirmed by Wikipedia and OEIS) - **Richard K. Guy**, "The Strong Law of Small Numbers", *American Mathematical Monthly* 95(8), 697–712 (1988) β€” [doi:10.1080/00029890.1988.11972074](https://doi.org/10.1080/00029890.1988.11972074){target=_blank} πŸ”’ (not read directly; details confirmed through Crossref) - **Marc Noy**, "A Short Solution of a Problem in Combinatorial Geometry", *Mathematics Magazine* 69(1), 52–53 (1996) β€” [doi:10.1080/0025570X.1996.11996383](https://doi.org/10.1080/0025570X.1996.11996383){target=_blank} πŸ”’ (not read directly; details confirmed through Crossref) - **John H. Conway and Richard K. Guy**, *The Book of Numbers*, "How Many Regions", pp. 76–79 (Springer, 1996) β€” [Springer](https://link.springer.com/book/10.1007/978-1-4612-4072-3){target=_blank} πŸ”’ - **Ross Honsberger**, *Mathematical Gems I*, ch. 9, "A Problem in Combinatorics", pp. 99–107 (MAA, 1973) β€” [Internet Archive](https://archive.org/details/mathematicalgems0001hons_m1e8){target=_blank} πŸ”“ *(borrow)* - **Martin Gardner**, *Mathematical Circus*, pp. 177, 180–181 (Knopf, 1979) β€” [Internet Archive](https://archive.org/details/mathematicalcirc00gard){target=_blank} πŸ”“ *(borrow)* - **Horace Freeland Judson**, *The Search for Solutions*, ch. 2, "Pattern" (1980), the reading due this session β€” [Internet Archive](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)* - **N. J. A. Sloane, ed.**, OEIS A000127, maximal number of regions from joining n points around a circle β€” [OEIS](https://oeis.org/A000127){target=_blank} πŸ”“ - **N. J. A. Sloane, ed.**, OEIS A006533, regions from n equally spaced points β€” [OEIS](https://oeis.org/A006533){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Circle Division by Chords", MathWorld β€” [MathWorld](https://mathworld.wolfram.com/CircleDivisionbyChords.html){target=_blank} πŸ”“ - **Wikipedia contributors**, "Moser's circle problem" β€” [Wikipedia](https://en.wikipedia.org/wiki/Moser%27s_circle_problem){target=_blank} πŸ”“ - **Grant Sanderson (3Blue1Brown)**, "This pattern breaks, but for a good reason | Moser's circle problem" (2023), a visual derivation β€” [YouTube](https://www.youtube.com/watch?v=YtkIWDE36qU){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Internet Archive](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/ --- title: "Presidents and States" description: "Winfree's own statement of this exercise is lost; this reconstruction takes two famous regularities from American history, the zero-year presidents and the bellwether states, and asks whether a run of confirmations makes a pattern, an accident or a law." type: Activity tags: [course, student-facing, problem, section-4, patterns, induction, coincidences, bellwether-states] status: stable problem: section: 4 session: 19 identification: unknown kind: discussion generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: wikipedia-curse-of-tippecanoe resource: "https://en.wikipedia.org/wiki/Curse_of_Tippecanoe" title: "Curse of Tippecanoe" author: "Wikipedia contributors" - id: wikipedia-as-maine-goes resource: "https://en.wikipedia.org/wiki/As_Maine_goes,_so_goes_the_nation" title: "As Maine goes, so goes the nation" author: "Wikipedia contributors" - id: wikipedia-missouri-bellwether resource: "https://en.wikipedia.org/wiki/Missouri_bellwether" title: "Missouri bellwether" author: "Wikipedia contributors" - id: wikipedia-bellwether-politics resource: "https://en.wikipedia.org/wiki/Bellwether_(politics)" title: "Bellwether (politics)" author: "Wikipedia contributors" - id: diaconis-mosteller-1989 resource: "https://www.stat.berkeley.edu/~aldous/157/Papers/diaconis_mosteller.pdf" title: "Methods for Studying Coincidences" author: "Persi Diaconis and Frederick Mosteller" - id: wikipedia-presidents-by-home-state resource: "https://en.wikipedia.org/wiki/List_of_presidents_of_the_United_States_by_home_state" title: "List of presidents of the United States by home state" author: "Wikipedia contributors" - id: judson-1980 resource: "https://archive.org/details/searchforsolutio0000juds_r0s0" title: "The Search for Solutions" author: "Horace Freeland Judson" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Presidents and States Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 19. Discussed together with [n dots on the rim of a circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's handout for this exercise has not survived. The two patterns below were chosen by the editors as examples of the kind the session is about, not as his. **Presidents.** Every president elected in a year divisible by twenty from 1840 to 1960 died in office: Harrison, Lincoln, Garfield, McKinley, Harding, Franklin Roosevelt and Kennedy. Only one other president had died in office: Zachary Taylor, elected in 1848, who died in 1850. **States.** Maine voted for state offices in September, two months before the presidential vote. From 1820 to 1932 that result matched the party of the next president in 21 of 28 cycles: "As Maine goes, so goes the nation". Missouri voted for the winner in every presidential election from 1904 through 2004 except 1956. **Your task.** Is each regularity a pattern, an accident or a law, and what mechanism could produce it? Imagining it is 1970, give odds that the president elected in 1980 will die in office, and say how you got them. How many other rules of this shape could you have searched for? ![Timeline from 1840 to 2020 with a marker for each presidential election held in a year divisible by twenty; the markers for 1840 through 1960 are filled, showing that each of those presidents died in office, and those for 1980, 2000 and 2020 are open, showing that the pattern failed](https://tyson-swetnam.github.io/aosd/assets/images/problems/presidents-and-states.svg){ width="560" } *Drawn for this site (CC BY 4.0). Small squares mark the other elections; the filled one is Taylor's.* ## Why it is in the course Session 19 opens Section 4, "Patterns, Empirical Generalizations", with Judson's chapter *Pattern*, and pairs this exercise with [n dots on the rim of a circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md). The disk gives a number pattern you can test by drawing the next case; history offers no next case on demand. Both ask how many confirmations a generalization needs. Session 4 asked for "facts before explanations of facts" and for "distinguishing things we know vs only imagine". That seven zero-year presidents died in office is a fact. A curse is an explanation, and accepting it is a separate step from checking the record. ## Where it comes from *Ripley's Believe It or Not!* noted the zero-year pattern in 1931 and again in 1948. It came to be blamed on a curse laid by the Shawnee leader Tenskwatawa after his brother's defeat by Harrison at Tippecanoe in 1811, hence the name "Curse of Tippecanoe". Maine's reputation dates to 1840. In 1936 Maine voted Republican in September, and in November Landon carried only Maine and Vermont, which prompted James Farley's quip "As Maine goes, so goes Vermont". A 1959 law moved Maine's elections to November. Missouri then inherited the reputation. ![An 1848 lithograph banner: oval portraits of earlier presidents around a central portrait of Lewis Cass, Democratic candidate](https://tyson-swetnam.github.io/aosd/assets/images/problems/presidents-and-states-1848.jpg){ width="560" } *Popular Graphic Arts, "The Presidents of the United States" (1848), Library of Congress, via Wikimedia Commons. Public domain. A campaign banner for Lewis Cass.* ??? tip "Hints" - How many presidents had there been by 1970, and how many had died in office? Only then ask how remarkable the zero-year list is. - State the rule so exactly that someone in 1970 could apply it. Lincoln and Roosevelt each died in a term won in a later election. Does the rule still count them? - A pattern found by searching many possible rules must be judged against the number of rules searched. - One state has a story for why its record worked, and the story predicts when it should stop working. Which state? ??? success "Resolution" **The presidents.** The run broke in 1980: Reagan was shot in March 1981 and survived, and Bush (2000) and Biden (2020) served out their terms. It should not have been very surprising. About one in five of the presidents before Reagan had died in office. The rule was stated after the fact, with a free choice of interval, starting year and criterion. Let the zero year be a year of death as well as of election, and Taylor, who died in 1850, fits too. Nobody publishes the patterns that fail. Diaconis and Mosteller name the two traps the multiplicity of endpoints and the law of truly large numbers: a rule that may count many outcomes as hits is far less improbable than it looks. A rule with no mechanism earns no credit for the next case, so honest odds in 1970 were near the base rate, not near certainty. **The states.** Maine's record had a mechanism: its early election sampled the same national mood as the November vote. That explains the record and its end, since the 1936 landslide showed the sample could mislead and the 1959 move to November removed it. Missouri had no such mechanism beyond resembling the country for a while, and its run ended with the 2008 election without any change in the rule. **The moral.** Rules that survive, such as Kepler's third law in [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md), rest on a mechanism, not on a longer run of confirmations. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Internet Archive](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Wikipedia contributors**, "Curse of Tippecanoe" β€” [Wikipedia](https://en.wikipedia.org/wiki/Curse_of_Tippecanoe){target=_blank} πŸ”“ - **Wikipedia contributors**, "As Maine goes, so goes the nation" β€” [Wikipedia](https://en.wikipedia.org/wiki/As_Maine_goes,_so_goes_the_nation){target=_blank} πŸ”“ - **Wikipedia contributors**, "Missouri bellwether" β€” [Wikipedia](https://en.wikipedia.org/wiki/Missouri_bellwether){target=_blank} πŸ”“ - **Wikipedia contributors**, "Bellwether (politics)" β€” [Wikipedia](https://en.wikipedia.org/wiki/Bellwether_(politics)){target=_blank} πŸ”“ (the end of Missouri's run) - **Persi Diaconis and Frederick Mosteller**, "Methods for Studying Coincidences", *Journal of the American Statistical Association* 84(408), 853–861 (1989) β€” [PDF](https://www.stat.berkeley.edu/~aldous/157/Papers/diaconis_mosteller.pdf){target=_blank} πŸ”“ - **Wikipedia contributors**, "List of presidents of the United States by home state" β€” [Wikipedia](https://en.wikipedia.org/wiki/List_of_presidents_of_the_United_States_by_home_state){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions* (1980), chapter 2, "Pattern" β€” [Internet Archive](https://archive.org/details/searchforsolutio0000juds_r0s0){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure. The syllabus and Winfree's archived handout give only the name and the session, and no other source names the exercise. Candidate readings: - **Bellwether states** such as Maine and Missouri, which join both nouns in the title. - **The zero-year presidents**, the closest match to the disk problem's run of confirmations followed by failure, but with no obvious "states". - **A table of presidents against states** to search for patterns. Eight presidents were born in Virginia, yet by home state New York leads with seven, so the headline depends on the column you read. - **Two datasets side by side**, one about presidents and one about states. The reconstruction above takes this reading. --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/ --- title: "Cell Shapes Lab" description: "A three-session group lab in counting the sides of cells in a flat soap froth, a leaf peel or a drawn mosaic, pooling the class's data and hunting for the empirical rules hidden in it." type: Activity tags: [course, student-facing, problem, section-4, foams, cell-packing, euler-formula, empirical-laws] status: stable problem: section: 4 session: 20 identification: probable kind: lab generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: thompson-1917 resource: "https://www.gutenberg.org/ebooks/55264" title: "On Growth and Form (first edition, 1917), Chapters VII-VIII, 'The Forms of Tissues or Cell-Aggregates'" author: "D'Arcy Wentworth Thompson" - id: hooke-1665 resource: "https://commons.wikimedia.org/wiki/File:Micrographia_Schem_11.jpg" title: "Micrographia (1665), Observation XVIII, 'Of the Schematisme or Texture of Cork, and of the Cells and Pores of some other such frothy Bodies' (Scheme XI plate)" author: "Robert Hooke" - id: wikipedia-plateaus-laws resource: "https://en.wikipedia.org/wiki/Plateau%27s_laws" title: "Plateau's laws" author: "Wikipedia contributors" - id: lewis-1928 resource: "https://doi.org/10.1002/ar.1090380305" title: "The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of Cucumis" author: "Frederic T. Lewis" - id: matzke-1946 resource: "https://doi.org/10.1002/j.1537-2197.1946.tb10347.x" title: "The three-dimensional shape of bubbles in foam - an analysis of the role of surface forces in three-dimensional cell shape determination" author: "Edwin B. Matzke" - id: aboav-1970 resource: "https://doi.org/10.1016/0026-0800(70)90038-8" title: "The arrangement of grains in a polycrystal" author: "D. A. Aboav" - id: mullins-1956 resource: "https://doi.org/10.1063/1.1722511" title: "Two-dimensional motion of idealized grain boundaries" author: "W. W. Mullins" - id: weaire-rivier-1984 resource: "https://doi.org/10.1080/00107518408210979" title: "Soap, cells and statistics - random patterns in two dimensions" author: "Denis Weaire and Nicolas Rivier" - id: roth-jones-durian-2012 resource: "https://arxiv.org/abs/1206.2293" title: "Coarsening of Two Dimensional Foam on a Dome" author: "A. E. Roth, C. D. Jones and D. J. Durian" - id: kelvin-1887 resource: "https://doi.org/10.1080/14786448708628135" title: "On the division of space with minimum partitional area" author: "Sir William Thomson (Lord Kelvin)" - id: weaire-phelan-1994 resource: "https://doi.org/10.1080/09500839408241577" title: "A counter-example to Kelvin's conjecture on minimal surfaces" author: "Denis Weaire and Robert Phelan" - id: graner-riveline-2017 resource: "https://doi.org/10.1242/dev.151233" title: "'The Forms of Tissues, or Cell-aggregates': D'Arcy Thompson's influence and its limits" author: "FranΓ§ois Graner and Daniel Riveline" - id: fischer-bassel-kollmannsberger-2023 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC10369035/" title: "Tissues as networks of cells: towards generative rules of complex organ development" author: "Sabine C. Fischer, George W. Bassel and Philip Kollmannsberger" - id: altman-winfree-1977 resource: "https://doi.org/10.1002/cne.901710102" title: "Postnatal development of the cerebellar cortex in the rat. V. Spatial organization of Purkinje cell perikarya" author: "Joseph Altman and Arthur T. Winfree" - id: winfree-chronological-2002 resource: "https://web.archive.org/web/20020110220211/http://eebweb.arizona.edu:80/Faculty/Winfree/chronological.html" title: "Winfree publications: chronological (archived lab page)" author: "Arthur T. Winfree" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Cell Shapes Lab Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 20, continuing through sessions 21 and 22. Session 21 also deals with [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md) and [Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md); session 22 with the [Egg Pouches](https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/index.md) lab and [Platonic Solids](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md).* !!! abstract "The problem" Reconstructed from the syllabus, which says only "Start Cell Shapes lab in class", then "Collaborative experiments" and "Further experiments on cell shapes". No lab sheet survives; the recipe is the editors' suggestion. **The material.** A flat patch of cells, with no gaps and no overlaps. Different groups can use different ones: a *froth* (bubbles blown into a saucer of dish-soap solution and squashed into one layer under clear plastic), a *living sheet* (onion skin, a leaf peel, a micrograph), or a *drawn sheet* (forty scattered dots, each given the points nearer to it than to any other dot). **The task.** Count before you theorise. Record each cell's sides and area, and how many walls meet at each junction, at what angles. Pool the class's counts and look for regularities: the average number of sides, and any link between size and sides, or between a cell's sides and its neighbours'. If you made a froth, revisit the same cells next session: which grew, which vanished? Then try to prove whichever rule survives. ![Photograph of a single layer of soap bubbles seen from above: bright thin walls outline an irregular mosaic of polygonal cells on a dark background](https://tyson-swetnam.github.io/aosd/assets/images/problems/cell-shapes-lab-foam.jpg){ width="560" } *A two-dimensional foam, the kind of patch a group would count. Photograph by Klaus-Dieter Keller. Public domain, via Wikimedia Commons.* ## Why it is in the course Section 4 is "Patterns, Empirical Generalizations". Nobody announces the law: the class finds it by counting, then tries to explain it, in session 4's order, "facts before explanations of facts". It is one of the course's puzzles "for simple lab manipulation", and the syllabus gives it three sessions, the second for "Collaborative experiments". It also sets a trap. A froth *looks* hexagonal, so people say "hexagons". Some of what the counts show is forced by topology, and some is only roughly true; telling those apart echoes session 4's "distinguishing things we know vs only imagine". Session 22 pairs the last experiments with [Platonic Solids](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md), carrying the question into three dimensions. ## Where it comes from Robert Hooke named the biological "cell" in *Micrographia* (1665), describing cork under a heading that already speaks of "frothy Bodies". Joseph Plateau set out the rules of soap films in the nineteenth century; D'Arcy Thompson's *On Growth and Form* (1917) saw the same froth in plant tissue, with cell walls "everywhere meeting, by threes, at angles of 120 degrees, irrespective of the size of the individual cells". Then came the counters: F. T. Lewis on cucumber epidermis (1928), Edwin Matzke on 600 foam bubbles (1946), D. A. Aboav on grains in metal (1970), von Neumann and W. W. Mullins on how a flat froth coarsens (1952, 1956). Weaire and Rivier reviewed the field in 1984, and Graner and Riveline (2017) test Thompson's analogy against living tissue. Winfree knew this ground: his publication list includes a 1961 Westinghouse Science Talent Search award project, "The physics and chemistry of soap bubbles and films", and a 1977 study with Joseph Altman of how Purkinje cells are spaced. ??? tip "Hints" - Settle your definitions first: what is a side, a vertex, a rim cell? Write the rules down. - Look at the junctions before the cells. If four walls seem to meet, look closer. - Compute the mean and the whole distribution, not just the commonest value. - Plot area against number of sides, and a cell's sides against its neighbours' average. Which relation only looks convincing? - Count vertices, edges and faces and try Euler's V - E + F = 2. Could your average have come out any other way? ??? success "Resolution" - **Three walls to a junction, at 120 degrees.** Plateau's rule in a froth; a four-way junction splits into two three-way ones. - **Six sides on average, a theorem.** If three edges meet at every vertex, 3V = 2E, and Euler's formula gives F = E/3 + 2. Each edge borders two cells, so the mean number of sides is 2E/F, which tends to 6 as the patch grows. Topology forces it, not soap or biology. - **Not hexagons.** Five-, six- and seven-sided cells dominate. In Thompson's words, "the cells will be *on the average* hexagonal, but some will have fewer and some more sides than six". - **Lewis's law (approximate).** Cells with more sides tend to be larger; a rule of thumb that fails in several real tissues (Fischer and colleagues, 2023). - **Aboav-Weaire law (approximate).** Many-sided cells tend to have few-sided neighbours. - **Von Neumann's law, in a froth watched over days.** A cell's area changes at a rate proportional to (n - 6): cells with fewer than six sides shrink and vanish, those with more grow, and the average stays at six (tested on a real foam by Roth, Jones and Durian, 2012). ![Left: a patch of seven regular hexagons, each marked 6, with one inner junction circled where three walls meet at 120 degrees. Right: the same patch after two neighbouring cells stop touching and the cells above and below them meet instead, giving two 5-sided cells with arrows pointing inward and two 7-sided cells with arrows pointing outward; the sides still total 42 over 7 cells](https://tyson-swetnam.github.io/aosd/assets/images/problems/cell-shapes-lab.svg){ width="560" } *Drawn for this site (CC BY 4.0). Schematic, with straight walls.* **In three dimensions** no Platonic solid appears: Matzke's bubbles averaged about 13.7 faces, and Kelvin's 1887 candidate for the least-area partition of space was beaten in 1994 by the Weaire-Phelan structure. ## Sources - **D'Arcy Wentworth Thompson**, *On Growth and Form*, 1st ed., Chapters VII-VIII (1917) β€” [Project Gutenberg](https://www.gutenberg.org/ebooks/55264){target=_blank} πŸ”“ - **Robert Hooke**, *Micrographia*, Observation XVIII and Scheme XI (1665) β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Micrographia_Schem_11.jpg){target=_blank} πŸ”“ - **Wikipedia contributors**, "Plateau's laws" β€” [Wikipedia](https://en.wikipedia.org/wiki/Plateau%27s_laws){target=_blank} πŸ”“ - **Frederic T. Lewis**, "The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of Cucumis", *Anatomical Record* 38, 341–376 (1928) β€” [doi:10.1002/ar.1090380305](https://doi.org/10.1002/ar.1090380305){target=_blank} πŸ”’ - **Edwin B. Matzke**, "The three-dimensional shape of bubbles in foam", *American Journal of Botany* 33, 58–80 (1946) β€” [doi:10.1002/j.1537-2197.1946.tb10347.x](https://doi.org/10.1002/j.1537-2197.1946.tb10347.x){target=_blank} πŸ”’ - **D. A. Aboav**, "The arrangement of grains in a polycrystal", *Metallography* 3, 383–390 (1970) β€” [doi:10.1016/0026-0800(70)90038-8](https://doi.org/10.1016/0026-0800(70)90038-8){target=_blank} πŸ”’ - **W. W. Mullins**, "Two-dimensional motion of idealized grain boundaries", *Journal of Applied Physics* 27, 900–904 (1956) β€” [doi:10.1063/1.1722511](https://doi.org/10.1063/1.1722511){target=_blank} πŸ”’ - **Denis Weaire and Nicolas Rivier**, "Soap, cells and statistics - random patterns in two dimensions", *Contemporary Physics* 25, 59–99 (1984) β€” [doi:10.1080/00107518408210979](https://doi.org/10.1080/00107518408210979){target=_blank} πŸ”’ - **A. E. Roth, C. D. Jones and D. J. Durian**, "Coarsening of Two Dimensional Foam on a Dome", *Physical Review E* 86, 021402 (2012) β€” [arXiv:1206.2293](https://arxiv.org/abs/1206.2293){target=_blank} πŸ”“ - **Sir William Thomson (Lord Kelvin)**, "On the division of space with minimum partitional area", *Philosophical Magazine* 24, 503–514 (1887) β€” [doi:10.1080/14786448708628135](https://doi.org/10.1080/14786448708628135){target=_blank} πŸ”’ - **Denis Weaire and Robert Phelan**, "A counter-example to Kelvin's conjecture on minimal surfaces", *Philosophical Magazine Letters* 69, 107–110 (1994) β€” [doi:10.1080/09500839408241577](https://doi.org/10.1080/09500839408241577){target=_blank} πŸ”’ - **FranΓ§ois Graner and Daniel Riveline**, "'The Forms of Tissues, or Cell-aggregates': D'Arcy Thompson's influence and its limits", *Development* 144, 4226–4237 (2017) β€” [doi:10.1242/dev.151233](https://doi.org/10.1242/dev.151233){target=_blank} πŸ”’ - **Sabine C. Fischer, George W. Bassel and Philip Kollmannsberger**, "Tissues as networks of cells: towards generative rules of complex organ development", *Journal of the Royal Society Interface* 20, 20230115 (2023) β€” [PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC10369035/){target=_blank} πŸ”“ - **Joseph Altman and Arthur T. Winfree**, "Postnatal development of the cerebellar cortex in the rat. V. Spatial organization of Purkinje cell perikarya", *Journal of Comparative Neurology* 171, 1–16 (1977) β€” [doi:10.1002/cne.901710102](https://doi.org/10.1002/cne.901710102){target=_blank} πŸ”’ - **Arthur T. Winfree**, publications in chronological order (archived lab page, 2002) β€” [Internet Archive](https://web.archive.org/web/20020110220211/http://eebweb.arizona.edu:80/Faculty/Winfree/chronological.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery* course handout (archived 20 April 2002) β€” [Internet Archive](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Sure of the subject, not of the method. The syllabus gives the name and three sessions; Winfree's archived handout lists the same entries. No description of the exercise survives, so the medium is unknown. - **A flat soap froth**, watched as it coarsens: cheap, changing between classes, and close to Winfree's 1961 soap-film project. Medium confidence. - **Biological cell sheets**, such as leaf or onion epidermis, the material of Lewis's counts. Needs microscopes. Medium confidence. - **A paper construction**, such as cells drawn around scattered dots, following session 19's [slicing of a disk](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md). Needs only pencils, but "experiments" suggest something that changes. Low confidence. --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/neutrinos/ --- title: "Neutrinos" description: "An editors' reconstruction of the solar neutrino problem, in which every detector counted too few neutrinos from the Sun for thirty years: state the pattern, sort the explanations, and find the measurement that decides between them." type: Activity tags: [course, student-facing, problem, section-4, solar-neutrinos, empirical-generalizations, anomalies, physics] status: stable problem: section: 4 session: 21 identification: probable kind: case-study generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: bahcall-2004-arxiv resource: "https://arxiv.org/abs/physics/0406040" title: "Solving the Mystery of the Missing Neutrinos" author: "John N. Bahcall" - id: bahcall-2004-nobel resource: "https://www.nobelprize.org/prizes/themes/solving-the-mystery-of-the-missing-neutrinos/" title: "Solving the mystery of the missing neutrinos (Nobel Foundation web version)" author: "John N. Bahcall" - id: davis-harmer-hoffman-1968 resource: "https://doi.org/10.1103/PhysRevLett.20.1205" title: "Search for Neutrinos from the Sun" author: "Raymond Davis Jr., Don S. Harmer, Kenneth C. Hoffman" - id: sno-2001 resource: "https://arxiv.org/abs/nucl-ex/0106015" title: "Measurement of the rate of nu_e + d -> p + p + e^- interactions produced by 8B solar neutrinos at the Sudbury Neutrino Observatory" author: "SNO Collaboration (Q. R. Ahmad et al.)" - id: sno-2002 resource: "https://arxiv.org/abs/nucl-ex/0204008" title: "Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory" author: "SNO Collaboration (Q. R. Ahmad et al.)" - id: bahcall-pena-garay-2004 resource: "https://arxiv.org/abs/hep-ph/0404061" title: "Solar models and solar neutrino oscillations" author: "John N. Bahcall and Carlos Pena-Garay" - id: nobel-physics-2002 resource: "https://www.nobelprize.org/prizes/physics/2002/press-release/" title: "The Nobel Prize in Physics 2002: press release" author: "Royal Swedish Academy of Sciences" - id: nobel-physics-2015 resource: "https://www.nobelprize.org/prizes/physics/2015/press-release/" title: "The Nobel Prize in Physics 2015: press release" author: "Royal Swedish Academy of Sciences" - id: ehrlich-2001-ch6 resource: "https://doi.org/10.1515/9780691187839-007" title: "Nine Crazy Ideas in Science, chapter 6: The Solar System Has Two Suns" author: "Robert Ehrlich" - id: ehrlich-2001-book resource: "https://archive.org/details/ninecrazyideasin00ehrl" title: "Nine Crazy Ideas in Science: A Few Might Even Be True" author: "Robert Ehrlich" - id: publishers-weekly-ehrlich resource: "https://www.publishersweekly.com/9780691070018" title: "Review: Nine Crazy Ideas in Science: A Few Might Even Be True" author: "Publishers Weekly" - id: raup-sepkoski-1984 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC344925/" title: "Periodicity of extinctions in the geologic past" author: "David M. Raup and J. John Sepkoski Jr." - id: wikimedia-homestake-tank resource: "https://commons.wikimedia.org/wiki/File:U.S._Department_of_Energy_-_Science_-_390_002_007_(9952118384).jpg" title: "The underground tank where Brookhaven National Laboratory's solar neutrino experiment took place" author: "U.S. Department of Energy" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Neutrinos Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 21. The same session deals with [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md) and continues the [Cell Shapes](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) experiments.* !!! abstract "The problem" The syllabus gives only the name. No description of the problem survives, so this statement is the editors' reconstruction, built on the historical numbers. The Sun's fusion reactions release neutrinos, uncharged particles that pass through the Sun, the Earth and you almost without a trace. A model of the solar interior predicts how many should reach the Earth, and at what energies. Detectors began counting them in 1968. Each is blind below some energy, so each samples a different slice of the spectrum. (1 SNU is one capture per second per 10^36^ target atoms.) | Detector | Sees neutrinos above | Measured | Predicted | | :-- | :-- | :-- | :-- | | Chlorine, Homestake mine, South Dakota (1968–1994) | about 0.8 MeV | 2.6 Β± 0.2 SNU | about 8.5 SNU | | Gallium, GALLEX/GNO (Italy) and SAGE (Russia), from 1991 | about 0.23 MeV | 69 Β± 4 SNU | about 131 SNU | | Water, Kamiokande and Super-Kamiokande (Japan), from 1987 | roughly 5 MeV | roughly half the prediction | β€” | Repeated for decades, the shortfall never went away. **Your task.** 1. State the pattern as precisely as the numbers allow. Does the shortfall depend on energy? 2. List every explanation you can and sort them into a few classes. 3. For each class, name an observation that would kill it. 4. What single measurement could decide among the classes without trusting the solar model at all? 5. Put yourself in the early 1990s with this table. Which class would you bet on? Write it down before reading "What happened". ## Why it is in the course Section 4 is "Patterns, Empirical Generalizations". Session 19's [n dots on a circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md) shows a pattern that betrays you (1, 2, 4, 8, 16, then 31). The neutrino deficit is the opposite: a pattern that survived every attempt to break it. The syllabus labels its session-04 problems "facts before explanations of facts" and "distinguishing things we know vs only imagine". Here the fact, too few neutrinos, stood for thirty years, while the explanation most people imagined, a faulty calculation or experiment, was wrong. The day's reading, Ehrlich's chapter 6, "The Solar System Has Two Suns", takes up a companion star proposed to explain a claimed periodicity in mass extinctions. And in Winfree's handout the link on [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md) points to a bookmark named `Keplers_Laws`. If that exercise was Kepler's laws, as the name suggests, the session set a pattern that meant what it looked like beside one that did not. ## Where it comes from In 1964 John Bahcall calculated how many argon-37 atoms solar neutrinos should make in a tank of chlorine-rich cleaning fluid. Raymond Davis Jr. built the tank a mile underground in the Homestake gold mine. His 1968 paper gave only an upper limit, already below the prediction; later runs found about a third of it. ![A curved pale steel tank with walkways over it, in a rock cavern lined with safety netting; a worker stands on the walkway](https://tyson-swetnam.github.io/aosd/assets/images/problems/neutrinos-homestake-tank.jpg){ width="560" } *The chlorine tank of Brookhaven National Laboratory's solar neutrino experiment in the Homestake mine. U.S. Department of Energy photograph, public domain, via Wikimedia Commons.* Bahcall later named three classes of explanation: the calculation was wrong, the experiment was wrong, or, "the most daring and least discussed possibility", physicists misunderstood neutrinos. Gribov and Pontecorvo had proposed that in 1969; few took it seriously. In 1997 helioseismology measured the speed of sound inside the Sun and found it matched the solar model's to within 0.1 percent. The schedule places session 21 on Tuesday 30 October 2001, four months after the Sudbury Neutrino Observatory (SNO) submitted its first result in June 2001. But the syllabus calls the schedule "a retrospective syllabus of Spring 2001, with dates changed to reflect the future", so the problem may first have been set before that result. ??? tip "Hints" - Chlorine sees about a third, gallium about half. Could one mistake, or a Sun dimmer by a fixed factor, explain both? - Sort explanations by what they attack: the Sun, the detector, or the particles on their eight-minute trip. Which was almost unmentionable? - Is there a measurement that compares neutrinos with neutrinos, rather than with a prediction? ## What happened The data were right; a hidden assumption was wrong. Neutrinos have mass, so one born as an electron-type neutrino in the Sun can arrive as a muon- or tau-type neutrino, a change amplified inside the Sun by the matter effect of Mikheyev, Smirnov and Wolfenstein. The detectors saw mostly or only electron-type neutrinos, so all counted low, by an amount that depends on energy. That is why chlorine and gallium disagree with each other as well as with the model. ![Bar chart: chlorine 0.31, gallium 0.53, water about 0.45 and SNO 2001 electron-type 0.35 of the solar-model prediction; SNO 2002 all neutrino types about 1.0](https://tyson-swetnam.github.io/aosd/assets/images/problems/neutrinos.svg){ width="560" } *Drawn for this site (CC BY 4.0). SNO fluxes are compared with about 5.05 million per cmΒ² per second predicted for the boron-8 branch.* SNO's heavy water allows one reaction that only electron-type neutrinos drive and another that all three types drive. In 2001 its electron-neutrino flux, 1.75 million per cmΒ² per second, differed from Super-Kamiokande's rate by 3.3 sigma. In 2002 it measured all three types together: 5.09 million, as the solar model predicted. The missing two-thirds had changed identity. Davis and Masatoshi Koshiba shared half of the 2002 Nobel Prize in Physics; Takaaki Kajita and Arthur McDonald shared the 2015 prize. Why so slow? Bahcall quotes Sidney Drell: "the success of the Standard Model (of particle physics) was too dear to give up." ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ (session 21 line and date; the Keplers_Laws bookmark on Paired Observations) - **John N. Bahcall**, "Solving the Mystery of the Missing Neutrinos" (2004) β€” [arXiv:physics/0406040](https://arxiv.org/abs/physics/0406040){target=_blank} πŸ”“; also at [nobelprize.org](https://www.nobelprize.org/prizes/themes/solving-the-mystery-of-the-missing-neutrinos/){target=_blank} πŸ”“ - **Raymond Davis Jr., Don S. Harmer and Kenneth C. Hoffman**, "Search for Neutrinos from the Sun", *Physical Review Letters* 20, 1205–1209 (1968) β€” [doi:10.1103/PhysRevLett.20.1205](https://doi.org/10.1103/PhysRevLett.20.1205){target=_blank} πŸ”’ - **SNO Collaboration (Q. R. Ahmad et al.)**, charged-current rate from boron-8 solar neutrinos, *Physical Review Letters* 87, 071301 (2001) β€” [arXiv:nucl-ex/0106015](https://arxiv.org/abs/nucl-ex/0106015){target=_blank} πŸ”“ - **SNO Collaboration (Q. R. Ahmad et al.)**, "Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory", *Physical Review Letters* 89, 011301 (2002) β€” [arXiv:nucl-ex/0204008](https://arxiv.org/abs/nucl-ex/0204008){target=_blank} πŸ”“ - **John N. Bahcall and Carlos PeΓ±a-Garay**, "Solar models and solar neutrino oscillations", *New Journal of Physics* 6, 63 (2004) β€” [arXiv:hep-ph/0404061](https://arxiv.org/abs/hep-ph/0404061){target=_blank} πŸ”“ (the capture rates in the table) - **Royal Swedish Academy of Sciences**, Nobel Prize in Physics press releases β€” [2002](https://www.nobelprize.org/prizes/physics/2002/press-release/){target=_blank} πŸ”“, [2015](https://www.nobelprize.org/prizes/physics/2015/press-release/){target=_blank} πŸ”“ - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True* (2001), chapter 6, "The Solar System Has Two Suns" β€” [doi:10.1515/9780691187839-007](https://doi.org/10.1515/9780691187839-007){target=_blank} πŸ”’; whole book at the [Internet Archive](https://archive.org/details/ninecrazyideasin00ehrl){target=_blank} πŸ”’ *(print-disabled readers only)* - **Publishers Weekly**, review of *Nine Crazy Ideas in Science* (2001) β€” [publishersweekly.com](https://www.publishersweekly.com/9780691070018){target=_blank} πŸ”“ - **David M. Raup and J. John Sepkoski Jr.**, "Periodicity of extinctions in the geologic past", *PNAS* 81, 801–805 (1984) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC344925/){target=_blank} πŸ”“ (the pattern behind the session's Ehrlich chapter) - **U.S. Department of Energy**, photograph of the Homestake tank β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:U.S._Department_of_Energy_-_Science_-_390_002_007_(9952118384).jpg){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus says only "Deal with Neutrinos", and no surviving Winfree material describes the problem. Identification is **probable**: - **The solar neutrino problem** (high confidence): what "the neutrino problem" meant in 2001, the year SNO's first result began to settle it. - **Ehrlich's case that neutrinos travel faster than light** (low): neutrinos are Ehrlich's own field (Publishers Weekly notes it), but that is his chapter 9, which the syllabus never assigns. - **A claimed periodicity in the neutrino counts** (low): it would echo chapter 6, but no Winfree material mentions it. --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/paired-observations/ --- title: "Paired Observations" description: "An editors' reconstruction of a lost Winfree exercise filed under Kepler's laws: find the rule hidden in six pairs of numbers, one distance and one period per planet, then test it on planets and moons it never saw." type: Activity tags: [course, student-facing, problem, section-4, kepler, empirical-laws, power-laws, data-analysis] status: stable problem: section: 4 session: 21 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: kepler-harmonices-mundi-1619 resource: "https://archive.org/details/ioanniskepplerih00kepl" title: "Ioannis Keppleri Harmonices mundi libri V (Linz, 1619)" author: "Johannes Kepler" - id: kepler-astronomia-nova-1609 resource: "https://archive.org/details/Astronomianovaa00Kepl" title: "Astronomia nova (1609)" author: "Johannes Kepler" - id: langley-1981 resource: "https://doi.org/10.1111/j.1551-6708.1981.tb00869.x" title: "Data-Driven Discovery of Physical Laws" author: "Pat Langley" - id: langley-simon-bradshaw-zytkow-1987 resource: "https://doi.org/10.7551/mitpress/6090.001.0001" title: "Scientific Discovery: Computational Explorations of the Creative Processes" author: "Patrick W. Langley, Herbert A. Simon, Gary Bradshaw, Jan M. Zytkow" - id: qin-simon-1990 resource: "https://doi.org/10.1207/s15516709cog1402_4" title: "Laboratory Replication of Scientific Discovery Processes" author: "Yulin Qin and Herbert A. Simon" - id: ehrlich-2001 resource: "https://archive.org/details/ninecrazyideasin00ehrl" title: "Nine Crazy Ideas in Science: A Few Might Even Be True" author: "Robert Ehrlich" - id: koestler-1960 resource: "https://archive.org/details/watershedbiograp00koes" title: "The Watershed: A Biography of Johannes Kepler" author: "Arthur Koestler" - id: nasa-planetary-fact-sheets resource: "https://nssdc.gsfc.nasa.gov/planetary/factsheet/" title: "Planetary Fact Sheets, NASA Space Science Data Coordinated Archive" author: "David R. Williams, NASA Goddard Space Flight Center" - id: nasa-jovian-satellite-fact-sheet resource: "https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html" title: "Jovian Satellite Fact Sheet, NASA Space Science Data Coordinated Archive" author: "David R. Williams, NASA Goddard Space Flight Center" - id: gordon-scientific-problem-solving resource: "https://scientificproblemsolving.com/Mods/ModsIndex.html" title: "Scientific Problem Solving: Modules index" author: "Herman Gordon" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Paired Observations Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 21. The same session continues the collaborative experiments on [Cell Shapes](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) and deals with [Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's problem sheet has not survived, so this is the editors' reading of the one clue in his handout (see below). The numbers are modern NASA values, not Winfree's. **Part 1. A table with no labels.** Each row is a pair of measurements made on one object. | Object | x | y | | :-- | --: | --: | | A | 57.909 | 87.969 | | B | 108.210 | 224.701 | | C | 149.598 | 365.256 | | D | 227.956 | 686.980 | | E | 778.479 | 4332.589 | | F | 1432.041 | 10755.699 | y grows with x, but not in proportion. Find a rule y = f(x) that fits every row to better than one per cent, and record how you searched, including the rules you threw away. Do this before reading on. **Part 2. The labels.** x is a planet's mean distance from the Sun (million km); y is its orbital period (days). The rows are Mercury, Venus, Earth, Mars, Jupiter and Saturn. Does knowing this change your confidence in the rule? **Part 3. Tests the rule did not see.** Two planets unknown in 1619 lie at 2867.043 and 4514.953 million km from the Sun. Predict their periods before looking them up. Then try Jupiter's four large moons, with distance measured from Jupiter: | Moon | distance (thousand km) | period (days) | | :-- | --: | --: | | Io | 421.8 | 1.769138 | | Europa | 671.1 | 3.551181 | | Ganymede | 1070.4 | 7.154553 | | Callisto | 1882.7 | 16.689017 | Does the same form of rule work, with the same constant? And what does your rule leave unexplained? ## Why it is in the course Section 4 is "Patterns, Empirical Generalizations", and this is the model case of one. Kepler's rule for the planets was found in a table of numbers, and it was right for decades before anyone could say why. The exercise trains three habits. Search the space of possible rules on purpose: ratios, powers and logarithms make a hidden regularity show up as a constant or a straight line. Treat a rule fitted to six rows as unproven until it predicts rows it has not seen. And remember that a correct empirical generalization says *that*, not *why*. (The day's reading, Ehrlich's Chapter 6, "The Solar System Has Two Suns", is also about orbits; that link is the editors' observation.) ## Where it comes from In Winfree's handout, the link on this item points to a bookmark named `Keplers_Laws`. No archived capture contains the target, so the link text and the bookmark name are all that survive of the exercise. Kepler was also on his reading list: the syllabus puts "Arthur Koestler, *The Watershed*, biography of Johannes Kepler (a chapter of *The Sleepwalkers*)" on reserve. From Tycho Brahe's observations of Mars, Johannes Kepler found his first two laws in *Astronomia nova* (1609). The third came a decade later. In Book V of *Harmonices mundi* (Linz, 1619) he states an exact proportion between the periodic times of any two planets and their mean distances, found in the spring of 1618 after a false start. It stood as an empirical law with no accepted explanation until Newton's gravitation. ![Title page of Kepler's Harmonices mundi libri V, printed at Linz in 1619](https://tyson-swetnam.github.io/aosd/assets/images/problems/paired-observations-harmonices.jpg){ width="560" } *Johannes Kepler, Harmonices mundi libri V (Linz, 1619), title page; scan from the Posner Library, Carnegie Mellon University. Public domain, via Wikimedia Commons.* The law later became a benchmark for studying discovery: the BACON programs of Pat Langley and colleagues rediscovered versions of it from data (1981, 1987), and when Yulin Qin and Herbert Simon gave fourteen people the numbers unlabelled (1990), four found it. ??? tip "Hints" - By what factor do x and y each grow from first row to last? - For two rows, find p with (ratio of y) = (ratio of x) to the power p. Same p for another pair? - Logarithms turn powers into slopes: plot log y against log x. - Build a quantity from x and y that should be the same in every row, and check it. ??? success "Resolution" **The rule.** A least-squares line through the six points (log x, log y) has slope 1.4985, very close to 3/2. So y goes as x to the power 3/2: the period squared is proportional to the distance cubed. That is Kepler's third law. In these units, x cubed divided by y squared stays between 25.09 and 25.39 for all six rows. Scaling from the Earth's row, the rule predicts every period to within 0.6 per cent. **The tests.** Scaling from the Earth, y = 365.256 (x / 149.598)^(3/2) predicts 30,645 days for Uranus (actual 30,685.4) and 60,560 for Neptune (actual 60,189.0), errors of 0.13 and 0.6 per cent. Jupiter's moons give slope 1.5002, the same form, but a constant of about 0.024 in the planets' units instead of 25.1. Why each centre has its own constant, the law cannot say; that waited for Newton. ![On logarithmic axes, the periods of the eight planets against their distances from the Sun fall on a straight dashed line of slope three halves; the six planets known to Kepler are filled dots, Uranus and Neptune open dots on the same line](https://tyson-swetnam.github.io/aosd/assets/images/problems/paired-observations.svg){ width="560" } *Period against distance for the eight planets, log scales, NASA fact-sheet values. Drawn for this site (CC BY 4.0).* ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579), course handout; the session-21 schedule line and its link markup β€” [Wayback Machine capture, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Johannes Kepler**, *Harmonices mundi libri V* (Linz, 1619), Book V, chapter 3 β€” [Internet Archive](https://archive.org/details/ioanniskepplerih00kepl){target=_blank} πŸ”“ - **Johannes Kepler**, *Astronomia nova* (1609) β€” [Internet Archive](https://archive.org/details/Astronomianovaa00Kepl){target=_blank} πŸ”“ - **Pat Langley**, "Data-Driven Discovery of Physical Laws", *Cognitive Science* 5(1), 31–54 (1981) β€” [doi:10.1111/j.1551-6708.1981.tb00869.x](https://doi.org/10.1111/j.1551-6708.1981.tb00869.x){target=_blank} πŸ”’ - **Patrick W. Langley, Herbert A. Simon, Gary Bradshaw and Jan M. Zytkow**, *Scientific Discovery: Computational Explorations of the Creative Processes* (MIT Press, 1987) β€” [doi:10.7551/mitpress/6090.001.0001](https://doi.org/10.7551/mitpress/6090.001.0001){target=_blank} πŸ”’ - **Yulin Qin and Herbert A. Simon**, "Laboratory Replication of Scientific Discovery Processes", *Cognitive Science* 14(2), 281–312 (1990) β€” [doi:10.1207/s15516709cog1402_4](https://doi.org/10.1207/s15516709cog1402_4){target=_blank} πŸ”’ - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True* (Princeton University Press, 2001), Chapter 6 β€” [Internet Archive](https://archive.org/details/ninecrazyideasin00ehrl){target=_blank} πŸ”’ *(print-disabled readers only)* - **Arthur Koestler**, *The Watershed: A Biography of Johannes Kepler* (Anchor Books, 1960) β€” [Internet Archive](https://archive.org/details/watershedbiograp00koes){target=_blank} πŸ”“ *(borrow)* - **David R. Williams**, NASA Space Science Data Coordinated Archive, Planetary Fact Sheets (per-planet sheets, sidereal periods) β€” [nssdc.gsfc.nasa.gov](https://nssdc.gsfc.nasa.gov/planetary/factsheet/){target=_blank} πŸ”“ - **David R. Williams**, NASA Space Science Data Coordinated Archive, Jovian Satellite Fact Sheet β€” [nssdc.gsfc.nasa.gov](https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html){target=_blank} πŸ”“ - **Herman Gordon**, *Scientific Problem Solving*, the successor course to Winfree's: Modules index β€” [scientificproblemsolving.com](https://scientificproblemsolving.com/Mods/ModsIndex.html){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The handout's link names a topic, not a task, and no other document checked (including Herman Gordon's successor course) describes the exercise. Identification is therefore **probable**. The candidates: - **Finding Kepler's third law from paired distances and periods** (likeliest): the data are pairs, one distance and one period per planet, and the task needs only arithmetic or a log-log plot. - **Kepler's sightings of Mars taken 687 days apart** in *Astronomia nova*: also under Kepler's laws, but geometrically heavy and built on sets of three or four sightings. - **The class's paired GamesWorth judgements** (unlikely): a paired design in Winfree's own course, but his link points to Kepler. - **A matched-pairs statistics exercise** (unlikely): standard vocabulary, with nothing linking it to Winfree. --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/ --- title: "Egg Pouches Lab" description: "An in-class lab known only by its name, reconstructed here as counting and pooling data along a worm-like chain of egg pouches such as a whelk egg-case string, to find out what a whole class can generalize that one specimen cannot." type: Activity tags: [course, student-facing, problem, section-4, empirical-generalizations, pooling-data, egg-cases, segmentation] status: stable problem: section: 4 session: 22 identification: unknown kind: lab generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: wikipedia-busycon resource: "https://en.wikipedia.org/wiki/Busycon" title: "Busycon" author: "Wikipedia contributors" - id: wikipedia-annelid resource: "https://en.wikipedia.org/wiki/Annelid" title: "Annelid" author: "Wikipedia contributors" - id: wikipedia-cestoda resource: "https://en.wikipedia.org/wiki/Cestoda" title: "Cestoda" author: "Wikipedia contributors" - id: wikipedia-egg-case-chondrichthyes resource: "https://en.wikipedia.org/wiki/Egg_case_(Chondrichthyes)" title: "Egg case (Chondrichthyes)" author: "Wikipedia contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Egg Pouches Lab Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 22. The same session continues the [Cell Shapes](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) lab and deals with [Platonic Solids](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md).* !!! abstract "The problem" This is a reconstruction. The schedule says only "Do Egg Pouches lab in class", and Winfree's lab sheet is lost. What follows is a candidate lab built to fit that line and the bookmark discussed at the end of this page. It is not his wording. Each group gets a long chain of egg pouches joined end to end, such as a dried whelk egg-case string from a beach. **First, write down what you expect.** Is this one animal, or a container made by one? Which end came first? How do the pouches differ along the row? **Then count.** Number every pouch from one end, counting twice in opposite directions. Measure the size of each pouch, using the same measurement every time. Open a sample of pouches spread along the whole length and count what is inside each; record empty ones as zero. **Then pool.** Plot every group's specimens on one graph, contents against pouch number, and state a generalization in one sentence. Does it hold for every chain, or only on average? Could anything you counted tell an animal made of segments from a row of containers? ![Two panels. Left: a curved chain of eighteen lens-shaped pouches on one stalk, unequal in size but not ordered from small to large, numbered 1, 9 and 18. Right: contents counted against pouch number for four specimens of different lengths, one rising, one falling, one flat and one humped, with no trend line](https://tyson-swetnam.github.io/aosd/assets/images/problems/egg-pouches-lab.svg){ width="560" } *One chain, and four chains pooled. The sizes and counts are illustrative, not measured. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 4 is "Patterns, Empirical Generalizations". Here you repeat one measurement along a series and end with a sentence no single pouch could support. The syllabus explains why such work happens in class: "Class meetings will also prove essential for some problems in which no one individual can collect enough data, but if we pool data, reality will come into focus." One specimen is an anecdote. Twenty make a distribution. The session's reading is "Adams Chapter 6: Alternative thinking languages", and this lab cannot be done in words: you number, count and plot, then look at the shape of the points. ## Where it comes from Apart from the bookmark discussed at the end of this page, the name is the only evidence. "Egg pouches" is the popular name for the capsules on the egg-case string of a *Busycon* whelk. Wikipedia's article on the genus says the dried strings are sometimes called "mermaid's necklaces" because they look like a necklace strung with "medallion-shaped egg pouches", and that each pouch holds "numerous protoconchs (baby whelks)". A string is a ready-made series with countable contents, and at a glance it looks like a worm. ![A dried whelk egg-case string lying on beach sand: a curving row of flat, disc-shaped capsules that looks like a segmented worm](https://tyson-swetnam.github.io/aosd/assets/images/problems/egg-pouches-lab-whelk.jpg){ width="560" } *A whelk egg-case string on a beach. Photo by Smallbones, via Wikimedia Commons, CC0.* The segmented worms proper are the annelids. In earthworms and leeches, a cocoon stores the fertilized eggs: one cocoon, not a chain of pouches to count. The animal whose body really is a chain of egg packets is the tapeworm: in Wikipedia's words, "Mature proglottids are essentially bags of eggs". Yet a footnote in the same article says tapeworms "are not formed of fixed body segments as are the annelids". ??? tip "Hints" - Write your predictions down before you open anything. What you can no longer deny having believed is the point. - If your two counts in opposite directions disagree, look at the ends. - Sample pouches along the whole length, not a handful from one end. - Specimens differ in length. Before deciding a pattern has failed, plot position as a fraction of the whole chain. - Ask what observation would tell one animal from a stack of containers. If nothing you counted can settle it, say so. That is a result too. ??? success "Resolution" No resolution of Winfree's survives. The result is the class's own pooled graph: a pattern may appear only when many chains are plotted together, and may hold on average but not in every chain. A trend along a row records the order in which the row was made, so the data can argue for which end came first. Counting cannot say whether the object is an animal built of segments or a container built of compartments. Given the annelid and tapeworm facts above, something that looks like a segmented worm may be neither a worm nor segmented. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery (ECOL 479/579): course handout*, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Wikipedia contributors**, "Busycon" β€” [Wikipedia](https://en.wikipedia.org/wiki/Busycon){target=_blank} πŸ”“ (the "egg pouches" of whelk egg-case strings) - **Wikipedia contributors**, "Annelid" β€” [Wikipedia](https://en.wikipedia.org/wiki/Annelid){target=_blank} πŸ”“ (the segmented worms and their single cocoon) - **Wikipedia contributors**, "Cestoda" β€” [Wikipedia](https://en.wikipedia.org/wiki/Cestoda){target=_blank} πŸ”“ (proglottids as bags of eggs; the note on segmentation is an explanatory footnote) - **Wikipedia contributors**, "Egg case (Chondrichthyes)" β€” [Wikipedia](https://en.wikipedia.org/wiki/Egg_case_(Chondrichthyes)){target=_blank} πŸ”“ (the single-capsule "mermaid's purse", a rejected reading) - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure. The schedule line is Winfree's only text on this lab. In Winfree's handout, the link on "Egg Pouches" points to a bookmark named `Segmented_worm`. Its target was not archived. In the same handout, bookmark names often describe a problem's content rather than its classroom label (`Keplers_Laws` for "Paired Observations"), so the specimen was probably something that looked like a segmented worm. That is an inference, and it does not say which specimen. The candidates: - **A whelk egg-case string**, as above: its capsules are called "egg pouches" and it looks like a worm. No source ties it to Winfree. - **A tapeworm chain**, egg packets that look like segments. Less likely: it needs a preserved specimen and subject knowledge. - **Earthworm or leech cocoons, or segmentation itself as a pattern**, the most literal reading of the bookmark. Less likely: each cocoon is single, so there is no row of pouches to count. - **A packing lab**, squeezing soft eggs into polyhedra. Unlikely: it does not fit the bookmark, and [Cell Shapes](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) already covers that geometry. - **Shark or skate egg cases** ("mermaid's purses"). Rejected: each is a single capsule, not a row of pouches. --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/platonic-solids/ --- title: "Platonic Solids and Applications" description: "Count the corners, edges and faces of the five regular solids, find the rule that ties them together, test it until it breaks, and use it to explain soccer balls, viruses and radiolarian skeletons." type: Activity tags: [course, student-facing, problem, section-4, geometry, euler-characteristic, empirical-generalization, biological-form] status: stable problem: section: 4 session: 22 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: euclid-elements-xiii resource: "https://mathcs.clarku.edu/~djoyce/java/elements/bookXIII/bookXIII.html" title: "Euclid's Elements, Book XIII (Propositions 13-18 and the closing remark that only five regular solids exist)" author: "Euclid; D. E. Joyce (ed.)" - id: plato-timaeus-jowett resource: "https://www.gutenberg.org/ebooks/1572" title: "Timaeus (Jowett translation), Project Gutenberg eBook 1572" author: "Plato; Benjamin Jowett (trans.)" - id: euler-e230 resource: "https://scholarlycommons.pacific.edu/euler-works/230/" title: "Elementa doctrinae solidorum (E230)" author: "Leonhard Euler" - id: euler-e231 resource: "https://scholarlycommons.pacific.edu/euler-works/231/" title: "Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita (E231)" author: "Leonhard Euler" - id: thompson-growth-form-1917 resource: "https://archive.org/details/ongrowthform1917thom" title: "On Growth and Form (first edition)" author: "D'Arcy Wentworth Thompson" - id: lakatos-proofs-refutations resource: "https://doi.org/10.1017/CBO9781139171472" title: "Proofs and Refutations: The Logic of Mathematical Discovery" author: "Imre Lakatos (eds. John Worrall and Elie Zahar)" - id: richeson-eulers-gem resource: "https://doi.org/10.1515/9781400838561" title: "Euler's Gem: The Polyhedron Formula and the Birth of Topology" author: "David S. Richeson" - id: caspar-klug-1962 resource: "https://doi.org/10.1101/sqb.1962.027.001.005" title: "Physical Principles in the Construction of Regular Viruses" author: "D. L. D. Caspar and A. Klug" - id: kroto-1985 resource: "https://doi.org/10.1038/318162a0" title: "C60: Buckminsterfullerene" author: "H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl and R. E. Smalley" - id: mathworld-polyhedral-formula resource: "https://mathworld.wolfram.com/PolyhedralFormula.html" title: "Polyhedral Formula" author: "Eric W. Weisstein" - id: wikipedia-platonic-solid resource: "https://en.wikipedia.org/wiki/Platonic_solid" title: "Platonic solid" author: "Wikipedia contributors" - id: wikipedia-euler-characteristic resource: "https://en.wikipedia.org/wiki/Euler_characteristic" title: "Euler characteristic" author: "Wikipedia contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Platonic Solids and Applications Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 22. The same session holds further experiments in the [Cell Shapes](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) lab and the [Egg Pouches](https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/index.md) lab.* !!! abstract "The problem" Reconstructed from the syllabus, which says only "Platonic Solids and applications"; the worksheet is lost, and this is the editors' version of the standard exercise. Straws or cardboard help. **1. The five solids.** A regular (Platonic) solid is a convex solid whose faces are identical regular polygons, the same number meeting at every corner. Sketch or build all five and count each one's vertices *V*, edges *E* and faces *F*. Do not look the numbers up. **2. Find the pattern.** Find a rule connecting *V*, *E* and *F* that holds for all five. Test it on a prism, a pyramid, a cube with one corner sliced off. Then build a solid for which it fails; one exists. **3. Why only five?** Use your rule, or the angles that meet at a corner, to prove that no sixth regular solid can exist. **4. Applications.** - A soccer ball is sewn from pentagons and hexagons, three panels at each corner. How many pentagons must it have? - Can a closed shell be tiled with hexagons alone, three at each corner? - Some radiolarian skeletons, many virus coats and the C60 molecule have the symmetry of the icosahedron. Why might the regular dodecahedron and icosahedron appear among tiny skeletons but never as mineral crystals? - In 1596 Kepler proposed that the five solids, nested between the planetary spheres, explain the spacing of the six known planets. Judge it as an empirical generalization: what pattern was observed, what would count as a test, and what happened to the theory? ![Line drawings of the five Platonic solids, all edges shown with hidden edges dashed: tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons) and icosahedron (20 triangles)](https://tyson-swetnam.github.io/aosd/assets/images/problems/platonic-solids.svg){ width="560" } *The five regular solids, projected from exact coordinates. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 4 is "Patterns, Empirical Generalizations", and this exercise runs the cycle in miniature: count, notice a regularity, hunt for the exception, decide whether it kills the rule or only marks its limits, then prove the rule and use it. It also fits the syllabus: "The discovery exercises are mostly made from elementary mathematics so as to require no lab setup". The reading due that day is "Adams Chapter 6: Alternative thinking languages". Counting the edges of an icosahedron you cannot see all at once is where a sketch or a straw model beats words. A counting argument explains the soccer ball and the virus; Kepler's nested solids fit the numbers roughly and explain nothing. Applied to a flat sheet of cells meeting three at a corner, the same rule gives an average of six sides per cell. That may be why the session sits among the Cell Shapes experiments, but the link is the editors' inference. ## Where it comes from The regular solids are older than Plato, but his *Timaeus* (about 360 BC) builds four of them from triangles for fire, air, water and earth, adding that "There was yet a fifth combination which God used in the delineation of the universe" (Jowett's translation). Euclid's *Elements* ends with their construction and the remark, after Book XIII, Proposition 18, that "No other figure, besides the said five figures, can be constructed by equilateral and equiangular figures equal to one another." Kepler used them for his planetary model in the *Mysterium Cosmographicum* (1596). Euler found the counting rule in 1750 and published it in 1758, with a proof in a companion paper that was later found to be flawed. Lakatos's *Proofs and Refutations* (1976) turns its history of counterexamples into a classroom dialogue. D'Arcy Thompson's *On Growth and Form* (1917) found all five solids among Haeckel's radiolarians and noted that the regular dodecahedron and icosahedron never occur as crystals. Caspar and Klug (1962) explained why so many virus coats are icosahedral, and Kroto and colleagues (1985) proposed the soccer-ball shape for C60. ![Haeckel's engraving of the radiolarian Circogonia icosahedra: an icosahedral shell of fine hexagonal mesh with branched spines radiating from its corners](https://tyson-swetnam.github.io/aosd/assets/images/problems/platonic-solids-circogonia.jpg){ width="560" } *Ernst Haeckel, Circogonia icosahedra, from Kunstformen der Natur (1904), plate 1. CC0 / public domain, via Wikimedia Commons.* ??? tip "Hints" - Check each count a second way: every edge borders exactly two faces, so *E* = *F* Γ— (edges per face) Γ· 2. - Add and subtract the three columns in different combinations. The rule is linear and gives the same small number for all five solids. - Do not stop at solids that confirm the rule. Build the ugliest polyhedron you can, then one with a hole through it. - For the soccer ball, write *P* for pentagons and *H* for hexagons and count edges and corners two ways. - For "why only five", ask what the flat angles meeting at one corner must add up to if the corner is to stick out at all. ??? success "Resolution" **The rule.** Tetrahedron *V*, *E*, *F* = 4, 6, 4; cube 8, 12, 6; octahedron 6, 12, 8; dodecahedron 20, 30, 12; icosahedron 12, 30, 20. Each time *V* βˆ’ *E* + *F* = 2. It holds for any polyhedron whose surface is a deformed sphere, including the soccer ball (60 βˆ’ 90 + 32 = 2). It fails for a square picture frame of four bricks (16 βˆ’ 32 + 16 = 0); a surface with *g* holes gives 2 βˆ’ 2*g*. The counterexample marks the rule's domain rather than destroying it. **Why only five.** At least three faces meet at a corner, and their flat angles must total less than 360Β°. That allows 3, 4 or 5 triangles (60Β° each), 3 squares (90Β°) or 3 pentagons (108Β°), and nothing else: six triangles, four squares or three hexagons lie flat. With the rule instead: if each face has *p* edges and *q* faces meet at each corner, then *pF* = 2*E* = *qV*. Substituting into *V* βˆ’ *E* + *F* = 2 gives 1/*p* + 1/*q* = 1/2 + 1/*E* > 1/2, whose only solutions with *p*, *q* β‰₯ 3 are (3,3), (4,3), (3,4), (5,3) and (3,5): the five solids. **Twelve pentagons.** With *F* = *P* + *H* and 2*E* = 3*V* = 5*P* + 6*H*, the rule reduces to *P*/6 = 2, so *P* = 12 whatever the number of hexagons (20 on a standard ball, none on a dodecahedron). With *P* = 0 it reads 0 = 2: hexagons alone cannot close a shell. **Crystals and radiolaria.** A crystal lattice repeats by translation, and no such lattice can have a five-fold axis, so the regular dodecahedron and icosahedron are impossible crystal forms. A radiolarian skeleton is a single closed shell, not a lattice, so nothing forbids five-fold symmetry. Same shape, different constraints. **Kepler.** The nesting matched the orbits only roughly, offered no mechanism, and did not survive his own elliptical orbits or Uranus (1781): a striking pattern, not a law. ## Sources - **Euclid**, *Elements*, Book XIII, ed. D. E. Joyce (Clark University) β€” [Book XIII](https://mathcs.clarku.edu/~djoyce/java/elements/bookXIII/bookXIII.html){target=_blank} πŸ”“ - **Plato**, *Timaeus*, trans. Benjamin Jowett β€” [Project Gutenberg](https://www.gutenberg.org/ebooks/1572){target=_blank} πŸ”“ - **Leonhard Euler**, "Elementa doctrinae solidorum", *Novi Commentarii academiae scientiarum Petropolitanae* 4, 109–140 (1758) β€” [Euler Archive E230](https://scholarlycommons.pacific.edu/euler-works/230/){target=_blank} πŸ”“ - **Leonhard Euler**, "Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita", same volume, 140–160 (1758) β€” [Euler Archive E231](https://scholarlycommons.pacific.edu/euler-works/231/){target=_blank} πŸ”“ - **D'Arcy Wentworth Thompson**, *On Growth and Form* (1917), chapter IX, pp. 479–485 β€” [Internet Archive](https://archive.org/details/ongrowthform1917thom){target=_blank} πŸ”“ - **Imre Lakatos**, *Proofs and Refutations: The Logic of Mathematical Discovery*, ed. J. Worrall and E. Zahar (1976) β€” [doi:10.1017/CBO9781139171472](https://doi.org/10.1017/CBO9781139171472){target=_blank} πŸ”’ - **David S. Richeson**, *Euler's Gem: The Polyhedron Formula and the Birth of Topology* (2008) β€” [doi:10.1515/9781400838561](https://doi.org/10.1515/9781400838561){target=_blank} πŸ”’ (the formula's later history, from Euler's flawed proof to the Euler characteristic) - **D. L. D. Caspar and A. Klug**, "Physical Principles in the Construction of Regular Viruses", *Cold Spring Harbor Symposia on Quantitative Biology* 27, 1–24 (1962) β€” [doi:10.1101/sqb.1962.027.001.005](https://doi.org/10.1101/sqb.1962.027.001.005){target=_blank} πŸ”’ - **H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl and R. E. Smalley**, "C60: Buckminsterfullerene", *Nature* 318, 162–163 (1985) β€” [doi:10.1038/318162a0](https://doi.org/10.1038/318162a0){target=_blank} πŸ”’ - **Eric W. Weisstein**, "Polyhedral Formula", MathWorld β€” [MathWorld](https://mathworld.wolfram.com/PolyhedralFormula.html){target=_blank} πŸ”“ - **Wikipedia contributors**, "Platonic solid" β€” [Wikipedia](https://en.wikipedia.org/wiki/Platonic_solid){target=_blank} πŸ”“ - **Wikipedia contributors**, "Euler characteristic" β€” [Wikipedia](https://en.wikipedia.org/wiki/Euler_characteristic){target=_blank} πŸ”“ (the twelve-pentagon count for a football) - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Sure of the topic, not of the exercise. The syllabus says "Deal with Platonic Solids and applications"; no other Winfree text on it survives. The Euler-formula exercise is the editors' reconstruction, chosen to fit the section theme, the Adams reading and the Cell Shapes lab. The candidates: - **Count, find *V* βˆ’ *E* + *F* = 2, test and prove it, then apply it** to soccer balls, viruses, radiolaria and foams. Medium confidence. - **The biological applications**: radiolaria, virus coats, C60, crystals, foams. Medium confidence; probably part of the same session. - **Kepler's nested solids** as a beautiful pattern that proved false. Medium confidence; probably part rather than whole. - **A pure geometry proof** that exactly five regular solids exist. Low confidence as the whole session. --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/eleusis/ --- title: "Eleusis" description: "Play Robert Abbott's card game Eleusis, in which a dealer writes a secret rule and the players must discover it by playing cards and being told only right or wrong: induction from data, practised with your own hands." type: Activity tags: [course, student-facing, problem, section-4, induction, hypothesis-testing, card-games, scientific-method] status: stable problem: section: 4 session: 23 identification: confident kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: abbott-eleusis-page resource: "https://web.archive.org/web/20241120065735/http://www.logicmazes.com/games/eleusis/" title: "Eleusis and Eleusis Express (archived copy of logicmazes.com)" author: "Robert Abbott" - id: abbott-eleusis-publications resource: "https://web.archive.org/web/20240117163833/https://www.logicmazes.com/games/eleusis/eleusis2.html" title: "Eleusis: publication history (archived copy of logicmazes.com)" author: "Robert Abbott" - id: golden-eleusis-express resource: "https://web.archive.org/web/20250427175054/http://www.logicmazes.com/games/eleusis/express.html" title: "Eleusis Express (rules)" author: "John Golden, with Robert Abbott" - id: matuszek-new-eleusis resource: "https://matuszek.org/eleusis1.html" title: "New Eleusis (rules summary)" author: "David Matuszek" - id: gardner-1959 resource: "https://doi.org/10.1038/scientificamerican0659-160" title: "Mathematical Games (June 1959): An inductive card game" author: "Martin Gardner" - id: gardner-1977 resource: "https://doi.org/10.1038/scientificamerican1077-18" title: "Mathematical Games (October 1977): On playing New Eleusis, the game that simulates the search for truth" author: "Martin Gardner" - id: gardner-penrose-tiles resource: "https://archive.org/details/penrosetilestotr00gard" title: "Penrose Tiles to Trapdoor Ciphers" author: "Martin Gardner" - id: gardner-origami-eleusis resource: "https://books.google.com/books?vid=ISBN9780521735247" title: "Origami, Eleusis, and the Soma Cube: Martin Gardner's Mathematical Diversions" author: "Martin Gardner" - id: wikipedia-eleusis resource: "https://en.wikipedia.org/wiki/Eleusis_(card_game)" title: "Eleusis (card game)" author: "Wikipedia contributors" - id: commons-eleusis-photo resource: "https://commons.wikimedia.org/wiki/File:Eleusis_card_game.jpg" title: "Eleusis card game (photograph)" author: "Kevan Davis" - id: ehrlich-2001 resource: "https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science" title: "Nine Crazy Ideas in Science: A Few Might Even Be True" author: "Robert Ehrlich" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Eleusis Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 23. Played in class; the same session deals with [the Mirror Mystery](https://tyson-swetnam.github.io/aosd/problems/mirror-mystery/index.md).* !!! abstract "The problem" Eleusis is a card game in which one player invents a law of nature and the others discover it by experiment. (Editors' summary of Robert Abbott's rules.) **Setup.** The *dealer* secretly writes down a rule saying which card may follow the cards already accepted. The rule may depend only on accepted cards (for example the last one, or the last two). Shuffle two 52-card decks together, deal 14 cards to each other player (12 in the short version), and turn up one card as the *starter*. **Goal.** Get rid of your cards. A round ends when someone runs out; the fewer cards you still hold, the better you score. **Play.** In turn, each player lays down a card: an experiment. The dealer says only *right* or *wrong*, never why. - A right card extends the *mainline* to the right. - A wrong card goes *below* the last accepted card, in a *sideline*, and stays on the table; the player draws penalty cards. **Knowing the rule.** In Abbott's full game a player who thinks they know the rule may become *Prophet* and call other players' cards right or wrong; one wrong call and the Prophet is overthrown. In the short version (Eleusis Express) a player who has just played correctly may instead say a guess aloud; a correct guess ends the round. **Your task in class** (a reconstruction; nothing survives about how Winfree ran the session): play a few rounds, and log in your GamesWorth notebook each hypothesis you held, the card you played to test it, and how you got out of blind alleys. The log, not the score, is the exercise. ![Schematic Eleusis layout: five accepted cards in a row (7 of spades, 4 of hearts, jack of clubs, 2 of diamonds, 9 of spades) with rejected cards hanging in columns beneath the 4 of hearts (8 of clubs, 10 of diamonds), the 2 of diamonds (6 of spades) and the 9 of spades (king of hearts)](https://tyson-swetnam.github.io/aosd/assets/images/problems/eleusis.svg){ width="560" } *Drawn for this site (CC BY 4.0). A made-up round: the dealer's rule is given at the end of the page.* ## Why it is in the course Section 4 is "Patterns, Empirical Generalizations", and Eleusis is that section in its purest form: from a table of accepted and rejected cards you guess a rule, then choose the next card to test it. The syllabus says the puzzles are there "to slow you down for a few minutes so you can examine the working of your own mind", and asks you to "Write down your approaches, your lucky insights, how you got into and out of blind alleys." A game that never says why a card was wrong forces exactly that. The sidelines keep every mistake on the table, as Section 1's "Cherishing Mistakes" asks. Choosing the card that separates two rival rules is Chamberlin's *The Method of Multiple Working Hypotheses* (session 25) in miniature. The Prophet, brought down by one wrong call, anticipates Platt's *Strong Inference* (session 27). And among the syllabus's "Other good books on reserve" is George PΓ³lya's *Induction and Analogy in Mathematics*, the subject the game rehearses. ## Where it comes from The game inventor Robert Abbott devised Eleusis in 1956. Martin Gardner described it in his "Mathematical Games" column in *Scientific American* for June 1959; Abbott quotes Gardner's verdict that it "should be of special interest to mathematicians and other scientists because of its striking analogy with scientific method". From 1973 Abbott reworked the game, adding the sidelines and the Prophet, and Gardner presented the new version in his October 1977 column as "the game that simulates the search for truth". Abbott long refused to let players just announce the rule, because "if a scientist publishes a theory, then (unfortunately) the heavens do not part and God does not declare whether the theory is right or not." In 2006 the mathematician John Golden made a simpler version for elementary-school teachers, which Abbott named Eleusis Express; Abbott now writes that on guessing aloud he had been mistaken. Express came after the course, so Winfree cannot have used it. His handout does not say which of Abbott's versions the class played. ![Photograph of an Eleusis game on a red cloth: a row of five accepted cards with rejected cards beneath the first and third](https://tyson-swetnam.github.io/aosd/assets/images/problems/eleusis-layout.jpg){ width="560" } *A game of Eleusis in play. Photograph by Kevan Davis (2019), via Wikimedia Commons, CC0 1.0.* ??? tip "Hints" - Each wrong card rules out every rule that would have allowed it, so the sidelines are often stronger evidence than the mainline. - Before each turn, write down your favoured rule and one rival, then play the card that separates them. A card both rules allow teaches you nothing. - Dealers tend to use a small vocabulary: colour, suit, odd or even, high or low, arithmetic on the last card. Check each against the whole layout before inventing anything exotic. - "Consistent so far" is not proof; the Prophet who forgets this is overthrown. - If you deal, allow several legal cards at any moment but not most of the deck. In Golden's words, "whatever rule you come up with, it will always be harder than you think it will be." ## What happened Eleusis has no fixed answer: each round's answer is whatever the dealer wrote. Winfree's debrief does not survive; Abbott's and Gardner's commentary suggests what one can bring out: - The dealer plays Nature, answering yes or no, never why. The Prophet is a theorist staking a reputation on public predictions. - The sidelines exist because negative results are data. - Players who pick cards to confirm a favourite hypothesis learn slowly; players who pick cards to discriminate between hypotheses learn fast. - Dealers find that a rule they thought transparent is opaque to others. ??? success "The rule in the figure" Odd and even ranks alternate (ace 1, jack 11, queen 12, king 13), one of Golden's easy sample rules. The mainline runs 7, 4, 11, 2, 9. After the 4 an odd card was needed, so the 8 and 10 were wrong; after the 2, the 6 was wrong; after the 9 an even card was needed, so the king (13) was wrong. The mainline alone also fits "colours alternate". The sidelines rule that out: the 8 of clubs, the 6 of spades and the king of hearts each have the opposite colour to the card before them, yet all three were rejected. ## Sources - **Robert Abbott**, "Eleusis and Eleusis Express", logicmazes.com (archived) β€” [Wayback Machine](https://web.archive.org/web/20241120065735/http://www.logicmazes.com/games/eleusis/){target=_blank} πŸ”“ - **Robert Abbott**, "Eleusis": publication history, logicmazes.com (archived) β€” [Wayback Machine](https://web.archive.org/web/20240117163833/https://www.logicmazes.com/games/eleusis/eleusis2.html){target=_blank} πŸ”“ - **John Golden, with Robert Abbott**, "Eleusis Express" rules (2006) β€” [Wayback Machine](https://web.archive.org/web/20250427175054/http://www.logicmazes.com/games/eleusis/express.html){target=_blank} πŸ”“ - **David Matuszek**, "New Eleusis", a summary of Abbott's full rules (1995) β€” [matuszek.org](https://matuszek.org/eleusis1.html){target=_blank} πŸ”“ - **Martin Gardner**, "Mathematical Games: An inductive card game", *Scientific American* 200(6), 160–168 (June 1959) β€” [doi:10.1038/scientificamerican0659-160](https://doi.org/10.1038/scientificamerican0659-160){target=_blank} πŸ”’ - **Martin Gardner**, "Mathematical Games: On playing New Eleusis, the game that simulates the search for truth", *Scientific American* 237(4), 18–25 (October 1977) β€” [doi:10.1038/scientificamerican1077-18](https://doi.org/10.1038/scientificamerican1077-18){target=_blank} πŸ”’ - **Martin Gardner**, *Penrose Tiles to Trapdoor Ciphers* (W. H. Freeman, 1989), reprinting the 1977 column β€” [Internet Archive](https://archive.org/details/penrosetilestotr00gard){target=_blank} πŸ”“ *(borrow)* - **Martin Gardner**, *Origami, Eleusis, and the Soma Cube* (Cambridge University Press, 2008), reprinting the 1959 column β€” [Google Books](https://books.google.com/books?vid=ISBN9780521735247){target=_blank} πŸ”’ - **Wikipedia contributors**, "Eleusis (card game)" β€” [Wikipedia](https://en.wikipedia.org/wiki/Eleusis_(card_game)){target=_blank} πŸ”“ (overview; misdates the revised column to July 1977) - **Kevan Davis**, photograph "Eleusis card game" (2019), CC0 1.0 β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Eleusis_card_game.jpg){target=_blank} πŸ”“ - **Robert Ehrlich**, *Nine Crazy Ideas in Science: A Few Might Even Be True* (Princeton University Press, 2001), the session's assigned reading (Chapter 7) β€” [publisher](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/mirror-mystery/ --- title: "The Mirror Mystery" description: "Why does a mirror seem to swap left and right but not up and down? A household observation that everyone believes, and a lesson in checking the facts before explaining them." type: Activity tags: [course, student-facing, problem, section-4, mirrors, symmetry, handedness, empirical-generalization] status: stable problem: section: 4 session: 23 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: plato-timaeus-lamb resource: "http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0180%3Atext%3DTim.%3Asection%3D46b" title: "Timaeus 46a-c (on mirrors: why right appears as left)" author: "Plato, translated by W. R. M. Lamb" - id: lucretius-leonard resource: "https://www.gutenberg.org/ebooks/785" title: "On the Nature of Things (De Rerum Natura), Book IV" author: "Lucretius, translated by William Ellery Leonard" - id: kant-prolegomena-carus resource: "https://www.gutenberg.org/ebooks/52821" title: "Prolegomena to Any Future Metaphysics, Section 13" author: "Immanuel Kant, translated by Paul Carus" - id: gardner-1964 resource: "https://archive.org/details/ambidextrousuniv0000gard_i3y0" title: "The Ambidextrous Universe" author: "Martin Gardner" - id: block-1974 resource: "https://doi.org/10.2307/2024963" title: "Why do Mirrors Reverse Right/Left but not Up/Down?" author: "Ned Block" - id: denyer-1994 resource: "https://doi.org/10.1017/S0031819100046842" title: "Why do Mirrors Reverse Left/Right and not Up/Down?" author: "Nicholas Denyer" - id: gregory-1997 resource: "https://archive.org/details/mirrorsinmind0000greg" title: "Mirrors in Mind" author: "Richard L. Gregory" - id: corballis-2000 resource: "https://doi.org/10.3758/BF03210736" title: "Much ado about mirrors" author: "Michael C. Corballis" - id: feynman-fun-to-imagine resource: "https://archive.org/details/FunToImagine" title: "Richard Feynman: Fun to Imagine (BBC, 1983)" author: "Richard P. Feynman / BBC" - id: commons-mirror-writing resource: "https://commons.wikimedia.org/wiki/File:Mirror_writing2.jpg" title: "Ottoman calligraphic panel in mirror writing (Wikimedia Commons)" author: "Mahmoud Ibrahim" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: winfree-associativity resource: "https://web.archive.org/web/20030114050407/http://eebweb.arizona.edu/faculty/winfree/Associativity.htm" title: "How to use this in ASD (floating-point associativity and a broken mirror symmetry)" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # The Mirror Mystery Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md), session 23. In the same session the class plays [Eleusis](https://tyson-swetnam.github.io/aosd/problems/eleusis/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: the syllabus gives only the name, and this is the editors' statement of the classic puzzle it most probably means. None of the wording is Winfree's. Stand in front of a mirror and raise your right hand. The person in the glass seems to raise a left hand. Writing held up to the glass comes back backwards. Yet your head is still at the top and your feet at the bottom. **Why does a mirror reverse left and right but not up and down?** Before explaining, make sure you have the fact right. Check against a real mirror: - Which way does your reflection's nose point? - Point straight at the mirror. Which way does the reflected arm point? - Write a word on tracing paper. Hold it up with the writing facing you, not turned round, and read the word in the mirror through the paper. Where did the reversal go? - Lie on your side in front of a long mirror. What is swapped now? Record what you believed at the start, which observations survived, and what you believe now. ## Why it is in the course Section 4 is "Patterns, Empirical Generalizations", and its problems test generalizations people hold with confidence. "Mirrors reverse left and right" is a pure specimen: everyone believes it, and almost nobody has varied the observation. The same class period plays Eleusis, a card game that penalizes a guessed rule which only fits the cards you happen to have seen. It also echoes Section 1: "facts before explanations of facts" and "distinguishing things we know vs only imagine". Most people explain the reversal before checking that it happens. It needs no equipment, and the syllabus says the puzzles exist "to slow you down for a few minutes so you can examine the working of your own mind". Here you can watch yourself defend a wrong statement of the facts. ## Where it comes from The question is ancient. Plato's *Timaeus* (46b) says "Left appears as right, because contact takes place between opposite portions of the visual stream and opposite portions of the object, contrary to the regular mode of collision". Lucretius, in Book IV of *On the Nature of Things*, pictures the image flung back like a plaster mask thrown against a post before it dries, so that "now the right eye is the left, / The left the right". Kant made the neighbouring point famous in his *Prolegomena* (1783): a hand and its mirror image match in every part, yet "the glove of one hand cannot be used for the other". Martin Gardner opened *The Ambidextrous Universe* (1964) with the mirror question. Later it became a question about the mind: Ned Block (1974) argued that the answer lies in how we assign left and right to a body, and replies and surveys followed from Nicholas Denyer (1994), Richard Gregory (*Mirrors in Mind*, 1997) and Michael Corballis (2000). Feynman took it up in his 1983 BBC series *Fun to Imagine*. The difficulty was never the optics. ![An 18th-century Ottoman calligraphic panel in which a phrase is written twice, once as the mirror image of the other](https://tyson-swetnam.github.io/aosd/assets/images/problems/mirror-mystery-levha.jpg){ width="560" } *Mahmoud Ibrahim, Ottoman calligraphic panel in mirror writing (c. 1720-1730), Library of Congress, via Wikimedia Commons. Public domain.* ??? tip "Hints" - Say exactly what is reversed and what is not, without the words "left" and "right". - Change your own orientation: lie down, or put the mirror on the floor. Does the mirror care how you stand? - Point your arm across the mirror, then up, then straight at the glass. Which of the three reflected arms points the opposite way? - How do you decide which hand of the figure in the glass is *its* right hand? What do you do, in your head, to decide? ??? success "Resolution" A plane mirror reverses one direction only: the one perpendicular to its surface. With *x* across the mirror, *y* up it and *z* out of it, the point (*x*, *y*, *z*) appears at (*x*, *y*, βˆ’*z*). Left-right and up-down are both untouched; reversing front and back is enough to turn a right hand into a left one. The left-right swap is supplied by you. To name the reflection's left and right, you imagine yourself turned through 180 degrees about a vertical axis to stand where it stands, and that rotation exchanges left and right. Imagine turning head over heels instead and the same reflection reads as upside down, with left and right intact. ![Plan view from above. Left: a person facing a mirror and the image behind it, with the right hand and its image on the same side. Right: the same scene plus the person imagined turned round to face the way the image faces, which puts the right hand on the other side](https://tyson-swetnam.github.io/aosd/assets/images/problems/mirror-mystery.svg){ width="560" } *Seen from above; the red dot is your right hand. Left: the mirror reverses only the direction through the glass. Right: turning round in your head to face the way the image faces is what moves your right hand to the other side. Drawn for this site (CC BY 4.0).* Writing works the same way. Leave the tracing paper facing you and the word seen through the paper in the mirror reads normally. Turn the sheet round to face the glass and the reflection reads backwards, exactly as the sheet itself looks from behind. The reversal came from turning the page, not from the mirror. We make the imagined turn about a vertical axis because that is how we turn to face anyone, and gravity keeps the vertical fixed. Lie down beside the mirror and the mystery lies down with you. The lesson: a question can be unanswerable because it contains a false statement of the facts. Restate the facts rather than hunting harder for a cause. ## Sources - **Plato**, *Timaeus* 46a–c, translated by W. R. M. Lamb (1925) β€” [Perseus](http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0180%3Atext%3DTim.%3Asection%3D46b){target=_blank} πŸ”“ - **Lucretius**, *On the Nature of Things*, Book IV, translated by William Ellery Leonard (1916) β€” [Project Gutenberg](https://www.gutenberg.org/ebooks/785){target=_blank} πŸ”“ - **Immanuel Kant**, *Prolegomena to Any Future Metaphysics*, Section 13, translated by Paul Carus (1902) β€” [Project Gutenberg](https://www.gutenberg.org/ebooks/52821){target=_blank} πŸ”“ - **Martin Gardner**, *The Ambidextrous Universe* (1964) β€” [Internet Archive](https://archive.org/details/ambidextrousuniv0000gard_i3y0){target=_blank} πŸ”“ *(borrow)* - **Ned Block**, "Why do Mirrors Reverse Right/Left but not Up/Down?", *Journal of Philosophy* 71(9), 259–277 (1974) β€” [doi:10.2307/2024963](https://doi.org/10.2307/2024963){target=_blank} πŸ”’ - **Nicholas Denyer**, "Why do Mirrors Reverse Left/Right and not Up/Down?", *Philosophy* 69(268), 205–210 (1994) β€” [doi:10.1017/S0031819100046842](https://doi.org/10.1017/S0031819100046842){target=_blank} πŸ”’ - **Richard L. Gregory**, *Mirrors in Mind* (1997) β€” [Internet Archive](https://archive.org/details/mirrorsinmind0000greg){target=_blank} πŸ”“ *(borrow)* - **Michael C. Corballis**, "Much ado about mirrors", *Psychonomic Bulletin & Review* 7(1), 163–169 (2000) β€” [doi:10.3758/BF03210736](https://doi.org/10.3758/BF03210736){target=_blank} πŸ”’ - **Richard P. Feynman / BBC**, *Fun to Imagine* (1983) β€” [Internet Archive](https://archive.org/details/FunToImagine){target=_blank} πŸ”“ - **Mahmoud Ibrahim**, calligraphic panel in mirror writing (c. 1720–1730) β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Mirror_writing2.jpg){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, "How to use this in ASD" (floating-point arithmetic and a broken mirror symmetry; a candidate, not a source for this problem, see the note below) β€” [Wayback Machine](https://web.archive.org/web/20030114050407/http://eebweb.arizona.edu/faculty/winfree/Associativity.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Probable, not certain. The syllabus, like Winfree's archived handout, says only "Deal with the Mirror Mystery", and no other surviving Winfree page uses the name. The classic puzzle fits the name, needs no apparatus, and suits a section about generalizations. Other readings: - **Winfree's own broken symmetry.** His archived page "How to use this in ASD" describes a mirror-symmetric simulation that loses its symmetry because adding numbers in a different order gives a different sum. It never uses the name, needs a computer, and seems to belong with the error-checking sessions (6 or 12). - **The half-height mirror**: how much mirror you need to see all of yourself; usually a physics exercise. - **Mirror-image molecules in living things**: needs subject knowledge the syllabus avoids; untested. --- *Back to [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-4-patterns-empirical-generalizations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/ --- title: "Stacked Cantilevers Lab" description: "A three-session group lab on how far a pile of identical blocks can lean out over a table edge, moving from pooled measurements and wagers to a theory built on the harmonic series." type: Activity tags: [course, student-facing, problem, section-5, block-stacking, harmonic-series, statics, group-lab] status: stable problem: section: 5 session: 24 identification: probable kind: lab generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: walton-1855 resource: "https://archive.org/details/acollectionprob07waltgoog" title: "A Collection of Problems in Illustration of the Principles of Theoretical Mechanics, 2nd ed. (Miscellaneous Problems no. 27, p. 183)" author: "William Walton" - id: johnson-1955 resource: "https://doi.org/10.1119/1.1933957" title: "Leaning Tower of Lire" author: "Paul B. Johnson" - id: gamow-stern-1958 resource: "https://archive.org/details/puzzlemath0000unse" title: "Puzzle-Math" author: "George Gamow and Marvin Stern" - id: gardner-sixth-book resource: "https://archive.org/details/martingardnerssi0000gard" title: "Martin Gardner's Sixth Book of Mathematical Games from Scientific American" author: "Martin Gardner" - id: boas-1973 resource: "https://doi.org/10.1119/1.1987341" title: "Cantilevered Books" author: "R. P. Boas" - id: ainley-1979 resource: "https://doi.org/10.2307/3618049" title: "Finely Balanced" author: "S. Ainley" - id: paterson-zwick-2009 resource: "https://arxiv.org/abs/0710.2357" title: "Overhang" author: "Mike Paterson and Uri Zwick" - id: paterson-et-al-2009 resource: "https://arxiv.org/abs/0707.0093" title: "Maximum Overhang" author: "Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler and Uri Zwick" - id: mathworld-book-stacking resource: "https://mathworld.wolfram.com/BookStackingProblem.html" title: "Book Stacking Problem" author: "Eric W. Weisstein (MathWorld)" - id: dickau-book-stacking resource: "https://www.robertdickau.com/BookStacking.html" title: "The Book-Stacking Problem" author: "Robert M. Dickau" - id: wikipedia-block-stacking resource: "https://en.wikipedia.org/wiki/Block-stacking_problem" title: "Block-stacking problem" author: "Wikipedia contributors" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Stacked Cantilevers Lab Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 24. The lab starts in session 24, continues in session 25 and closes in session 26, which also takes up [Summing a Series](https://tyson-swetnam.github.io/aosd/problems/summing-a-series/index.md) and [Stalactites](https://tyson-swetnam.github.io/aosd/problems/stalactites/index.md).* !!! abstract "The problem" Reconstructed: the syllabus gives only the lab's name and schedule, so the setup and questions below are the editors'. You have many identical blocks (or books, or playing cards) and a table. Stack them one on another, the bottom block on the table, so that the pile leans out over the edge as far as it can without toppling. The *overhang* is the horizontal distance from the table edge to the far end of the top block, in block-lengths. 1. Can the top block lie entirely beyond the table edge? With how few blocks? 2. Can the overhang reach two block-lengths? Three? Is there any limit? 3. Write down a rule for the best overhang with *n* blocks and test it against the class's pooled measurements. An early printed version is in William Walton's mechanics problem book of 1855: "A pack of cards is laid on a table; each projects in the direction of the length of the pack beyond the one below it: if each projects as far as possible, prove that the distances between the extremities of the successive cards will form an harmonic progression." ## Why it is in the course This hands-on lab runs across three sessions. The syllabus schedules "Start Stacked Cantilevers lab" for session 24, the last session of Section 4 (Patterns, Empirical Generalizations); "Further collaborations on Stacked Cantilevers" for session 25, the day Chamberlin's *The Method of Multiple Working Hypotheses* is due; and "Theory of stacking cantilevers, resolution of wagers" for session 26. That sequence walks from measurement to explanation. How the class ran it is not recorded, so what follows is the editors' reading. One group's stacks give noisy numbers, but, as the syllabus says of class meetings, "if we pool data, reality will come into focus". With pooled data on the board, the next step, in Chamberlin's spirit, is to hold several candidate rules at once: does the overhang level off, or keep growing, and how fast? Wagers placed before the theory force everyone to commit to an intuition, and the theory then settles them. ## Where it comes from The problem has been a statics exercise in British mechanics texts since at least the mid-nineteenth century (Walton, 1855). Paul B. Johnson's note "Leaning Tower of Lire" (1955) gave it its best-known nickname, and R. P. Boas's "Cantilevered Books" (1973) is the published title closest to Winfree's name for the lab. Gamow and Stern's *Puzzle-Math* (1958) and Martin Gardner's *Scientific American* column of November 1964 made it widely known; Paterson and Zwick (2009) trace the history. ![Photograph of identical wooden blocks stacked so that each sits farther out than the one below, the pile stepping sideways as it rises](https://tyson-swetnam.github.io/aosd/assets/images/problems/stacked-cantilevers-lab-blocks.png){ width="560" } *Photo by Anton (German Wikipedia user), derived from File:Harmonischebrueckerp.jpg; CC BY 2.5, via [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:CantileverBlocks.png){target=_blank}.* ??? tip "Hints" - Start at the top. How far can one block stick out beyond the block it rests on? Where must its centre of mass be? - Treat the top *k* blocks as one rigid object: its combined centre of mass must lie above the block beneath. Find the best offsets for k = 1, 2, 3. - For the wagers, ask whether your offsets add up to something bounded as blocks are added. If the sum involves 1 + 1/2 + 1/3 + 1/4 + ..., group it as 1 + 1/2 + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... and estimate each bracket. - If real stacks fall short of your rule, ask what it assumes about the blocks, and about how many blocks may rest on any one block. ??? success "Resolution" Number the blocks from the top. The top block's centre of mass is at its middle, so it can project 1/2 beyond block 2. With the top block at that brink, the centre of mass of the top two sits 1/4 in from block 2's outer end, so the pair can project 1/4 beyond block 3. In general, suppose the top *k* blocks have their combined centre of mass exactly over the outer end of block *k*+1. Block *k*+1's own centre of mass is 1/2 behind that end, so the centre of mass of all *k*+1 blocks is (1/2)/(*k*+1) behind it. That is how far those *k*+1 blocks can project beyond the block below. The offsets are 1/2, 1/4, 1/6, ..., 1/(2*n*), and with *n* blocks the overhang is (1/2)(1 + 1/2 + 1/3 + ... + 1/*n*) block-lengths: 1/2, 3/4, 11/12, 25/24, ... This is Walton's harmonic progression. ![Four blocks over a table edge, each shifted farther than the one below by 1/2, 1/4, 1/6 and 1/8 of a block-length, so the top block lies entirely beyond the edge with a total overhang of 25/24 block-lengths](https://tyson-swetnam.github.io/aosd/assets/images/problems/stacked-cantilevers-lab.svg){ width="560" } *Drawn for this site (CC BY 4.0).* **The wagers.** With four blocks the top block already clears the table edge, by 1/24 of a block-length. Two block-lengths need 31 blocks (an overhang of 2.01362; 30 blocks give only 1.99749), and three need 227. There is no limit, because the harmonic series diverges: in the grouping 1 + 1/2 + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... each bracket is at least 1/2. The growth is only very slow. Real stacks fall a little short, because every partial stack sits exactly on the brink. **The hidden assumption.** The formula is best only if each block rests on a single block. Allow counterweights and you can do better: four blocks can reach about 1.16789 block-lengths instead of 25/24 (Ainley, 1979). Paterson and Zwick, after the course, showed that the overhang can then grow like the cube root of *n*. ## Sources - **William Walton**, *A Collection of Problems in Illustration of the Principles of Theoretical Mechanics*, 2nd ed., Miscellaneous Problems no. 27, p. 183 (Cambridge: Deighton, Bell and Co., 1855) β€” [Internet Archive](https://archive.org/details/acollectionprob07waltgoog){target=_blank} πŸ”“ - **Paul B. Johnson**, "Leaning Tower of Lire", *American Journal of Physics* 23(4), 240 (1955) β€” [doi:10.1119/1.1933957](https://doi.org/10.1119/1.1933957){target=_blank} πŸ”’ - **George Gamow and Marvin Stern**, *Puzzle-Math* (1958) β€” [Internet Archive](https://archive.org/details/puzzlemath0000unse){target=_blank} πŸ”“ *(borrow)* - **Martin Gardner**, *Martin Gardner's Sixth Book of Mathematical Games from Scientific American* (1971), pp. 167–169 β€” [Internet Archive](https://archive.org/details/martingardnerssi0000gard){target=_blank} πŸ”“ *(borrow)* - **R. P. Boas**, "Cantilevered Books", *American Journal of Physics* 41(5), 715 (1973) β€” [doi:10.1119/1.1987341](https://doi.org/10.1119/1.1987341){target=_blank} πŸ”’ - **S. Ainley**, "Finely Balanced", *The Mathematical Gazette* 63(426), 272 (1979) β€” [doi:10.2307/3618049](https://doi.org/10.2307/3618049){target=_blank} πŸ”’ - **Mike Paterson and Uri Zwick**, "Overhang", *American Mathematical Monthly* 116(1), 19–44 (2009) β€” [arXiv:0710.2357](https://arxiv.org/abs/0710.2357){target=_blank} πŸ”“ - **Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler and Uri Zwick**, "Maximum Overhang", *American Mathematical Monthly* 116(9), 763–787 (2009) β€” [arXiv:0707.0093](https://arxiv.org/abs/0707.0093){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Book Stacking Problem", MathWorld β€” [MathWorld](https://mathworld.wolfram.com/BookStackingProblem.html){target=_blank} πŸ”“ (the block counts 4, 31 and 227) - **Robert M. Dickau**, "The Book-Stacking Problem" β€” [robertdickau.com](https://www.robertdickau.com/BookStacking.html){target=_blank} πŸ”“ (the 30- and 31-block overhangs) - **Wikipedia contributors**, "Block-stacking problem" β€” [Wikipedia](https://en.wikipedia.org/wiki/Block-stacking_problem){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the name and three session entries. It is probably the classical block-stacking problem: "stacked cantilevers" describes such a pile, Boas's title is close, the lab needs only blocks, and its theory is a divergent series that people bet against. Session 26 also lists Summing a Series, but as a separate item, so that is a hint, not proof. Other candidates: - A cantilever-building contest with mixed materials and bets on whose reaches farthest. It fits "wagers" but not "stacking". - The multi-wide version, with counterweight blocks: better treated as an extension, since its modern theory postdates the course. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/stalactites/ --- title: "Stalactites" description: "Why does dripping water leave stone hanging from a cave ceiling, why is it shaped as it is, and how old is it? A reconstructed exercise in multiple working hypotheses and hidden assumptions." type: Activity tags: [course, student-facing, problem, section-5, geology, multiple-hypotheses, estimation, hidden-assumptions] status: stable problem: section: 5 session: 26 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: wikipedia-stalactite resource: "https://en.wikipedia.org/wiki/Stalactite" title: "Stalactite" author: "Wikipedia contributors" - id: nps-speleothems resource: "https://www.nps.gov/subjects/caves/speleothems.htm" title: "Speleothems - Caves and Karst" author: "U.S. National Park Service" - id: wikipedia-soda-straw resource: "https://en.wikipedia.org/wiki/Soda_straw" title: "Soda straw" author: "Wikipedia contributors" - id: wikipedia-kartchner-caverns resource: "https://en.wikipedia.org/wiki/Kartchner_Caverns_State_Park" title: "Kartchner Caverns State Park" author: "Wikipedia contributors" - id: dawkins-1874 resource: "https://archive.org/details/cavehuntingrese01dawkgoog" title: "Cave Hunting: Researches on the Evidence of Caves respecting the Early Inhabitants of Europe" author: "W. Boyd Dawkins" - id: short-2005-prl resource: "https://doi.org/10.1103/PhysRevLett.94.018501" title: "Stalactite Growth as a Free-Boundary Problem: A Geometric Law and Its Platonic Ideal" author: "M. B. Short, J. C. Baygents, J. W. Beck, D. A. Stone, R. S. Toomey III, R. E. Goldstein" - id: short-2005-pof resource: "https://doi.org/10.1063/1.2006027" title: "Stalactite growth as a free-boundary problem" author: "M. B. Short, J. C. Baygents, R. E. Goldstein" - id: commons-soda-straw-oregon-caves resource: "https://commons.wikimedia.org/wiki/File:Soda_Straw_Formation_(9940650705).jpg" title: "Soda Straw Formation" author: "Oregon Caves (National Park Service)" - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Stalactites Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 26. The same session settles the theory and the wagers on [Stacked Cantilevers](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) and deals with [Summing a Series](https://tyson-swetnam.github.io/aosd/problems/summing-a-series/index.md).* !!! abstract "The problem" Reconstructed from the syllabus, which says only "deal with" stalactites; Winfree's own write-up is lost. Stone icicles (stalactites) hang from cave ceilings, dripping slowly, and blunt mounds (stalagmites) rise beneath them. Keep several working hypotheses alive at once. 1. **Why is there stone at all?** Where does the limestone (calcium carbonate) come from, and what makes it leave the water on the ceiling? Give at least two mechanisms and a test for each. 2. **Why that shape?** A young stalactite is a hollow tube a few millimetres across (a "soda straw"); an old one is a cone tapering to a tip; the stalagmite below is broader, with a rounded top. Explain each. 3. **How old is it?** A metre-long stalactite drips once a minute. Estimate its age twice: from the drip rate and the mineral a drop could carry, and from any growth rate you can find or guess. Do they agree? Which assumption would you suspect first? 4. **Turn it around.** Could the thickness of a stalagmite floor date the bones buried beneath it? ![Cross-section of a stalactite with a hollow central canal and growth layers, a drop at its tip losing carbon dioxide, drops falling to a broad rounded stalagmite with no canal; an inset shows the young soda-straw tube, where each drop leaves a ring at its rim](https://tyson-swetnam.github.io/aosd/assets/images/problems/stalactites.svg){ width="560" } *Drawn for this site (CC BY 4.0). Schematic, not to scale.* ![Close-up of a translucent soda-straw stalactite, a hollow mineral tube with a drop of water at its open lower end](https://tyson-swetnam.github.io/aosd/assets/images/problems/stalactites-soda-straws.jpg){ width="560" } *A soda straw with a drop at its tip, Oregon Caves National Monument. Oregon Caves (National Park Service) photograph, CC BY 2.0, via Wikimedia Commons.* ## Why it is in the course Section 5 is "Inferences, Hypotheses, Explanations", and session 25 assigns Chamberlin's *The Method of Multiple Working Hypotheses*, an essay by a geologist. A stalactite suits that lesson: nobody is an expert, and the first explanation most people reach for, that the water dries up, is at best incomplete. The age question adds a second lesson: an inference is only as good as its assumptions. Two honest estimates can disagree wildly; finding the guilty assumption is the exercise. In the syllabus's words, "The purpose of the puzzles (many of them silly) is to slow you down for a few minutes so you can examine the working of your own mind." ## Where it comes from Dating by growth has an instructive history. At Ingleborough Cave in Yorkshire, James Farrer measured a stalagmite called the Jockey Cap in 1839 and 1845, and John Phillips put its age at 259 years, assuming that all or nearly all the lime in the dripping water was deposited. In 1873 William Boyd Dawkins measured it again: the gap to the roof had closed from 95.25 to 87 inches, about 0.29 inch a year. At that rate, he noted, it might be no more than 100 years old. He concluded that "the present rate of growth is not a measure of its past or future condition" (*Cave Hunting*, 1874). The shape was explained mathematically only in 2005, by Martin Short, Raymond Goldstein and four colleagues: five of the six worked at the University of Arizona, one at Kartchner Caverns State Park. Their growth law draws a broad range of starting shapes toward one ideal profile, close to the average of real stalactites. ??? tip "Hints" - Separate what you are sure of (stone hangs; water drips) from what you assume (the water dries up; the drip never changed). - Fizzy water left standing loses something. Compare soil air with cave air. - Watch a drop hang from a tap. Where would a deposit be left? - For the age, write the chain: drops per year, mineral per drop, mineral in the stalactite. Each link is a hypothesis. - If your estimates differ tenfold, do not average them. Ask what the drop does after it leaves the tip. ??? success "Resolution" **Why stone forms.** Rain picks up carbon dioxide in the soil and dissolves limestone on the way down. Cave air holds far less carbon dioxide, so the gas escapes from a hanging drop and calcium carbonate comes out of solution. To test "it evaporates", look at a cold, damp, still cave: stalactites still grow there. **Why the shape.** Each drop leaves a thin ring of calcite at its rim, and ring on ring builds the soda straw, about 4 to 5 mm across. When the tube plugs or water runs down the outside, the cone thickens where more water has passed. Drops splash and spread on the floor, so the stalagmite has no canal, is wider, and is rounded. **How old.** The routes disagree, which is the point. Growth rates alone give about 300 years at a fast 3 mm a year, 8,000 at the average 0.13 mm, and 60,000 at the sixteenth of an inch per century quoted at Kartchner Caverns. If every drop (about 530,000 a year) left its whole load, the cone would form in centuries, far faster than the average rate allows. That convicts the assumption that all the mineral stays on the stalactite: much is carried to the floor, and drip rate and chemistry change over time. Phillips's 259 years also came from one calculation that assumed complete deposition; Dawkins's re-measurement gave a different age, and he warned that the present rate is not a measure of the past. (Rough illustrations only.) **Turn it around.** Not by thickness alone, as Dawkins warned. Modern dating measures uranium-thorium or radiocarbon in the calcite itself. ## Sources - **Wikipedia contributors**, "Stalactite" β€” [Wikipedia](https://en.wikipedia.org/wiki/Stalactite){target=_blank} πŸ”“ (soda-straw diameter; growth rates) - **U.S. National Park Service**, "Speleothems" β€” [nps.gov](https://www.nps.gov/subjects/caves/speleothems.htm){target=_blank} πŸ”“ (carbon-dioxide loss; hollow tubes; stalagmite shape) - **Wikipedia contributors**, "Soda straw" β€” [Wikipedia](https://en.wikipedia.org/wiki/Soda_straw){target=_blank} πŸ”“ (ring deposition at the drop's edge) - **Wikipedia contributors**, "Kartchner Caverns State Park" β€” [Wikipedia](https://en.wikipedia.org/wiki/Kartchner_Caverns_State_Park){target=_blank} πŸ”“ (growth rate) - **W. Boyd Dawkins**, *Cave Hunting* (1874), pp. 39–40 and Appendix II β€” [Internet Archive](https://archive.org/details/cavehuntingrese01dawkgoog){target=_blank} πŸ”“ - **M. B. Short, J. C. Baygents, J. W. Beck, D. A. Stone, R. S. Toomey III and R. E. Goldstein**, "Stalactite Growth as a Free-Boundary Problem: A Geometric Law and Its Platonic Ideal", *Physical Review Letters* 94, 018501 (2005) β€” [doi:10.1103/PhysRevLett.94.018501](https://doi.org/10.1103/PhysRevLett.94.018501){target=_blank} πŸ”’ - **M. B. Short, J. C. Baygents and R. E. Goldstein**, "Stalactite growth as a free-boundary problem", *Physics of Fluids* 17, 083101 (2005) β€” [doi:10.1063/1.2006027](https://doi.org/10.1063/1.2006027){target=_blank} πŸ”’ - **Oregon Caves (National Park Service)**, "Soda Straw Formation", CC BY 2.0 β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Soda_Straw_Formation_(9940650705).jpg){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The topic is certain; the questions are not. The syllabus and archived handout give only "deal with stalactites"; the 2005 Arizona theory postdates the course and has no documented link to it. Candidates: - **Explain the phenomenon** (medium; used above): rival hypotheses for formation and shape. - **Estimate the age** (medium): expose the hidden assumptions, as Dawkins's re-measurement did for Phillips's estimate. - **A mathematical calculation** (low): the session's other items are mathematical. - **Pattern formation** (low): why dripping films produce one shape. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/summing-a-series/ --- title: "Summing a Series" description: "The theory of stacked blocks produces the sum 1 + 1/2 + 1/3 + ... + 1/n: does it have a limit? A reconstructed exercise in not trusting the first thousand terms." type: Activity tags: [course, student-facing, problem, section-5, harmonic-series, infinite-series, block-stacking] status: stable problem: section: 5 session: 26 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: wikipedia-block-stacking resource: "https://en.wikipedia.org/wiki/Block-stacking_problem" title: "Block-stacking problem" author: "Wikipedia contributors" - id: wikipedia-harmonic-series resource: "https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)" title: "Harmonic series (mathematics)" author: "Wikipedia contributors" - id: wikipedia-oresme resource: "https://en.wikipedia.org/wiki/Nicole_Oresme" title: "Nicole Oresme" author: "Wikipedia contributors" - id: mathworld-book-stacking resource: "https://mathworld.wolfram.com/BookStackingProblem.html" title: "Book Stacking Problem" author: "Eric W. Weisstein" - id: paterson-et-al-2009 resource: "https://arxiv.org/abs/0707.0093" title: "Maximum Overhang" author: "Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, Uri Zwick" - id: paterson-zwick-2009 resource: "https://arxiv.org/abs/0710.2357" title: "Overhang" author: "Mike Paterson, Uri Zwick" - id: johnson-1955 resource: "https://doi.org/10.1119/1.1933957" title: "Leaning Tower of Lire" author: "Paul B. Johnson" - id: hall-2005 resource: "https://doi.org/10.1119/1.2074007" title: "Fun with stacking blocks" author: "John F. Hall" - id: gardner-1964 resource: "https://en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns" title: "Some paradoxes and puzzles involving infinite series and the concept of limit (Mathematical Games, November 1964)" author: "Martin Gardner" - id: gardner-sixth-book resource: "https://archive.org/details/martingardnerssi0000gard" title: "Martin Gardner's Sixth Book of Mathematical Games from Scientific American" author: "Martin Gardner" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Summing a Series Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 26. The same session works out the theory of the [Stacked Cantilevers](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) lab and takes up [Stalactites](https://tyson-swetnam.github.io/aosd/problems/stalactites/index.md).* !!! abstract "The problem" *Reconstructed from the syllabus, which says only "Deal with Summing a Series", in the session on the theory of stacking cantilevers. The series below is the one that theory produces.* **The stack.** Identical uniform blocks, one unit long, are piled at a table edge, each resting on exactly one block below, with no glue and no counterweights. How far beyond the edge can the top block reach? Find the largest overhang for *n* blocks. **The series.** Show that the answer is half of S(*n*) = 1 + 1/2 + 1/3 + ... + 1/*n*. Then deal with this series. - How many blocks give an overhang of one full block length? Of two? - The terms shrink toward zero. Does the sum approach a limit, or pass any number you name? Prove your answer by more than one route. - Roughly how fast does S(*n*) grow? Estimate how many blocks an overhang of ten block lengths would need. - Compare 1 + 1/2 + 1/4 + 1/8 + ... and 1 + 1/4 + 1/9 + 1/16 + ... What is different? **The wager.** Had you bet on how far a stack could reach, which bets win, and what hidden assumption did each side rely on? ![Five blocks stacked at a table edge, each shifted out by one-half, one-quarter, one-sixth, one-eighth and one-tenth of a block length, so that the top block hangs entirely beyond the edge; the total overhang equals half the sum 1 + 1/2 + 1/3 + 1/4 + 1/5](https://tyson-swetnam.github.io/aosd/assets/images/problems/summing-a-series.svg){ width="560" } *Five blocks stacked one on one; the centre of mass of the whole stack sits exactly above the table edge. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 5 is "Inferences, Hypotheses, Explanations". Session 26 opens with "Theory of stacking cantilevers, resolution of wagers", after two sessions of the Stacked Cantilevers lab. If that lab stacked real blocks, as seems likely, the theory turns the stacks into a series, and whether the series has a limit settles the bets. It is a model case for the section. The first few terms, even the first few thousand, point the wrong way, and only an argument can decide. It also uses habits from earlier sessions: [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md) ("like a jig-saw puzzle of cross-checks") and session 04's "distinguishing things we know vs only imagine". And it shows how easily an inference overreaches: a 2005 paper in the *American Journal of Physics* improved the stack, then concluded wrongly that nothing better was possible. ## Where it comes from The overhang puzzle appears in nineteenth-century mechanics textbooks (Phear, 1850; Walton, 1855). It was posed in the *American Mathematical Monthly* in 1923. It resurfaced in Paul B. Johnson's note "Leaning Tower of Lire" (*American Journal of Physics*, 1955) and became widely known through Martin Gardner's *Scientific American* column of November 1964, "Some paradoxes and puzzles involving infinite series and the concept of limit", reprinted in his *Sixth Book of Mathematical Games* (1971). The series, 1 + 1/2 + 1/3 + ..., is the harmonic series. Whether it has a finite total was settled around 1350-1360 by the Paris scholar Nicole Oresme. His argument was forgotten, then found again by Pietro Mengoli (1650) and by Jacob and Johann Bernoulli (1689). ![Medieval manuscript miniature of Nicole Oresme seated at a writing desk, with an armillary sphere beside him](https://tyson-swetnam.github.io/aosd/assets/images/problems/summing-a-series-oresme.jpg){ width="560" } *Nicole Oresme at his studies, miniature from his* TraitΓ© de l'espΓ¨re *(c. 1400-1420), BibliothΓ¨que nationale de France, via Wikimedia Commons. Public domain.* ??? tip "Hints" - Start from the top. Where must the centre of mass of the top *k* blocks lie relative to block *k* + 1? - Tabulate S(*n*) for *n* = 1 to 10 and plot it against log *n*. - To decide about a limit, stop computing terms. Group them in blocks of 2, 4, 8, 16, ... and bound each block from below. - For a second route, compare 1/*k* with the area under y = 1/x from x = *k* to x = *k* + 1. - For the wager, list every assumption in the theory. Which could be dropped? ??? success "Resolution" **The stack.** The top *k* blocks together can project 1/(2*k*) beyond the block below them (or the table), so *n* blocks reach (1/2)(1 + 1/2 + ... + 1/*n*). Four blocks give 25/24, clearing the edge entirely; two block lengths need 31 blocks, three need 227. **The series.** It has no limit. Oresme grouped the terms: 1/3 + 1/4 > 1/2, 1/5 + ... + 1/8 > 1/2, and each later group of 8, 16, 32, ... terms adds more than 1/2, so the sum passes any number. The area comparison agrees, ln(*n* + 1) < S(*n*) ≀ 1 + ln *n*, and shows how slowly it grows: S(*n*) first exceeds 10 at *n* = 12,367, and an overhang of ten block lengths needs about 272 million blocks. By contrast 1 + 1/2 + 1/4 + ... = 2 and 1 + 1/4 + 1/9 + ... = π²/6. Terms going to zero is necessary for a finite sum, not sufficient. **The wagers.** A bet that the overhang is bounded loses in principle but wins in practice: the offsets soon shrink below the imperfections of real blocks. A bet that the top block cannot clear the edge loses at four blocks. And one-on-one stacking is an assumption, not a law. With counterweights, four blocks reach about 1.168 (Ainley, 1979). John F. Hall (2005) showed that counterweights asymptotically double the harmonic overhang, then inferred wrongly that no further gain was possible. Paterson and Zwick (2006) built stacks whose overhang grows like the cube root of *n*, exponentially further than the one-on-one stack's roughly (1/2) ln *n*. In 2009 Paterson, Peres, Thorup, Winkler and Zwick proved that order is the true one. ## Sources - **Wikipedia contributors**, "Block-stacking problem" β€” [Wikipedia](https://en.wikipedia.org/wiki/Block-stacking_problem){target=_blank} πŸ”“ - **Wikipedia contributors**, "Harmonic series (mathematics)" β€” [Wikipedia](https://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29){target=_blank} πŸ”“ (Oresme's grouping proof, Mengoli, the Bernoullis, the growth rate) - **Wikipedia contributors**, "Nicole Oresme" β€” [Wikipedia](https://en.wikipedia.org/wiki/Nicole_Oresme){target=_blank} πŸ”“ - **Eric W. Weisstein**, "Book Stacking Problem", MathWorld β€” [MathWorld](https://mathworld.wolfram.com/BookStackingProblem.html){target=_blank} πŸ”“ (the overhang formula and the counts 4, 31 and 227) - **Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler and Uri Zwick**, "Maximum Overhang", *American Mathematical Monthly* 116(9), 763–787 (2009) β€” [arXiv:0707.0093](https://arxiv.org/abs/0707.0093){target=_blank} πŸ”“ (history of the puzzle from 1850, Ainley 1979, Hall 2005) - **Mike Paterson and Uri Zwick**, "Overhang", *American Mathematical Monthly* 116(1), 19–44 (2009), first presented at SODA 2006 β€” [arXiv:0710.2357](https://arxiv.org/abs/0710.2357){target=_blank} πŸ”“ - **Paul B. Johnson**, "Leaning Tower of Lire", *American Journal of Physics* 23(4), 240 (1955) β€” [doi:10.1119/1.1933957](https://doi.org/10.1119/1.1933957){target=_blank} πŸ”’ - **John F. Hall**, "Fun with stacking blocks", *American Journal of Physics* 73(12), 1107–1116 (2005) β€” [doi:10.1119/1.2074007](https://doi.org/10.1119/1.2074007){target=_blank} πŸ”’ - **Martin Gardner**, "Some paradoxes and puzzles involving infinite series and the concept of limit", *Scientific American*, November 1964, 126–133 β€” [index of Gardner's columns](https://en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns){target=_blank} πŸ”“ (the link is Wikipedia's index, not the column, which is not freely online) - **Martin Gardner**, *Martin Gardner's Sixth Book of Mathematical Games from Scientific American* (1971), chapter 17, "Limits of Infinite Series" β€” [Internet Archive](https://archive.org/details/martingardnerssi0000gard){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Probable, not certain. The syllabus gives only the name, in session 26, beside the theory of stacking cantilevers and Stalactites. No surviving Winfree document names the series. Three readings: - **The harmonic series from the block stack** (likeliest): the classical theory of the stack is exactly this series, and whether the overhang is bounded is the natural wager. - **A separate cross-checking exercise** on some series, summed by several independent routes in the manner of session 12. This is compatible with the first reading. - **Paradoxes of infinite series**, such as 1/2 + 1/4 + 1/8 + ... = 1. Gardner's 1964 column treats these together with the block stack, but there is no direct evidence. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/loshu-lab/ --- title: "LoShu Lab" description: "A two-session lab built around the Lo Shu magic square: experiment with a pick-three-to-make-15 card game and with 3 x 3 magic squares, then build one theory that explains both." type: Activity tags: [course, student-facing, problem, section-5, magic-squares, tic-tac-toe, representation, strong-inference] status: stable problem: section: 5 session: 27 identification: probable kind: lab generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: simon-1969 resource: "https://archive.org/details/sciencesofartifi00simo" title: "The Sciences of the Artificial" author: "Herbert A. Simon" - id: simon-1996 resource: "https://books.google.com/books?id=k5Sr0nFw7psC&pg=PA131" title: "The Sciences of the Artificial, third edition" author: "Herbert A. Simon" - id: newell-simon-1972 resource: "https://archive.org/details/humanproblemsolv0000newe" title: "Human Problem Solving" author: "Allen Newell and Herbert A. Simon" - id: michie-1982 resource: "https://archive.org/details/machineintellige0000mich" title: "Machine Intelligence and Related Topics: An Information Scientist's Weekend Book" author: "Donald Michie" - id: saul-zelbo-2014 resource: "https://archive.org/details/camplogicweekofl0000saul" title: "Camp Logic: A Week of Logic Games and Activities for Young People" author: "Mark Saul and Sian Zelbo" - id: moler-2011 resource: "https://web.archive.org/web/20161220110529/https://people.sc.fsu.edu/~jburkardt/m_src/exm_pdf/tictactoe.pdf" title: "Chapter 11: TicTacToe Magic (Experiments with MATLAB)" author: "Cleve Moler" - id: wikipedia-tic-tac-toe resource: "https://en.wikipedia.org/wiki/Tic-tac-toe" title: "Tic-tac-toe" author: "Wikipedia contributors" - id: andrews-carus-1908 resource: "https://archive.org/details/magicsquarescube00andrrich" title: "Magic Squares and Cubes (with 'The Magic Square in China' by Paul Carus)" author: "W. S. Andrews and Paul Carus" - id: cammann-1961 resource: "https://doi.org/10.1086/462439" title: "The Magic Square of Three in Old Chinese Philosophy and Religion" author: "Schuyler Cammann" - id: wikipedia-luoshu resource: "https://en.wikipedia.org/wiki/Luoshu_Square" title: "Luoshu Square" author: "Wikipedia contributors" - id: platt-1964 resource: "https://doi.org/10.1126/science.146.3642.347" title: "Strong Inference" author: "John R. Platt" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # LoShu Lab Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 27. The lab finishes in session 28, which also deals with [The Miracle of FujiYama](https://tyson-swetnam.github.io/aosd/problems/miracle-of-fujiyama/index.md) and starts the [laws of a toy universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: no problem sheet survives, and this is the editors' wording, not Winfree's (see the note at the end). **Part 1: experiments (first class).** Work in pairs. *Experiment A, a game.* Lay nine cards numbered 1 to 9 face up. Take turns picking one card and keeping it. The first player to hold any three cards that add up to exactly 15 wins; if the cards run out first, it is a draw. Play at least ten games, log every pick, write hunches as testable statements ("5 is the best first pick"), and design games that could prove them wrong. *Experiment B, a square.* Place the digits 1 to 9 in a 3 x 3 grid so that all rows, columns and both diagonals have the same sum. Record every arrangement you find, and decide what counts as "different". **Part 2: a complete theory (second class).** Explain both experiments: the common sum and the centre digit, how many squares exist, every way to win the game, and the result of perfect play, with a strategy that guarantees it. ![Nine cards numbered 1 to 9 in a row above an empty three-by-three grid](https://tyson-swetnam.github.io/aosd/assets/images/problems/loshu-lab-setup.svg){ width="560" } *The two experiments. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 5 is "Inferences, Hypotheses, Explanations". Session 27, with Platt's *Strong Inference* due, reads "Start LoShu lab experiments in class". Session 28, with Judson's chapter "Strong Predictions" due, reads "Finish complete theory of LoShu." A card game is a cheap laboratory. A hunch such as "the first player can force a win" can be stated, tested and killed in minutes: Platt's cycle in miniature. The experiments-then-theory shape matches the lab just before it, [Stacked Cantilevers](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md), which ends with "Theory of stacking cantilevers, resolution of wagers". If the lab was built on the link between the square and tic-tac-toe (see the note at the end), it also carries Herbert Simon's lesson: how a problem is written down can decide whether it is hard. ## Where it comes from **The square.** The Lo Shu ("scroll of the River Luo") is an ancient Chinese magic square of the digits 1 to 9. Legend says Yu the Great saw it on a turtle's back. Paul Carus (1908) warned that the surviving diagram is "a reconstruction of an ancient document", not the document itself. How old the arrangement is remains disputed. Later scholarship, following Schuyler Cammann's 1961 study, places a clear description of the square at about 80 CE. **The game.** The link to tic-tac-toe is modern. In *The Sciences of the Artificial* (1969), Simon described a card game he called "number scrabble": the game above, played with the ace to nine of hearts. He used it to show problem solving as a change of representation, as did Newell and Simon in *Human Problem Solving* (1972). Donald Michie wrote that the number version is far harder for people than tic-tac-toe, although the two are logically identical. Cleve Moler recalls hearing of it, as Pick15, in the late 1960s. Who invented it is not established. ??? tip "Hints" - For every win in your log, note which three numbers made the 15. Can there be triples you have not seen? - List every set of three different numbers from 1 to 9 that adds to 15, and count how many sets each number is in. - Add 1 through 9. The three rows use every digit once, so what must each row add to? How many lines pass through the centre, a corner, an edge cell? - Compare the two sets of counts. Could you write each number in a cell so they agree everywhere? What would a winning triple look like then? ??? success "Resolution" **The square.** The digits 1 to 9 sum to 45, and the three rows use each once, so every line sums to 15. The four lines through the centre *c* cover every cell once and the centre three extra times: 4 x 15 = 45 + 3*c*, so *c* = 5. **The triples.** Exactly eight sets of three different digits sum to 15: {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}. The digit 5 is in four of them, each even digit in three, and each other odd digit in two. **Only one square.** A 3 x 3 grid has eight lines, so a magic square uses each triple once. The centre lies on four lines, so it holds 5; the corners lie on three, so they hold 2, 4, 6 and 8; the edges lie on two, so they hold 1, 3, 7 and 9. The top-left corner can be chosen 4 ways (fixing its opposite, which must add to 10), the other corners 2 ways, and the edges are then forced: 8 squares, the rotations and reflections of the Lo Shu. ![The Lo Shu square 4 9 2, 3 5 7, 8 1 6 with its three rows, three columns and two diagonals drawn as lines, each labelled with a sum of 15; the corner cells, holding the even numbers, are shaded](https://tyson-swetnam.github.io/aosd/assets/images/problems/loshu-lab.svg){ width="560" } *Drawn for this site (CC BY 4.0).* **The game is tic-tac-toe.** Holding three numbers that sum to 15 is holding three Lo Shu cells in a line, so every threat, block and fork carries over. Tic-tac-toe is a draw with best play, and so is the card game: picture the square (5 is the centre, evens are corners) and play ordinary tic-tac-toe. It only feels harder because sums, unlike lines, are not visible at a glance. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579) course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Herbert A. Simon**, *The Sciences of the Artificial*, "The Science of Design" (MIT Press, 1969) β€” [Internet Archive](https://archive.org/details/sciencesofartifi00simo){target=_blank} πŸ”“ *(borrow)* - **Herbert A. Simon**, *The Sciences of the Artificial*, 3rd ed., pp. 131-132 (1996) β€” [Google Books](https://books.google.com/books?id=k5Sr0nFw7psC&pg=PA131){target=_blank} πŸ”’ - **Allen Newell and Herbert A. Simon**, *Human Problem Solving*, Chapter 3 (1972) β€” [Internet Archive](https://archive.org/details/humanproblemsolv0000newe){target=_blank} πŸ”“ *(borrow)* - **Donald Michie**, *Machine Intelligence and Related Topics* (Gordon and Breach, 1982) β€” [Internet Archive](https://archive.org/details/machineintellige0000mich){target=_blank} πŸ”“ *(borrow)* - **Mark Saul and Sian Zelbo**, *Camp Logic* (2014) β€” [Internet Archive](https://archive.org/details/camplogicweekofl0000saul){target=_blank} πŸ”’ (print-disabled access only) - **Cleve Moler**, "TicTacToe Magic", *Experiments with MATLAB*, Chapter 11 (2011) β€” [Wayback Machine](https://web.archive.org/web/20161220110529/https://people.sc.fsu.edu/~jburkardt/m_src/exm_pdf/tictactoe.pdf){target=_blank} πŸ”“ - **Wikipedia contributors**, "Tic-tac-toe" β€” [Wikipedia](https://en.wikipedia.org/wiki/Tic-tac-toe){target=_blank} πŸ”“ - **W. S. Andrews**, *Magic Squares and Cubes*, with Paul Carus, "The Magic Square in China", pp. 122-123 (1908) β€” [Internet Archive](https://archive.org/details/magicsquarescube00andrrich){target=_blank} πŸ”“ - **Schuyler Cammann**, "The Magic Square of Three in Old Chinese Philosophy and Religion", *History of Religions* 1(1), 37-80 (1961) β€” [doi:10.1086/462439](https://doi.org/10.1086/462439){target=_blank} πŸ”’ - **Wikipedia contributors**, "Luoshu Square" (source of the 80 CE dating, citing Cammann) β€” [Wikipedia](https://en.wikipedia.org/wiki/Luoshu_Square){target=_blank} πŸ”“ - **John R. Platt**, "Strong Inference", *Science* 146, 347-353 (1964) β€” [doi:10.1126/science.146.3642.347](https://doi.org/10.1126/science.146.3642.347){target=_blank} πŸ”’ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only two session lines. In Winfree's [archived handout](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank}, the link on "LoShu" in both session lines points to a bookmark named `Tictactoe_LoShu`. Its target was in a companion document that was never archived. The label points to tic-tac-toe with the Lo Shu, but nothing found ties this exact exercise to him, so identification is probable. Candidates: - **Combined lab** (reconstructed above); Saul and Zelbo's *Camp Logic* has a classroom sequence of this shape. - **Game-first lab** in Simon's style: play cold, then find a representation that makes the game easy. - **Magic-square enumeration**, with tic-tac-toe as an aside. - **A different sum-to-15 tic-tac-toe** (odd numbers against even), which does not depend on the Lo Shu arrangement. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/ --- title: "Laws of a Toy Universe" description: "A class lab: watch a grid of cells change generation by generation, work out the hidden law that drives it, then design an experiment that could prove your law wrong." type: Activity tags: [course, student-facing, problem, section-5, cellular-automata, strong-inference, induction] status: stable problem: section: 5 session: 28 identification: confident kind: lab generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: gardner-1970-text resource: "https://www.ibiblio.org/lifepatterns/october1970.html" title: "Mathematical Games: The fantastic combinations of John Conway's new solitaire game 'life' (full text reproduction)" author: "Martin Gardner" - id: gardner-1970 resource: "https://www.scientificamerican.com/article/mathematical-games-1970-10/" title: "Mathematical Games (October 1970)" author: "Martin Gardner" - id: wikipedia-game-of-life resource: "https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life" title: "Conway's Game of Life" author: "Wikipedia contributors" - id: poundstone-1985 resource: "https://archive.org/details/recursiveunivers00poun" title: "The Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge" author: "William Poundstone" - id: berlekamp-conway-guy-1982 resource: "https://archive.org/details/winningwaysforyo02berl" title: "Winning Ways for Your Mathematical Plays" author: "Elwyn R. Berlekamp, John H. Conway and Richard K. Guy" - id: feynman-1965 resource: "https://archive.org/details/characterofphysi0000feyn" title: "The Character of Physical Law" author: "Richard P. Feynman" - id: golly resource: "https://golly.sourceforge.io/" title: "Golly: open-source cellular-automaton explorer" author: "Andrew Trevorrow, Tom Rokicki and others" - id: playgameoflife resource: "https://playgameoflife.com/" title: "Play John Conway's Game of Life (browser simulator)" author: "Edwin Martin" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Laws of a Toy Universe Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 28, finished in session 29. The same session deals with [The Miracle of FujiYama](https://tyson-swetnam.github.io/aosd/problems/miracle-of-fujiyama/index.md) and finishes the theory of [LoShu](https://tyson-swetnam.github.io/aosd/problems/loshu-lab/index.md).* !!! abstract "The problem" A reconstruction. The syllabus names the exercise and Winfree's handout identifies the universe (see "Where it comes from"), but how he ran it in class is not recorded. The runs below were computed for this page. A universe is a large grid of cells, each **occupied** or **empty**. Time runs in ticks and the whole grid changes at once: each generation fixes the next completely, with no chance and no outside input. You are not told the law, but you may ask for any starting pattern to be run. **Find The Laws**: a rule that predicts every next frame from the present one. Four runs (`#` occupied, `.` empty): ``` Run A gen 0 gen 1 gen 2 ..... ..... ..... ..... ..#.. ..... .###. ..#.. .###. ..... ..#.. ..... ..... ..... ..... Run B gen 0 gen 1 .... .... .##. .##. .##. .##. .... .... Run C gen 0 gen 1 gen 2 gen 3 gen 4 .#..... ....... ....... ....... ....... ..#.... #.#.... ..#.... .#..... ..#.... ###.... .##.... #.#.... ..##... ...#... ....... .#..... .##.... .##.... .###... ....... ....... ....... ....... ....... Run D gen 0 gen 1 gen 2 gen 3 ........ ........ ........ ........ ........ ...#.... ...#.... ..###... ..###... ..#.#... ..###... .#...#.. ..###... .#...#.. .##.##.. .#...#.. ..###... ..#.#... ..###... .#...#.. ........ ...#.... ...#.... ..###... ........ ........ ........ ........ ``` Answer in this order: 1. What is the law? State it so precisely that a stranger could compute generation 1 from generation 0 without asking you anything. 2. What would you have to see to know your law is wrong? Construct that test before you look. 3. Is your law the only one that fits everything you have seen? How would you tell? ![Two rows of square grids showing successive generations: a five-cell pattern that returns to its own shape one cell diagonally displaced after four steps, and a row of three cells that alternates between horizontal and vertical](https://tyson-swetnam.github.io/aosd/assets/images/problems/laws-of-toy-universe.svg){ width="560" } *Runs C and A from the box above; dark squares are occupied cells. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 5 is "Inferences, Hypotheses, Explanations". Its readings run from Chamberlin's multiple working hypotheses (session 25) through Platt's *Strong Inference* (27) to Judson's *Strong Predictions* (28). Session 28 says "Start discovering laws of a toy universe in class"; session 29, with Feynman's *The Character of Physical Law*, says "Finish collaborative discovery of The Laws". A toy universe is the cleanest laboratory there is: few, exact, local laws, free data, no instrument error. A class can hold rival rules at once, find the pattern they disagree about, run only that, and discard what fails. It is the kind of joint in-class problem the syllabus describes, one that needs "lots of data-collecting, best pooled from many sources". The lab also shows that no finite set of observations forces one law. A class can agree on The Laws and still be wrong about a case its patterns never visited. That is the gap the syllabus named back in session 4: "distinguishing things we know vs only imagine". ## Where it comes from The syllabus gives only the two schedule lines. The universe has a well-known name, and the name gives the answer away, so the history is folded below. Work on the runs first. ??? info "Which universe is it? (names the answer)" In Winfree's course handout (Wayback Machine capture, April 2002), the links on "toy universe" (session 28) and "The Laws" (session 29) both point to a bookmark named `Game_of_Life`. The bookmarks led into a problem document that was not archived, so Winfree's own write-up of the exercise does not survive. John Horton Conway devised the Game of Life in 1970. Martin Gardner's "Mathematical Games" column in *Scientific American* (October 1970) brought it to the public, setting out Conway's rules for survival, death and birth. The column started an amateur research programme: Bill Gosper's group at MIT won Conway's $50 prize for showing that a pattern can grow without limit, with the glider gun. Conway's own treatment is in *Winning Ways for Your Mathematical Plays* (1982), and William Poundstone's *The Recursive Universe* (1985) uses Life to ask how much its laws let an observer know. ??? tip "Hints" - Assume the law is local and the same everywhere; check that against Run D. - Is a neighbour one of four cells (edges) or eight (with corners)? Run A settles it: with four, the empty cell above the row's middle and the one beyond its end have equal counts, yet only one fills. - Ask separately when an occupied cell survives and when an empty cell fills. - Tabulate neighbour count against what happened next. The rule appears in the table, not the pictures. - When two rules survive, run the pattern they disagree about. ??? success "Resolution" **The Laws.** Each cell has eight neighbours (edge or corner). All cells update at once from the previous generation: - an occupied cell with **two or three** occupied neighbours stays occupied; - an occupied cell with fewer than two, or more than three, becomes empty (isolation, overcrowding); - an empty cell with **exactly three** occupied neighbours becomes occupied. In shorthand, **B3/S23**: birth on 3, survival on 2 or 3. This is Conway's Game of Life. Run A is the *blinker*: the end cells die with one neighbour, the middle survives on two, and the cells above and below are born on three. Run B, the *block*, never changes. Run C, the *glider*, regains its shape after four generations one cell along the diagonal. Run D's square loses its centre to overcrowding and opens into a ring. Two points matter more than the rule. 1. *The data never forced it.* Any rule agreeing with B3/S23 on the cases these runs visited fits equally well; only a test of an unvisited case decides. These runs never show an empty cell with six or seven occupied neighbours, or an occupied cell with none, six or seven. 2. *Knowing the law is not knowing the universe.* Life is Turing-complete: anything a computer can calculate can be calculated inside it. So there is no general shortcut for predicting whether a pattern dies, settles or grows forever; usually you have to run it. To stage it, show frames on a simulator such as Golly, never the rule, and name the game only after The Laws are agreed. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579), course handout; the session 28 and 29 lines and their bookmark β€” [Wayback Machine capture, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Martin Gardner**, "Mathematical Games: The fantastic combinations of John Conway's new solitaire game 'life'", *Scientific American* 223(4), 120–123 (October 1970) β€” [full-text reproduction](https://www.ibiblio.org/lifepatterns/october1970.html){target=_blank} πŸ”“ Β· [publisher page](https://www.scientificamerican.com/article/mathematical-games-1970-10/){target=_blank} πŸ”’ - **William Poundstone**, *The Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge* (1985) β€” [Internet Archive](https://archive.org/details/recursiveunivers00poun){target=_blank} πŸ”“ *(borrow)* - **Elwyn R. Berlekamp, John H. Conway and Richard K. Guy**, *Winning Ways for Your Mathematical Plays*, vol. 2 (1982) β€” [Internet Archive](https://archive.org/details/winningwaysforyo02berl){target=_blank} πŸ”’ - **Richard P. Feynman**, *The Character of Physical Law* (1965), the session 29 reading β€” [Internet Archive](https://archive.org/details/characterofphysi0000feyn){target=_blank} πŸ”’ - **Wikipedia contributors**, "Conway's Game of Life" β€” [Wikipedia](https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life){target=_blank} πŸ”“ - **Andrew Trevorrow, Tom Rokicki and others**, *Golly*, open-source cellular-automaton explorer β€” [golly.sourceforge.io](https://golly.sourceforge.io/){target=_blank} πŸ”“ - **Edwin Martin**, *Play John Conway's Game of Life*, browser simulator β€” [playgameoflife.com](https://playgameoflife.com/){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/miracle-of-fujiyama/ --- title: "The Miracle of FujiYama" description: "Winfree's statement is lost; an editors' reconstruction asks where a row of eight coupled reactors that stores three patterns must settle, and where that strong prediction goes silent." type: Activity tags: [course, student-facing, problem, section-5, strong-predictions, pattern-recognition, counting, chemical-oscillators] status: stable problem: section: 5 session: 28 identification: unknown kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL/EEB 479/479H/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: hohmann-kraus-schneider-1999 resource: "https://doi.org/10.1021/jp991480n" title: "Pattern Recognition by Electrical Coupling of Eight Chemical Reactors" author: "W. Hohmann, M. Kraus, F. W. Schneider" - id: outlook-1916 resource: "https://archive.org/details/sim_new-outlook_1916-01-26" title: "The Outlook, 26 January 1916" author: "The Outlook" - id: judson-1980 resource: "https://archive.org/details/searchforsolutio0000juds_r0s0" title: "The Search for Solutions (chapter 7, listed in Winfree's handout as 'Strong Predictions')" author: "Horace Freeland Judson" - id: platt-1964 resource: "https://doi.org/10.1126/science.146.3642.347" title: "Strong Inference" author: "John R. Platt" - id: wikipedia-natural-reactor resource: "https://en.wikipedia.org/wiki/Natural_nuclear_fission_reactor" title: "Natural nuclear fission reactor" author: "Wikipedia contributors" - id: wikipedia-paul-kuroda resource: "https://en.wikipedia.org/wiki/Paul_Kuroda" title: "Paul Kuroda" author: "Wikipedia contributors" - id: gyibc-2005 resource: "https://archive.org/details/gyibcannualrepor2005grea" title: "GYIBC annual report 2005" author: "Greater Yellowstone Interagency Brucellosis Committee" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # The Miracle of FujiYama Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 28. The same session finishes the theory of [LoShu](https://tyson-swetnam.github.io/aosd/problems/loshu-lab/index.md) and starts on the [laws of a toy universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md).* !!! abstract "The problem" A reconstruction, not Winfree's text, which is lost. The editors built this substitute on one reading of the only surviving clue (see *Where it comes from*); the patterns are their own. Eight chemical reactors stand in a row, numbered 1 to 8. Each is either **P** (oscillating) or **N** (at rest). A row of eight states is a *pattern*. The reactors are wired together so that the row has *stored* three patterns: - A = PPPPNNNN - B = PNPNPNPN - C = PPNNPPNN Started in any other pattern, the row changes until it settles. The builders claim it settles into **the stored pattern that differs from the start in the fewest places.** The figure works one case. 1. How many patterns can the row show? How many are not stored? 2. For each pair of stored patterns, count the places where they differ. 3. Say in advance where the row must end, or that the claim makes no prediction, for (a) PNPPPNNN, (b) NNNNPPPP, (c) NNNNNNNN. 4. For how many unstored patterns is the prediction definite? Which would you test first? 5. A run started one step from A ends in B. What has been refuted, and what has not? ![Eight reactors in a row, three oscillating and five at rest, schematically wired together, above a table of the three stored patterns A, B and C with the number of places each differs from the starting pattern: 1, 3 and 3](https://tyson-swetnam.github.io/aosd/assets/images/problems/miracle-of-fujiyama.svg){ width="560" } *The worked case from the reconstruction, not Winfree's problem. Drawn for this site (CC BY 4.0).* ## Why it is in the course Session 28 assigns chapter 7 of Judson's *The Search for Solutions*, listed in the schedule as "Strong Predictions", one session after Platt's *Strong Inference*. In the reconstruction, "the row settles into the nearest stored pattern" is a strong claim in Platt's sense: it names the end state before the run, so one run can refute it. Working the numbers also exposes its silent zones, the ties where it predicts nothing. Knowing where your theory says nothing is the first step to designing the test that could embarrass it. Reduced to a table of P and N, the exercise fits the syllabus's promise that exercises are "mostly made from elementary mathematics so as to require no lab setup". ## Where it comes from The syllabus gives only the title. In Winfree's handout, the link on this item points to a bookmark named `Eight_Reactors`. The bookmarks pointed into a companion document that was never archived, so the problem text is lost. Elsewhere a playful title hides a plain label: "Paired Observations" links to `Keplers_Laws`. Which reactors is not recorded. The closest match lies in Winfree's own field, the Belousov-Zhabotinsky oscillating reaction. In 1999 W. Hohmann, M. Kraus and F. W. Schneider wired eight such reactors together like a Hopfield neural network, each periodic (P) or at rest (N). The network stored three of the 256 patterns and carried some of the others to the stored pattern they differed from least. No document links Winfree to this paper. Nothing found explains "FujiYama". The phrase is no known puzzle name, though a 1916 *Outlook* article uses it plainly of the mountain seen "against the western sky". ??? tip "Hints" - Count patterns as you would count eight coin tosses. - Write A, B and C one above another and compare them column by column. - For each test pattern, count its mismatches with A, B and C. A prediction exists only when one count is strictly smallest. - Each of the eight possible columns of three P/N symbols appears exactly once, and columns 1 and 8, 2 and 7, 3 and 6, 4 and 5 are opposites (every symbol flipped). Working pair by pair lets you count ties without listing every pattern. ??? success "Resolution" This resolves the reconstruction only. Counts were checked by listing all 256 patterns. 1. 2^8 = 256 patterns; 253 are not stored. 2. Every pair differs in exactly 4 places, so none is favoured. 3. Mismatches with A, B and C: (a) 2, 2, 4: A and B tie, no prediction. (b) 8, 4, 4: B and C tie, no prediction. It is A with every state flipped; do not predict A because it "looks like" A. (c) 4, 4, 4: a three-way tie. 4. Only 129 of the 253 have a single nearest pattern (43 each); 84 tie between two and 40 among all three. Test first the 24 patterns one step from a stored pattern, where a failure hurts most, then the ties, where the row must do something the claim does not describe. 5. "Fewest mismatches wins" is refuted, since A was strictly nearest. That the row stores nothing is not shown; ask what rule it obeys. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL/EEB 479/479H/579), course handout; the session-28 line and its bookmark β€” [Wayback Machine capture, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **W. Hohmann, M. Kraus, F. W. Schneider**, "Pattern Recognition by Electrical Coupling of Eight Chemical Reactors", *Journal of Physical Chemistry A* 103(38), 7606-7611 (1999) β€” [DOI](https://doi.org/10.1021/jp991480n){target=_blank} πŸ”’ - **Eliza Ruhamah Scidmore**, "Japan's Coronation Season", *The Outlook*, 26 January 1916 β€” [Internet Archive](https://archive.org/details/sim_new-outlook_1916-01-26){target=_blank} πŸ”“ - **Horace Freeland Judson**, *The Search for Solutions* (1980), chapter 7, listed in Winfree's handout as "Strong Predictions" β€” [Internet Archive](https://archive.org/details/searchforsolutio0000juds_r0s0){target=_blank} πŸ”“ *(borrow)* - **John R. Platt**, "Strong Inference", *Science* 146(3642), 347-353 (1964) β€” [DOI](https://doi.org/10.1126/science.146.3642.347){target=_blank} πŸ”’ - **Wikipedia contributors**, "Natural nuclear fission reactor" β€” [Wikipedia](https://en.wikipedia.org/wiki/Natural_nuclear_fission_reactor){target=_blank} πŸ”“ - **Wikipedia contributors**, "Paul Kuroda" β€” [Wikipedia](https://en.wikipedia.org/wiki/Paul_Kuroda){target=_blank} πŸ”“ - **Greater Yellowstone Interagency Brucellosis Committee**, *GYIBC Annual Report 2005* β€” [Internet Archive](https://archive.org/details/gyibcannualrepor2005grea){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" Not sure. Which kind of reactor the item concerned is unknown. The readings weighed, all low confidence: - **Eight coupled chemical reactors**, as in the 1999 experiment: the reconstruction above. Nothing ties Winfree to the paper. - **Nuclear reactors.** The best nuclear fit is the Japanese-born chemist Paul Kuroda's 1956 proposal of natural fission reactors, confirmed at Oklo, Gabon, in 1972: a strong prediction, but with no eight and no Fuji. - **Diagnostic "reactors"**, subjects that test positive, as in a 2005 brucellosis report of eight reactors in one herd. Nothing links it to Winfree. Earlier guesses from the title alone (a sky wonder at the summit, the monk-on-the-mountain puzzle) are disfavoured: none involves reactors. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/antigen-invasions/ --- title: "Antigen Invasions" description: "A lost Section 5 problem, reconstructed: from a record of repeated invasions by foreign substances, state the rules the body follows, invent rival mechanisms, and find the observation that decides between them." type: Activity tags: [course, student-facing, problem, section-5, immunology, multiple-hypotheses, clonal-selection] status: stable problem: section: 5 session: 29 identification: unknown kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: jerne-1955 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC534292/" title: "The Natural-Selection Theory of Antibody Formation" author: "Niels K. Jerne" - id: burnet-1957 resource: "https://doi.org/10.3322/canjclin.26.2.119" title: "A modification of Jerne's theory of antibody production using the concept of clonal selection" author: "F. Macfarlane Burnet" - id: pauling-1940 resource: "https://doi.org/10.1021/ja01867a018" title: "A Theory of the Structure and Process of Formation of Antibodies" author: "Linus Pauling" - id: landsteiner-specificity resource: "https://archive.org/details/specificityofser0000land" title: "The Specificity of Serological Reactions" author: "Karl Landsteiner" - id: janeway-2001-ch1 resource: "https://www.ncbi.nlm.nih.gov/books/NBK10757/" title: "Immunobiology: The Immune System in Health and Disease, 5th ed., Chapter 1: Basic Concepts in Immunology" author: "Charles A. Janeway Jr., Paul Travers, Mark Walport, Mark Shlomchik" - id: janeway-2001-memory resource: "https://www.ncbi.nlm.nih.gov/books/NBK27158/" title: "Immunobiology: The Immune System in Health and Disease, 5th ed.: Immunological memory" author: "Charles A. Janeway Jr., Paul Travers, Mark Walport, Mark Shlomchik" - id: frank-2002 resource: "https://www.ncbi.nlm.nih.gov/books/NBK2405/" title: "Immunology and Evolution of Infectious Disease, Chapter 3: Benefits of Antigenic Variation" author: "Steven A. Frank" - id: wikipedia-clonal-selection resource: "https://en.wikipedia.org/wiki/Clonal_selection" title: "Clonal selection (overview)" author: "Wikipedia contributors" - id: wikipedia-antigenic-variation resource: "https://en.wikipedia.org/wiki/Antigenic_variation" title: "Antigenic variation (overview)" author: "Wikipedia contributors" - id: feynman-1965 resource: "https://archive.org/details/characterofphysi00feyn" title: "The Character of Physical Law" author: "Richard P. Feynman" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Antigen Invasions Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 29. Dealt with in the same session as [Martian DNA](https://tyson-swetnam.github.io/aosd/problems/martian-dna/index.md), while the class finishes discovering [the laws of a toy universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: only the name survives, so this exercise is the editors' own, not Winfree's text. An animal is invaded, on a schedule, by foreign substances **A**, **B** and **C**. A is a small synthetic chemical that no ancestor of this animal ever met. (A molecule this small provokes nothing on its own, so A is coupled to a carrier protein for injection; take that as given.) The animal's blood is drawn at intervals and tested for anything that binds each substance. The units are schematic: only the shapes matter. | Day | Injected | binds A | binds B | binds C | | ---: | :-- | ---: | ---: | ---: | | 0 | A | 0 | 0 | 0 | | 7 | - | 3 | 0 | 0 | | 14 | - | 30 | 0 | 0 | | 28 | A + B | 15 | 0 | 0 | | 31 | - | 200 | 0 | 0 | | 35 | - | 900 | 3 | 0 | | 42 | - | 1000 | 30 | 0 | Also: - C was never injected. - A close chemical relative of A is bound by the day-42 blood about ten times less tightly. - The animal makes nothing that binds its own proteins, and tissue from another animal, introduced early in life, is later accepted as its own. **Your tasks** 1. **State the rules.** Write down what this record forces you to believe, and no more. Compare the two rises in the "binds A" column, then compare the second rise with what B does on the same days. 2. **Explain them.** Propose at least two mechanisms, as different as you can make them: say, the invader is a mould the body shapes its answer around; or the body already carries a vast variety of answers and the invader picks out and multiplies the ones that fit. 3. **Find the test that decides.** For each mechanism, name an observation it predicts and the other forbids. Which rule is awkward for the mould? 4. **Count.** Suppose the animal must be ready for about 10⁸ distinguishable invaders, and its genome has 10⁴ to 10⁡ genes. Under the second mechanism, how many kinds of binder must exist in advance? Does that arithmetic kill the mechanism, or can you repair it? ![Schematic log-scale graph of antibody over 56 days. Antigen A is injected on day 0 and again on day 28 with a new antigen B. Anti-A rises slowly to about 30, sags to 15, then after the second injection climbs within a week to nearly 1000. Anti-B, from day 28, repeats the slow small first rise. Anti-C, never injected, stays flat at none.](https://tyson-swetnam.github.io/aosd/assets/images/problems/antigen-invasions.svg){ width="560" } *The table above as a graph. The dots are the table's values; the lines between and after them are schematic. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 5 is "Inferences, Hypotheses, Explanations", and session 29 sets Feynman's *The Character of Physical Law* beside two problems, "Deal with Antigen Invasions" and "Deal with Martian DNA". Earlier the class read Chamberlin's *The Method of Multiple Working Hypotheses* and Platt's *Strong Inference*; this reconstruction puts them to work. Several explanations fit some of the facts, and the job is to find the observation that separates them. The course promises exercises that "depend as little as possible on knowledge of any particular subject area" and that are "mostly made from elementary mathematics". Nothing here needs immunology, and task 4 shows that an explanation surviving the qualitative facts can still be killed, or rescued, by counting. ## Where it comes from How a body answers an invader it has never met is a classic biological puzzle with a history of rival hypotheses. Paul Ehrlich's side-chain theory (1900) had cells already carrying many kinds of receptor, over-producing and shedding the kind a toxin fits; it was later abandoned. Karl Landsteiner then showed that animals make specific antibodies against synthetic chemicals, which evolution cannot have prepared them for. ![Four drawings of a cell with differently shaped side chains; a dark toxin molecule binds one kind, the cell grows many more of that kind, and the excess breaks away into the surroundings](https://tyson-swetnam.github.io/aosd/assets/images/problems/antigen-invasions-ehrlich.jpg){ width="560" } *Diagrams illustrating Paul Ehrlich's side-chain theory, from his Croonian Lecture "On Immunity with Special Reference to Cell Life" (1900). Wellcome Collection, CC BY 4.0, via Wikimedia Commons.* In 1940 Linus Pauling proposed that an antibody folds around the antigen and takes its shape from it. The template theory dominated for fifteen years. Niels Jerne (1955), then David Talmage and F. Macfarlane Burnet (1957), turned the logic around; Susumu Tonegawa, from 1976, settled the counting. The name fits a second story too: antigenic variation, in which a parasite changes its coat and invades the same host again. ??? tip "Hints" - Specificity is easy for a mould. What does a mould say about the *second* invasion, or about why the body spares itself? - If the variety exists beforehand, something must be thrown away. What happens to binders that fit the body's own molecules, and when? - One gene per specificity is the assumption doing the damage. What if a binder is built from parts picked from several small boxes? - Look for a test at the level of a single cell. ??? success "Resolution" *For the reconstruction only; Winfree's answer is not known. This is how biology settled the question.* **Selection, not instruction.** The body carries a huge, randomly generated repertoire before any invasion; the invader picks the cells whose answer already fits and drives them to multiply. Jerne proposed selection; Talmage and Burnet made it clonal, one lymphocyte, one specificity, and Nossal and Lederberg (1958) confirmed that one cell makes one antibody. - **Memory** follows: clones selected by the first invasion are still there, enlarged, so a second dose of A is met faster and harder while B, on the same day, starts from scratch. - **Self-tolerance** follows: self-reactive clones are removed or silenced during development, so early-introduced tissue counts as self. - **Counting**: Tonegawa showed antibody genes are spliced in each developing cell from separate pools of segments; with two chains and mutation, a few hundred parts give well over 10⁸ combinations. Pauling's template explained specificity but not memory or tolerance. For a changing invader the answer is antigenic variation: trypanosomes switch among more than a thousand coat genes, so each cleared wave is followed by a variant the immune system has not met. ## Sources - **Niels K. Jerne**, "The Natural-Selection Theory of Antibody Formation", *PNAS* 41(11), 849–857 (1955) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC534292/){target=_blank} πŸ”“ - **F. Macfarlane Burnet**, "A modification of Jerne's theory of antibody production using the concept of clonal selection" (*Australian Journal of Science*, 1957; reprinted in *CA: A Cancer Journal for Clinicians* 26(2), 119–121, 1976) β€” [DOI](https://doi.org/10.3322/canjclin.26.2.119){target=_blank} πŸ”’ - **Linus Pauling**, "A Theory of the Structure and Process of Formation of Antibodies", *Journal of the American Chemical Society* 62, 2643–2657 (1940) β€” [DOI](https://doi.org/10.1021/ja01867a018){target=_blank} πŸ”’ - **Karl Landsteiner**, *The Specificity of Serological Reactions* (Dover, 1962) β€” [Internet Archive](https://archive.org/details/specificityofser0000land){target=_blank} πŸ”“ *(borrow)* - **Charles A. Janeway Jr. et al.**, *Immunobiology*, 5th ed. (2001), Chapter 1: Basic Concepts in Immunology β€” [NCBI Bookshelf](https://www.ncbi.nlm.nih.gov/books/NBK10757/){target=_blank} πŸ”“ - **Charles A. Janeway Jr. et al.**, *Immunobiology*, 5th ed. (2001), "Immunological memory" β€” [NCBI Bookshelf](https://www.ncbi.nlm.nih.gov/books/NBK27158/){target=_blank} πŸ”“ - **Steven A. Frank**, *Immunology and Evolution of Infectious Disease* (2002), Chapter 3: Benefits of Antigenic Variation β€” [NCBI Bookshelf](https://www.ncbi.nlm.nih.gov/books/NBK2405/){target=_blank} πŸ”“ - **Wikipedia contributors**, "Clonal selection" (overview) β€” [Wikipedia](https://en.wikipedia.org/wiki/Clonal_selection){target=_blank} πŸ”“ - **Wikipedia contributors**, "Antigenic variation" (overview) β€” [Wikipedia](https://en.wikipedia.org/wiki/Antigenic_variation){target=_blank} πŸ”“ - **Richard P. Feynman**, *The Character of Physical Law* (1965), the session 29 reading β€” [Internet Archive](https://archive.org/details/characterofphysi00feyn){target=_blank} πŸ”“ *(borrow)* - **Arthur T. Winfree**, *The Art of Scientific Discovery: original course syllabus* (2001) β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" We are not: the identification is **unknown**. The syllabus gives only the name and the session, no statement of the problem survives, and searches found no puzzle by that name. Candidates, the first two merged above: - **Explain the immune response**: propose and test mechanisms for specificity, memory and self-tolerance, with counting as a check. - **Read the rules off a data set** of successive injections; the plural "Invasions" suggests a series of exposures. - **Repeated invasions by a changing invader**: why relapsing fever, trypanosomes or influenza come back (antigenic variation). Winfree worked on dynamical systems, but nothing favours this reading. If the problem sheet surfaces, this page should be rewritten around it. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/martian-dna/ --- title: "Martian DNA" description: "Winfree's write-up is lost; this reconstruction hands you a picture captioned as Martian hereditary material and asks which of your conclusions came from the picture and which came from the caption." type: Activity tags: [course, student-facing, problem, section-5, inference, pattern, hypotheses, observation] status: stable problem: section: 5 session: 29 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: winfree-lab-home-2006 resource: "https://web.archive.org/web/20060911203252/http://eebweb.arizona.edu/faculty/winfree/index.html" title: "Winfree lab home page (Wayback Machine capture, 2006)" author: "Arthur T. Winfree" - id: feynman-1965 resource: "https://archive.org/details/characterofphysi0000feyn" title: "The Character of Physical Law" author: "Richard P. Feynman" - id: wikipedia-chargaff resource: "https://en.wikipedia.org/wiki/Chargaff%27s_rules" title: "Chargaff's rules" author: "Wikipedia contributors" - id: watson-crick-1953 resource: "https://doi.org/10.1038/171737a0" title: "Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid" author: "J. D. Watson and F. H. C. Crick" - id: mckay-1996 resource: "https://doi.org/10.1126/science.273.5277.924" title: "Search for Past Life on Mars: Possible Relic Biogenic Activity in Martian Meteorite ALH84001" author: "D. S. McKay, E. K. Gibson Jr., K. L. Thomas-Keprta, H. Vali, C. S. Romanek, S. J. Clemett, X. D. F. Chillier, C. R. Maechling, R. N. Zare" - id: commons-old-cobbles resource: "https://commons.wikimedia.org/wiki/File:Old_cobbles_stones.jpg" title: "Old cobbles stones (photograph)" author: "Titus Tscharntke" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Martian DNA Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 29. Dealt with together with [Antigen Invasions](https://tyson-swetnam.github.io/aosd/problems/antigen-invasions/index.md); the same session finishes the collaborative discovery of [the laws of a toy universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md).* !!! abstract "The problem" Reconstructed from the syllabus: Winfree's write-up is lost. This is the editors' version, consistent with the surviving evidence (see the note at the foot of the page), not his text. You are handed a photograph with no scale bar. Its only caption: *"A high-resolution image of a stretch of Martian hereditary material. Deal with it."* It shows units of a few kinds packed edge to edge in rough rows, with occasional misfits. In your GamesWorth book: 1. **Facts first.** Describe only what a camera records: kinds of unit, proportions, arrangement, where the arrangement fails. 2. **Rules.** Propose at least two rules that could have produced it, one involving no code and no heredity at all. 3. **Strong predictions.** For each rule, name a feature it forbids in the part you have not seen. Ask to see the part where your rules disagree. 4. **Scale.** Which conclusions change if a unit is a nanometre, a centimetre or a metre across? 5. **The label.** Mark each statement in your notes as coming from the picture or from the words *Martian* and *DNA*. Which set is larger? ![Two panels. Above, a band of irregular units in three shades packed edge to edge in three rough rows, with the middle row marked and one split unit labelled misfit. Below, the middle row transcribed as eighteen letters, A C G A C G A C G A C T A C G A C G, with brackets marking a repeat of period 3 and the odd letter T circled and queried](https://tyson-swetnam.github.io/aosd/assets/images/problems/martian-dna.svg){ width="560" } *Drawn for this site (CC BY 4.0). A made-up specimen, not Winfree's picture: reading its middle row unit by unit turns a mosaic into a "sequence" with a repeat and one odd letter.* ## Why it is in the course Section 5 is "Inferences, Hypotheses, Explanations". Session 29 reads: "Deal with Antigen Invasions. Deal with Martian DNA. Finish collaborative discovery of The Laws. ..." All three ask what rule lies behind something. In the editors' reading, the trap in this one is the caption. Session 4 asked for "facts before explanations of facts" and for "distinguishing things we know vs only imagine". The syllabus also asks students to "generate several alternative guesses, and test them for workability". A picture labelled as alien genetic material tests all three. Much of what a student writes in the first two minutes is not in the picture; it is in the word *DNA*. The day's reading, Feynman's *The Character of Physical Law*, makes the same demand: guess a law, work out its consequences, compare them with experiment. A guess is worth having only if something computed from it could come out wrong. ## Where it comes from The editors know of no published puzzle called "Martian DNA". The phrase appears on no archived page of Winfree's lab site except the course handout. In the handout's HTML the words are a link to an internal bookmark whose name suggests what the object was. The write-up it pointed to does not survive in any archived copy. The bookmark is named in the note at the foot of the page, so try the exercise first. Period background only: the 1996 claim that the Martian meteorite ALH84001 held traces of past life was still argued over in 2001, and Chargaff's base ratios had helped Watson and Crick find DNA's structure in 1953. ??? tip "Hints" - Write down only what a camera records. "Martian" and "DNA" are not in the picture, and neither is the scale. - Count kinds, then ratios. For real DNA, Chargaff's counts of how often each unit occurs were an early clue to its structure. - For each regularity, ask whether it is a rule about meaning or about fitting. Things packed edge to edge obey constraints that carry no information. - A hypothesis earns its keep by forbidding something. Name what should be absent from the unseen part, then ask for exactly that part. - Which conclusions survive if the object is not Martian, not genetic and not alive? ??? success "Resolution" *What can honestly be said, given that Winfree's write-up is lost.* Winfree's bookmark name (see the note below) points to a cobblestone pavement. If that reading is right, the object billed as Martian DNA was a street. This is the editors' inference from the name, not a statement Winfree left behind. ![A cobblestone pavement seen from above: stones of several sizes and colours packed edge to edge in rough rows, with gaps and misfits](https://tyson-swetnam.github.io/aosd/assets/images/problems/martian-dna-cobbles.jpg){ width="560" } *A stand-in for Winfree's photograph, which does not survive. Photograph by Titus Tscharntke, public domain, via Wikimedia Commons.* Everything that can really be inferred from such a picture is about laying stones: a few kinds of unit, sizes in a narrow range, rough rows, defects where two patches of rows meet. None of it supports a sentence about coding, heredity or life; that came from the caption. A base-pairing rule for the stones is fine reasoning on a premise nobody tested. The productive moves were the ones that could have failed: counting ratios, predicting the unseen part, asking the scale, and imagining the picture if the caption were false. How Winfree ran the exercise is not recoverable. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579): course handout, archived 20 April 2002 β€” [Wayback Machine](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ (view the page source for the session 29 link) - **Arthur T. Winfree**, lab home page, Wayback Machine capture of 2006 β€” [Wayback Machine](https://web.archive.org/web/20060911203252/http://eebweb.arizona.edu/faculty/winfree/index.html){target=_blank} πŸ”“ (the archived site: no page other than the handout mentions the problem) - **Richard P. Feynman**, *The Character of Physical Law* (1965), the lecture "Seeking New Laws" β€” [Internet Archive](https://archive.org/details/characterofphysi0000feyn){target=_blank} πŸ”“ *(borrow)* - **Wikipedia contributors**, "Chargaff's rules" β€” [Wikipedia](https://en.wikipedia.org/wiki/Chargaff%27s_rules){target=_blank} πŸ”“ (period background) - **J. D. Watson and F. H. C. Crick**, "Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid", *Nature* 171, 737–738 (1953) β€” [doi:10.1038/171737a0](https://doi.org/10.1038/171737a0){target=_blank} πŸ”’ (period background) - **D. S. McKay et al.**, "Search for Past Life on Mars: Possible Relic Biogenic Activity in Martian Meteorite ALH84001", *Science* 273, 924–930 (1996) β€” [doi:10.1126/science.273.5277.924](https://doi.org/10.1126/science.273.5277.924){target=_blank} πŸ”’ (period background) - **Titus Tscharntke**, "Old cobbles stones", public domain β€” [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Old_cobbles_stones.jpg){target=_blank} πŸ”“ - **Arthur T. Winfree**, *The Art of Scientific Discovery*: original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the name and session. In Winfree's handout, the link on "Martian DNA" points to a bookmark named `Pohang_cobblestones` (Pohang is probably the port city in South Korea). Many of his bookmark names simply repeat the title, but several name the content instead ("Paired Observations" points to `Keplers_Laws`, "Superposed filters" to `Polaroids`). Read the same way, this one makes a cobbled pavement the very likely object. His wording and procedure are lost, and no independent web search for the phrase was possible while this page was researched. Candidate readings, which can overlap: - **Pattern reading with a sting** (likeliest): a close-up of the pavement presented as Martian hereditary material; separate what is in the picture from what the label supplies. - **Sequence decoding**: a run of stones transcribed as a symbol string, analysed for alphabet, ratios, repeats and pairing rules. - **Tiling rules**: infer how the paving was laid, and which patterns need a rule-maker rather than packing alone. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/problems/bacterial-hybrids/ --- title: "Bacterial Hybrids" description: "Two strains that cannot grow alone give colonies when mixed: list every explanation, read the linkage hidden in a table of colony counts, and design the experiment that kills the rival. An editors' reconstruction; Winfree's own problem sheet is lost." type: Activity tags: [course, student-facing, problem, section-5, genetics, bacteria, multiple-working-hypotheses, linkage] status: stable problem: section: 5 session: 30 identification: probable kind: puzzle generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout-2002 resource: "https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm" title: "The Art of Scientific Discovery (ECOL 479/579): course handout, archived 20 April 2002" author: "Arthur T. Winfree" - id: lederberg-tatum-1946 resource: "https://doi.org/10.1038/158558a0" title: "Gene Recombination in Escherichia coli" author: "Joshua Lederberg and Edward L. Tatum" - id: tatum-lederberg-1947 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC518375/" title: "Gene Recombination in the Bacterium Escherichia coli" author: "Edward L. Tatum and Joshua Lederberg" - id: lederberg-1947 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC1209393/" title: "Gene Recombination and Linked Segregations in Escherichia Coli" author: "Joshua Lederberg" - id: davis-1950 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC385908/" title: "Nonfiltrability of the Agents of Genetic Recombination in Escherichia coli" author: "Bernard D. Davis" - id: hayes-1953 resource: "https://doi.org/10.1099/00221287-8-1-72" title: "Observations on a Transmissible Agent Determining Sexual Differentiation in Bacterium coli" author: "William Hayes" - id: wollman-jacob-hayes-1956 resource: "https://doi.org/10.1101/sqb.1956.021.01.012" title: "Conjugation and Genetic Recombination in Escherichia coli K-12" author: "Γ‰lie L. Wollman, FranΓ§ois Jacob and William Hayes" - id: lederberg-1996 resource: "https://pmc.ncbi.nlm.nih.gov/articles/PMC1207540/" title: "Genetic Recombination in Escherichia coli: Disputation at Cold Spring Harbor, 1946–1996" author: "Joshua Lederberg" - id: almquist-1924 resource: "https://doi.org/10.1093/infdis/35.4.341" title: "Investigations on Bacterial Hybrids" author: "E. Almquist" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Bacterial Hybrids Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *[Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md), session 30, the last session in the schedule. The reading for the day is Judson, Chapter 9: Theory.* !!! abstract "The problem" This statement is the editors' reconstruction. Winfree's own problem sheet does not survive (see the note at the foot of the page). The numbers were made up for this page; they are not anyone's lab data. Two strains of a microbe, **A** and **B**, cannot grow on a bare nutrient plate. Strain A cannot make two substances it needs; strain B cannot make two different ones. Each strain also carries two traits you can read from a colony: **Lac**, whether it ferments lactose (Lac+) or not (Lacβˆ’), and **Vir**, whether it resists (VirΚ³) or is killed by (VirΛ’) a certain virus. | | needs | Lac | Vir | | :-- | :-- | :-- | :-- | | strain A | substances 1 and 2 | Lac+ | VirΛ’ | | strain B | substances 3 and 4 | Lacβˆ’ | VirΚ³ | **Experiment 1.** Two billion cells of A alone on a bare plate: nothing grows. B alone: nothing. Two billion of each, mixed, washed and spread: **312 colonies**. **Experiment 2.** The 312 colonies, scored for Lac and Vir: | colony type | count | | :-- | --: | | Lac+ VirΛ’ (like A) | 128 | | Lacβˆ’ VirΚ³ (like B) | 141 | | Lac+ VirΚ³ | 21 | | Lacβˆ’ VirΛ’ | 22 | **Experiment 3.** A third trait, **Gal** (galactose fermentation), is added. The cross is repeated with strains that differ in each pair of traits, and each time the share of colonies with a mixed combination (one trait from each parent) is counted: Lac and Vir, about 14%; Lac and Gal, about 5%; Gal and Vir, about 9%. **Your task.** 1. List every explanation for the 312 colonies, with an observation that would rule each out. 2. What would Experiment 2 show if the two traits were inherited independently? What does it actually show? 3. What relation do the three percentages of Experiment 3 obey, and what picture does that relation force on you? 4. Do the cells pass material by touching, or through the broth? Design one apparatus that decides. 5. Predict one thing your explanation says must happen that nobody has looked at yet. ![Three petri dishes of bare growth medium: strain A alone is empty, strain B alone is empty, and the dish spread with a mixture of both strains carries scattered colonies](https://tyson-swetnam.github.io/aosd/assets/images/problems/bacterial-hybrids.svg){ width="560" } *Experiment 1 as three plates. Schematic: the dots stand for colonies and are not a count. Drawn for this site (CC BY 4.0).* ## Why it is in the course Section 5, "Inferences, Hypotheses, Explanations", opens with Chamberlin's *The Method of Multiple Working Hypotheses* (session 25) and Platt's *Strong Inference* (session 27). It ends in session 30 with Judson's chapter on theory and this problem, the last one in the schedule. The version reconstructed here asks for both readings on one set of numbers. Colonies where there should be none have several honest explanations. "Facts before explanations of facts" (the syllabus's phrase for session 04) means listing them all before choosing one. Then traits nobody selected narrow the choice, and three measurements obey a relation that none of them shows alone. No biology is needed. The syllabus says the exercises "depend as little as possible on knowledge of any particular subject area". ## Where it comes from The phrase "bacterial hybrids" was already in print in 1924, in the title of a paper by E. Almquist. Until the mid-1940s, though, bacteria were widely assumed to lie outside genetics: no visible chromosomes, no sex, nothing to cross. In 1946 Joshua Lederberg, then a graduate student, and Edward Tatum mixed two strains of *Escherichia coli* K-12, each unable to make several substances it needed. They recovered rare colonies that grew where neither parent could. What those colonies meant was disputed for years. Lederberg shared the 1958 Nobel Prize in Physiology or Medicine for this work, and in 1996 he looked back on how sharply it had been contested. How the dispute was settled is in the Resolution below. ![Scanning electron micrograph of rod-shaped Escherichia coli cells, digitally coloured](https://tyson-swetnam.github.io/aosd/assets/images/problems/bacterial-hybrids-ecoli.jpg){ width="560" } *Escherichia coli at about 10,000Γ—. Photo by Eric Erbe, digital colourization by Christopher Pooley, USDA Agricultural Research Service. Public domain, via Wikimedia Commons.* ??? tip "Hints" - First explain the two plates that grew nothing. What do they rule out? - Independent traits behave like two coin flips. What counts would that give? - Add two of the percentages and compare with the third. - For question 4, you need a vessel where broth is shared but cells are not. ??? success "Resolution" **Linkage.** Independence predicts about 78 in each class. Instead the parental types make up 269 of 312 (86%): Lac and Vir usually travel together. They are *linked*. In 1947 Lederberg reported this kind of linkage among unselected traits, including lactose fermentation and phage resistance. **Order.** 5 + 9 = 14, and no other pairing works. Frequencies that add like distances lie on a line, with Gal between Lac and Vir. What passes between the cells is an ordered arrangement, not a bag of separate factors. **The rivals.** The control plates all but exclude back-mutation. Re-streaking one colony alone on a bare plate tests cross-feeding: if it grows by itself, it is a new kind of cell. The diffusible substance fell to Bernard Davis's 1950 U-tube: a sintered glass filter let broth through but no cell, and no recombinants appeared. The cells must touch. **What it turned out to be.** Transfer by contact, now called bacterial conjugation. William Hayes showed in 1953 that it runs one way, from a donor to a recipient. Γ‰lie Wollman, FranΓ§ois Jacob and Hayes then interrupted matings at timed intervals (1956) and saw markers arrive in sequence. The map eventually closed on itself: the *E. coli* chromosome is a circle. **A caveat.** The tidy sum was built into the made-up data; real recombination frequencies add only approximately. ## Sources - **Arthur T. Winfree**, *The Art of Scientific Discovery* (ECOL 479/579), course handout; the session-30 schedule line and its link markup β€” [Wayback Machine capture, 20 April 2002](https://web.archive.org/web/20020420212713/http://eebweb.arizona.edu/Faculty/Winfree/handout_479.htm){target=_blank} πŸ”“ - **Joshua Lederberg and Edward L. Tatum**, "Gene Recombination in Escherichia coli", *Nature* 158, 558 (1946) β€” [doi:10.1038/158558a0](https://doi.org/10.1038/158558a0){target=_blank} πŸ”’ - **Edward L. Tatum and Joshua Lederberg**, "Gene Recombination in the Bacterium Escherichia coli", *Journal of Bacteriology* 53(6), 673–684 (1947) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC518375/){target=_blank} πŸ”“ - **Joshua Lederberg**, "Gene Recombination and Linked Segregations in Escherichia Coli", *Genetics* 32(5), 505–525 (1947) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC1209393/){target=_blank} πŸ”“ - **Bernard D. Davis**, "Nonfiltrability of the Agents of Genetic Recombination in Escherichia coli", *Journal of Bacteriology* 60(4), 507–508 (1950) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC385908/){target=_blank} πŸ”“ - **William Hayes**, "Observations on a Transmissible Agent Determining Sexual Differentiation in Bacterium coli", *Journal of General Microbiology* 8(1), 72–88 (1953) β€” [doi:10.1099/00221287-8-1-72](https://doi.org/10.1099/00221287-8-1-72){target=_blank} πŸ”’ - **Γ‰lie L. Wollman, FranΓ§ois Jacob and William Hayes**, "Conjugation and Genetic Recombination in Escherichia coli K-12", *Cold Spring Harbor Symposia on Quantitative Biology* 21, 141–162 (1956) β€” [doi:10.1101/sqb.1956.021.01.012](https://doi.org/10.1101/sqb.1956.021.01.012){target=_blank} πŸ”’ - **Joshua Lederberg**, "Genetic Recombination in Escherichia coli: Disputation at Cold Spring Harbor, 1946–1996", *Genetics* 144(2), 439–443 (1996) β€” [PubMed Central](https://pmc.ncbi.nlm.nih.gov/articles/PMC1207540/){target=_blank} πŸ”“ - **E. Almquist**, "Investigations on Bacterial Hybrids", *Journal of Infectious Diseases* 35(4), 341–346 (1924) β€” [doi:10.1093/infdis/35.4.341](https://doi.org/10.1093/infdis/35.4.341){target=_blank} πŸ”’ (cited only for the phrase in its title) - **Arthur T. Winfree**, *The Art of Scientific Discovery*, original course syllabus β€” [PDF](https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf){target=_blank} πŸ”“ !!! note "How sure are we that this is Winfree's problem?" The syllabus gives only the name, the session and the reading. In Winfree's handout, the link on this item points to a bookmark named `Lac_Vir`; its target was never archived. Such bookmarks elsewhere name the subject behind a playful title ("Paired Observations" points to `Keplers_Laws`). Lac (lactose fermentation) and virus resistance were the traits scored in the early *E. coli* crosses, so the subject is fairly secure. That reading is the editors' inference, not Winfree's text. What is lost is the exercise itself: his wording, his data and its form. Identification: **probable**. - **Reconstructing the Lederberg–Tatum reasoning** (high): the reading used on this page. - **The same episode as a multiple-working-hypotheses exercise** (medium): compatible with the first. - **Later *E. coli* genetics** (not weighed): Lac as the lac operon and Vir as a virulent phage such as lambda vir. - **An invented "toy genetics" table** (low): does not explain why Lac and Vir are named. --- *Back to [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md) Β· [All problems](https://tyson-swetnam.github.io/aosd/problems/index.md) Β· [The schedule](https://tyson-swetnam.github.io/aosd/syllabus/index.md#section-5-inferences-hypotheses-explanations)* ---8<--- https://tyson-swetnam.github.io/aosd/gamesworth/ --- title: "The GamesWorth Method" description: "Professor Winfree's GamesWorth, taken from his syllabus: a daily uninterrupted hour of focused thought, the rules for the bound notebook and its left-hand pages, daily notebook swaps and emailed comparisons, and how the course was graded." type: Guide tags: [course, student-facing, gamesworth, daily-practice, notebook, focused-thinking, grading] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" - id: winfree-adventures-intro resource: "https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html" title: "Adventures in Discovery: About This Column (Society for Amateur Scientists E-Bulletin, 2001), Web Archive capture of 2003-01-14" author: "Arthur T. Winfree" --- # The GamesWorth Method Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *A daily hour of focused thought, written down so that you and others can learn from it* Every rule on this page comes from Professor Winfree's [syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md). ## What a GamesWorth is A **GamesWorth** (Winfree writes "gw") is one sitting of serious, focused thought: about as much thinking as one game of serious chess. The course problems give you something to think about. The daily habit is what Winfree hoped would last. !!! quote "Winfree, syllabus" Use these homework puzzles to nucleate a habit of doing a daily 'Game's-Worth' (henceforth, "gw") of focused thought, as in the first hand-out (Platt: *The Art of Creative Thinking*: the allusion is to how much thought it takes to play one game of serious chess). This might be the most important (potentially enduring) effect of the course. That first handout, John R. Platt's "The Art of Creative Thinking", is a chapter of his book *The Excitement of Science* (1962) and is not freely online (see the [reading list](https://tyson-swetnam.github.io/aosd/readings/index.md#platt-creative-thinking)). A blog post, [a "GamesWorth" of reasoning](https://www.datadeluge.com/2011/10/gamesworth-of-reasoning.html){target=_blank} πŸ”“, explains the idea; it is not Platt's text. Winfree practised what he taught. In the [introduction](https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html){target=_blank} πŸ”“ to his column *Adventures in Discovery*, he writes that its puzzles were ones he explored during 2001 in the spirit of Platt's essay, and that he had kept up Platt's daily Gamesworth since reading it. ## The daily hour The syllabus asks for a gw daily, "or at least 5 times a week, realistically", outside class, ideally at the same hour each day. It also admits that "hardly anyone actually does this, whatever good intentions they may have resolved on the first day". The hour must be unbroken, because an interruption costs far more than its own minutes: > Remember that in such efforts an hour's work punctuated with three 5-minute interruptions has lost you not 15 minutes, as might be the case were you shining shoes or digging a ditch: it has instead prevented you getting warmed up to 60-minute heat, replacing that experience by 3 times getting up to 15-minute heat. You can probably think of other experiences like this. Reserve that sacred daily uninterrupted hour to yourself (or self and partner, if you need to talk to think). Winfree himself worked alone, away from people, TV and music, in "a big open space that becomes littered with drawings and with verbal notes, usually in outline form, and with stacks of worked-out simple examples." But: "Not everyone works best this way. Some people think productively only in dialogue with a partner. Others only in a coffee shop. If you are that kind, do it that way." Try a few ways, find the best, then "log in and out at that place and time each time, many times in a row." ## The notebook ### Physical rules - **Bound, not loose-leaf.** A wire spiral is enough. No loose pages. - **Right-hand pages only** for your working. The left-hand pages have other uses (below). - **Numbered pages**, so you can write "go to page xxx" or "continued from page". - **Legible**, "like an industrial research notebook, as though to be notarized at intervals: not a collection of scraps", so you can resume after an interruption without rummaging through a jumble. - **Enough pages.** The semester has 15 weeks of 5 days, or 75 days, so you "will need at least 150 pages (times 2 because there is also a left side)". "Likely this will require two such notebooks." Readings go in a separate loose-leaf binder, with a page of your own thoughts on each. ### Logging in and out Winfree insists that you "formally log your brains in and out of each session". For each session, write: - the date; - the start and end times; - where you are working (the idea is to find the place that works best); - the numbers of the pages used. ### What to write Write the process: "Write down your approaches, your lucky insights, how you got into and out of blind alleys. This is the main thing, not the 'answers'." The syllabus calls mistakes "often the most available doors to discovery", so false starts belong on the page. Before you log out, "tidily summarize, as though for notarizing in an industrial research lab (but really so your classmates can read it and glean the same harvest as you did.)" The syllabus also recommends a checklist of good ideas from the readings: tick an idea when you try it and note the gw page, to see which ones help. Winfree adds that "so far as I am aware, no student has ever done this. I still recommend it." ## The left-hand pages "Left" means the back of each sheet, on the left when the book lies open. These pages have two uses. **1. In-class notes.** **2. Morning-afters.** After the first few weeks, go back to a problem you struggled with weeks earlier, once you have moved on to others (Winfree also calls this "Monday-morning quarterbacking"). Ask: - Where could I have taken a different approach? - How well did the methods I tried actually work? - What did classmates discover that I didn't, and why? - Why did I get stuck, or how did I avoid the snag that tripped up everyone else? "We are here not so much to solve the problems as to see how we didn't or see how some quirky habit of thought saved the day, and benefit from those recognitions." And: "A thoughtful thorough morning-after is a perfectly valid substitute for working a new problem in your gw notebook." ## Working with others **Daily swaps.** At each class meeting students swap gw notebooks to "share in their harvest of insights", which works only if your pages are clear. When you read a neighbour's book, remind them of the logging rules, and leave signed, helpful comments on their work. **Dual GamesWorths.** You may also work with a partner: "dual GamesWorths often work pretty well, like cutting wood with a long 2-handled band saw." Note your partner's name and contributions along with your log-in and log-out times. **The emailed comparison.** After each class, email Winfree "a couple lines, no more" on "why either your gw or the one you received in swap during the class meeting was the better of the two." Decide: "do not cop out by reporting a 'tie'". Swap in pairs, with at most one three-way swap on days with an odd number of students, so he can see whether both partners agree on whose work was better. ## How it was graded In this course "actually solving the practice problems is way less important than learning how to try (and, for grading, demonstrating real effort.)" - **Blue books.** Each class opened with a warm-up quiz in a blue exam book, collected at the end of class; it doubled as the attendance record. Late arrivals still logged into the blue book, noting the time. - **The gw books** were collected at the end of term ("On the last class day", says the text; the schedule lists it at session 29, the next-to-last meeting) and skimmed to check them against the swap emails, with "particular attention to stuff on the left-hand sides": morning-afters and comments on classmates' work. - **The final exam**, in exam week, was "a collection of problems from which you will choose some, exhibit as many distinct approaches as you can, and as many cross-checking distinct solutions as you can." Winfree leaned toward a take-home. The grade came from a spreadsheet formula combining six inputs: 1. Lively participation every session (absence only in dire emergency, with advance notice by email). 2. Performance on the daily blue-book warm-ups. 3. The daily emails comparing two gw's. 4. His reading of your gw book, stressing morning-afters, and of your comments in others' books. 5. Reading each assignment no later than its class meeting. 6. Extra work for Honors and graduate students. A daily uninterrupted hour plus vigorous participation "will likely develop an A"; because most students let the habit slip, "about half the class ends up with a B". ## Common difficulties **"I'm too busy this week."** If you set the course aside "just temporarily", the syllabus warns, "you will find yourself backlogged and despairing", and catching up in a rush fails: "Hurrying under pressure is guaranteed to abort the special opportunities you confront in this course." Protect the hour first. **"I'm stuck," or "I have nothing to work on."** Write down where and why you are stuck: a recorded blind alley is material for a later morning-after. Or do a morning-after now. **"My work isn't good enough to swap."** Swap it anyway. The grade rests on "demonstrating real effort", not on answers. Write clearly so your partner can follow what you tried. --- *"The purpose of making this a formal 'course' is to provide you a legitimate (regularly scheduled, graded) escape from the usual pressures, during which to consciously cultivate skills and personal style in problem solving."* ---8<--- https://tyson-swetnam.github.io/aosd/readings/ --- title: "Recommended Readings" description: "Annotated reading list for The Art of Scientific Discovery: a session-by-session table of what Winfree's syllabus assigns, entries for the three course books, every short handout and the books he put on reserve, then a bibliography of Winfree's own work, each link marked open, borrowable or paywalled." type: Resource List tags: [course, student-facing, readings, bibliography, open-access] status: stable generated: by: "claude/opus-5" at: "2026-09-16T00:00:00Z" sources: - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" --- # Recommended Readings Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *Every reading the syllabus assigns, in the order the course meets it* !!! note "About the links" πŸ”“ = open access (free full text) Β· πŸ”“ *(borrow)* = free to read via Internet Archive controlled lending (a free account is needed; some copies offer only a one-hour browse) Β· πŸ”’ = paywalled (subscription or purchase). Where a free copy exists alongside the publisher version, the open link is given first and the canonical DOI noted in the annotation. Readings are the first of the course's three tools. The syllabus names two books you buy, Ehrlich's *Nine Crazy Ideas in Science* and Adams's *Conceptual Blockbusting*, "And a lot of xeroxed handouts, not listed here, many from current periodicals." Each reading is due on the day of its class meeting; the [original syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md) has the full schedule. ## How to read for this course The syllabus asks for more than reading: - **Mark it up.** Comment on each reading, "at least by underlining and making marginal notations." - **Write a page.** "I expect you to commit to paper a page per class session to celebrate the best things you find or think of while involved with these readings." - **Keep a checklist.** Save a list of good ideas from the readings, or from your reactions to them, and use it "to joggle your brains during every problem-solving session." Put a check mark beside an idea when you try it, with the page number in your GamesWorth book, and "See which ones help you." - **File it.** Keep the readings and your comments on them in a separate loose-leaf binder, not tucked into your notebook. - **Read on time.** Readings are due "no later than the corresponding class meeting date"; apart from Adams and Ehrlich, Winfree handed them out a week ahead. On the checklist Winfree added: "By the way, so far as I am aware, no student has ever done this. I still recommend it." ## Readings by session Built from the schedule in the [syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md). Each reading links to its entry below. Sessions with no reading listed are left blank, as in the syllabus. | Session | Section | Reading | | :-- | :-- | :-- | | 01 | [1](https://tyson-swetnam.github.io/aosd/section1/index.md) | Introduction and handouts; [Adams](#adams), preface and Chapter 1 | | 02 | [1](https://tyson-swetnam.github.io/aosd/section1/index.md) | [Adams](#adams), preface and Chapter 1
[Platt, *The Art of Creative Thinking*](#platt-creative-thinking)
[Feynman, *Cargo Cult Science*](#feynman-cargo-cult-science) | | 03 | [1](https://tyson-swetnam.github.io/aosd/section1/index.md) | [Judson](#judson), Chapter 1: *The Rage to Know* | | 04 | [1](https://tyson-swetnam.github.io/aosd/section1/index.md) | | | 05 | [1](https://tyson-swetnam.github.io/aosd/section1/index.md) | [The N-rays affair](#n-rays)
[Langmuir, *Pathological Science*](#langmuir) | | 06 | [1](https://tyson-swetnam.github.io/aosd/section1/index.md) | [Packet of readings: valuing mistakes](#valuing-mistakes)
[Ehrlich](#ehrlich), Chapter 1 (Introduction) | | 07 | [2](https://tyson-swetnam.github.io/aosd/section2/index.md) | [Adams](#adams), Chapter 2: Perceptual blocks | | 08 | [2](https://tyson-swetnam.github.io/aosd/section2/index.md) | [Calandra, *The Barometer Story*](#barometer-story)
[Adams](#adams), Chapter 3: Emotional blocks | | 09 | [2](https://tyson-swetnam.github.io/aosd/section2/index.md) | [Adams](#adams), Chapter 4: Cultural blocks
[Platt, *Diversity*](#platt-diversity) | | 10 | [2](https://tyson-swetnam.github.io/aosd/section2/index.md) | [Adams](#adams), Chapter 5: Intellectual blocks | | 11 | [2](https://tyson-swetnam.github.io/aosd/section2/index.md) | [Adams](#adams), Chapter 7: Blockbusters | | 12 | [2](https://tyson-swetnam.github.io/aosd/section2/index.md) | [Dyson, *Unfashionable Pursuits*](#dyson)
[Capecchi one-page biography](#capecchi)
[Narlikar on venture funding](#narlikar) | | 13 | [3](https://tyson-swetnam.github.io/aosd/section3/index.md) | | | 14 | [3](https://tyson-swetnam.github.io/aosd/section3/index.md) | [Judson](#judson), Chapter 4: *Chance*
[Anderson on research strategy](#anderson) | | 15 | [3](https://tyson-swetnam.github.io/aosd/section3/index.md) | [Judson](#judson), Chapter 8: *Evidence* | | 16 | [3](https://tyson-swetnam.github.io/aosd/section3/index.md) | [Ehrlich](#ehrlich), Chapter 2 | | 17 | [3](https://tyson-swetnam.github.io/aosd/section3/index.md) | [Ehrlich](#ehrlich), Chapter 3 | | 18 | [3](https://tyson-swetnam.github.io/aosd/section3/index.md) | [Ehrlich](#ehrlich), Chapter 4 | | 19 | [4](https://tyson-swetnam.github.io/aosd/section4/index.md) | [Judson](#judson), Chapter 2: *Pattern* | | 20 | [4](https://tyson-swetnam.github.io/aosd/section4/index.md) | [Ehrlich](#ehrlich), Chapter 5 | | 21 | [4](https://tyson-swetnam.github.io/aosd/section4/index.md) | [Ehrlich](#ehrlich), Chapter 6 | | 22 | [4](https://tyson-swetnam.github.io/aosd/section4/index.md) | [Adams](#adams), Chapter 6: Alternative thinking languages | | 23 | [4](https://tyson-swetnam.github.io/aosd/section4/index.md) | [Ehrlich](#ehrlich), Chapter 7 | | 24 | [4](https://tyson-swetnam.github.io/aosd/section4/index.md) | [Ehrlich](#ehrlich), Chapter 10 | | 25 | [5](https://tyson-swetnam.github.io/aosd/section5/index.md) | [Chamberlin, *The Method of Multiple Working Hypotheses*](#chamberlin) | | 26 | [5](https://tyson-swetnam.github.io/aosd/section5/index.md) | TBA | | 27 | [5](https://tyson-swetnam.github.io/aosd/section5/index.md) | [Platt, *Strong Inference*](#platt-strong-inference) | | 28 | [5](https://tyson-swetnam.github.io/aosd/section5/index.md) | [Judson](#judson), Chapter 7: *Strong Predictions* | | 29 | [5](https://tyson-swetnam.github.io/aosd/section5/index.md) | [Feynman, *The Character of Physical Law*](#feynman-character-of-physical-law) | | 30 | [5](https://tyson-swetnam.github.io/aosd/section5/index.md) | [Judson](#judson), Chapter 9: *Theory* | Beveridge, PΓ³lya, Gleick and Koestler are not assigned to any session; they were on library reserve (see [Books on reserve](#books-on-reserve)). ## The course books The syllabus says "Two required books await you in the local bookstores": Adams and Ehrlich. A third, Judson, was out of print, so Winfree provided copies. ### James L. Adams, *Conceptual Blockbusting* { #adams } [*Conceptual Blockbusting: A Guide to Better Ideas*, 5th edition](https://www.hachettebookgroup.com/titles/james-l-adams/conceptual-blockbusting/9781541674059/?lens=basic-books){target=_blank} πŸ”’ *Basic Books, 2019.* Adams, a consulting engineer in Winfree's description, sorts the things that stop good ideas into kinds of block and then offers ways around them. Section 2 of the course is built on his chapters. The syllabus assigns them in this order, using its own chapter labels (the 2001 course used an earlier edition, so chapter numbers in the current edition may differ): - Preface and Chapter 1: sessions 01 and 02 - Chapter 2, Perceptual blocks: session 07 - Chapter 3, Emotional blocks: session 08 - Chapter 4, Cultural blocks: session 09 - Chapter 5, Intellectual blocks: session 10 - Chapter 7, Blockbusters: session 11 - Chapter 6, Alternative thinking languages: session 22 No legitimate open-access edition exists; libraries hold the earlier editions. ### Robert Ehrlich, *Nine Crazy Ideas in Science* { #ehrlich } [*Nine Crazy Ideas in Science: A Few Might Even Be True*](https://press.princeton.edu/books/paperback/9780691094953/nine-crazy-ideas-in-science){target=_blank} πŸ”’ *Princeton University Press, 2001.* A physicist takes a series of unpopular claims and asks how far each survives the evidence. The first chapter is an introduction on how to judge a crazy idea; each later chapter takes up one idea. It gives the course a steady supply of claims to weigh in Sections 3 and 4. The syllabus assigns: - Chapter 1 (Introduction): session 06 - Chapter 2: session 16 - Chapter 3: session 17 - Chapter 4: session 18 - Chapter 5: session 20 - Chapter 6: session 21 - Chapter 7: session 23 - Chapter 10: session 24 ### Horace Freeland Judson, *The Search for Solutions* { #judson } [*The Search for Solutions*](https://archive.org/details/searchforsolutio00juds){target=_blank} πŸ”“ *(borrow)*. *Holt, Rinehart and Winston, 1980.* The Internet Archive copy is the Johns Hopkins University Press edition of 1987. {: #additional-core-reading } A historian's account of how scientists actually find things out, told through short case studies. The syllabus notes it was out of print, so Winfree provided copies and put it on Main Library Reserve. The chapters assigned, with their titles as the syllabus gives them: - Chapter 1, *The Rage to Know*: session 03 - Chapter 2, *Pattern*: session 19 - Chapter 4, *Chance*: session 14 - Chapter 7, *Strong Predictions*: session 28 - Chapter 8, *Evidence*: session 15 - Chapter 9, *Theory*: session 30 ## The handouts The short readings, in the order the schedule assigns them. The syllabus names most of them in a few words; where it names only an author and a topic, the entry says which piece the editors take it to be. {: #on-scientific-method-and-discovery } ### Platt, *The Art of Creative Thinking* { #platt-creative-thinking } John R. Platt, "The Art of Creative Thinking," a chapter of [*The Excitement of Science*](https://archive.org/details/excitementofscie0000john){target=_blank} πŸ”“ *(borrow)*. *Houghton Mifflin, 1962.* Session 02. The first handout of the course and the source of its central habit. The syllabus explains the name of the daily notebook: "Platt: *The Art of Creative Thinking*: the allusion is to how much thought it takes to play one game of serious chess." The introduction to Winfree's column [Adventures in Discovery](#writing-for-amateurs) names the same essay as the source of his own daily Gamesworth habit. See the [GamesWorth page](https://tyson-swetnam.github.io/aosd/gamesworth/index.md) for how the course uses it. A third-party explainer: [a "GamesWorth" of reasoning](https://www.datadeluge.com/2011/10/gamesworth-of-reasoning.html){target=_blank} πŸ”“. ### Feynman, *Cargo Cult Science* { #feynman-cargo-cult-science } Richard P. Feynman, [*Cargo Cult Science*](https://resolver.caltech.edu/CaltechES:37.7.CargoCult){target=_blank} πŸ”“ Caltech commencement address, 1974; *Engineering and Science* 37(7), 10–13 (1974). Session 02. Feynman on research that has the outward form of science without its honesty, and on the extra care it takes not to fool yourself. It sets up Section 1's theme of detecting nonsense. ### The N-rays affair { #n-rays } [N-ray: overview and guide to the primary sources](https://en.wikipedia.org/wiki/N-ray){target=_blank} πŸ”“ Session 05. In 1903 RenΓ© Blondlot reported a new radiation that other laboratories then "saw" too; R. W. Wood's visit, in which he secretly removed a key part of the apparatus, showed the effect was imaginary. Primary sources: R. W. Wood, "The n-Rays," *Nature* 70, 530–531 (1904), [doi:10.1038/070530a0](https://doi.org/10.1038/070530a0){target=_blank} πŸ”’; I. M. Klotz, "The N-Ray Affair," *Scientific American* 242(5), 168–175 (1980), [doi:10.1038/scientificamerican0580-168](https://doi.org/10.1038/scientificamerican0580-168){target=_blank} πŸ”’. The syllabus names only "N-Rays", so which text Winfree handed out is not recorded. The case is also a course problem: [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md). ### Langmuir, *Pathological Science* { #langmuir } Irving Langmuir, [*Pathological Science*](https://www.cs.princeton.edu/~ken/Langmuir/langmuir.htm){target=_blank} πŸ”“ Colloquium at General Electric, 18 December 1953, transcribed by R. N. Hall; published in *Physics Today* 42(10), 36–48 (1989), [doi:10.1063/1.881205](https://doi.org/10.1063/1.881205){target=_blank} πŸ”’. Session 05. Langmuir's case studies of capable scientists who fooled themselves, N-rays among them, and his list of symptoms by which such work can be recognized. ### Packet of readings: valuing mistakes { #valuing-mistakes } Session 06. The syllabus lists only "Packet of readings: valuing mistakes"; its contents were not recorded, and no copy is known. The syllabus lists among the course's aims "learning a positive attitude toward mistakes, because they are often the most available doors to discovery." ### Calandra, *The Barometer Story* { #barometer-story } Alexander Calandra, [*The Barometer Story* ("Angels on the Head of a Pin")](https://www.stephenhicks.org/2022/07/10/angels-on-the-head-of-a-pin-by-alexander-calandra/){target=_blank} πŸ”“ *Saturday Review*, 21 December 1968. Session 08. A student asked to find a building's height with a barometer gives every answer except the one the examiner wants. Read with Adams on emotional blocks; the course treats it as a problem too: [The Barometer Story](https://tyson-swetnam.github.io/aosd/problems/barometer-story/index.md). ### Platt, *Diversity* { #platt-diversity } John R. Platt, [*Diversity*](https://doi.org/10.1126/science.154.3753.1132){target=_blank} πŸ”’ *Science* 154, 1132–1139 (1966). Session 09. On the value of many different approaches and styles in science. Assigned alongside Adams on cultural blocks. ### Dyson, *Unfashionable Pursuits* { #dyson } Freeman Dyson, [*Unfashionable Pursuits*](https://doi.org/10.1007/BF03026573){target=_blank} πŸ”’ *The Mathematical Intelligencer* 5(3), 47–54 (1983); collected in *From Eros to Gaia* (1992). Session 12. A case for working on problems that are out of fashion. ### Capecchi, one-page biography { #capecchi } Session 12. A one-page biography of Mario Capecchi, handed out in class. The syllabus says only "Capecchi one-page biography"; the piece itself has not been identified, so no link is given. ### Narlikar, *Venture Funding for New Ideas* { #narlikar } Jayant V. Narlikar, [*Venture funding for new ideas*](https://doi.org/10.1038/35008158){target=_blank} πŸ”’ *Nature* 404, 707 (2000). Session 12. A short argument that the way research is funded shapes which new ideas get a hearing. The syllabus assigns it with Dyson and the Capecchi biography in the last session of Section 2. ### Anderson, *More Is Different* { #anderson } Philip W. Anderson, [*More Is Different*](https://www.rpgroup.caltech.edu/embl_pboc_2023/assets/pdfs/anderson1972.pdf){target=_blank} πŸ”“ *Science* 177, 393–396 (1972). Canonical: [doi:10.1126/science.177.4047.393](https://doi.org/10.1126/science.177.4047.393){target=_blank} πŸ”’. Session 14. The syllabus says only "Anderson on research strategy"; this essay is the editors' identification, not Winfree's. Anderson argues that each level of complexity needs its own concepts and cannot simply be derived from the level below. ### Chamberlin, *The Method of Multiple Working Hypotheses* { #chamberlin } T. C. Chamberlin, [*The Method of Multiple Working Hypotheses*](https://www.whoi.edu/cms/files/chamberlin65sci_72744.pdf){target=_blank} πŸ”“ *Science* 148, 754–759 (1965), a reprint of the paper first published in *Science* ns-15, 92–96 (1890). DOIs: [1890 original](https://doi.org/10.1126/science.ns-15.366.92){target=_blank} πŸ”’, [1965 reprint](https://doi.org/10.1126/science.148.3671.754){target=_blank} πŸ”’. Session 25. {: #on-multiple-hypotheses-and-objectivity } The classic warning against falling in love with one explanation: keep several in play at once so that no single favourite bends what you see. It opens Section 5. ### Platt, *Strong Inference* { #platt-strong-inference } John R. Platt, [*Strong Inference*](https://www.whoi.edu/cms/files/platt64sci_72743.pdf){target=_blank} πŸ”“ *Science* 146, 347–353 (1964). Canonical: [doi:10.1126/science.146.3642.347](https://doi.org/10.1126/science.146.3642.347){target=_blank} πŸ”’. Session 27. Platt builds on Chamberlin: list the alternative hypotheses, design the experiment whose outcome rules some of them out, run it, and repeat. ### Feynman, *The Character of Physical Law* { #feynman-character-of-physical-law } Richard P. Feynman, *The Character of Physical Law* (first published 1965; MIT Press edition: [publisher](https://mitpress.mit.edu/9780262060165/the-character-of-physical-law/){target=_blank} πŸ”’). The 1964 Cornell Messenger Lectures on which the book is based are [free to watch on the Internet Archive](https://archive.org/details/the-messenger-lectures){target=_blank} πŸ”“. Session 29. Feynman on what physical laws are like and how they are found. The syllabus names the book without a chapter, so which part was handed out is not known. ## Books on reserve { #books-on-reserve } Not assigned to any session. Winfree put these on library reserve "for your attention if you consider it important to acquire better skills in problem solving." ### Beveridge, *The Art of Scientific Investigation* { #beveridge } W. I. B. Beveridge, [*The Art of Scientific Investigation*](https://archive.org/details/artofscientifici00beve){target=_blank} πŸ”“ *W. W. Norton, 1957 (first published 1950).* Free full text on the Internet Archive (no known copyright restrictions, as determined by the scanning institution). A British physician's study of how discoveries are made: chance, intuition, hypothesis, and the habits of productive scientists. Winfree dropped it from the requirements "because I find that no one reads it anyhow, I think because archaic attitudes offend," but added: "I consider it the best of the lot." ### PΓ³lya, *Mathematics and Plausible Reasoning* { #polya } George PΓ³lya, *Mathematics and Plausible Reasoning*, Princeton University Press, 1954, in two volumes: - Vol. I, [*Induction and Analogy in Mathematics*](https://archive.org/details/inductionanalogy00pl){target=_blank} πŸ”“ *(borrow)* - Vol. II, [*Patterns of Plausible Inference*](https://archive.org/details/patternsofplausi0000gpol){target=_blank} πŸ”“ *(borrow)* In the syllabus's words, "using elementary math as example material for general principles": how to guess well, generalize, specialize and test before you can prove. ### Gleick, *Genius* { #gleick } James Gleick, [*Genius: The Life and Science of Richard Feynman*](https://www.penguinrandomhouse.com/books/60762/genius-by-james-gleick/){target=_blank} πŸ”’ *Pantheon, 1992.* The syllabus describes it as a "biography of Richard Feynman, a great problem solver." ### Koestler, *The Watershed* { #koestler } Arthur Koestler, [*The Watershed: A Biography of Johannes Kepler*](https://archive.org/details/watershed0000unse){target=_blank} πŸ”“ *(borrow)*. *Doubleday Anchor, 1960.* Kepler's long struggle toward his laws of planetary motion, false starts included. As the syllabus notes, it is a chapter of Koestler's *The Sleepwalkers*. ## Professor Arthur T. Winfree's own work Not course readings: a short guide to the scientist who designed the course. **J. J. Tyson & L. Glass**, [*Arthur T. Winfree (1942–2002)*](https://doi.org/10.1016/j.jtbi.2004.04.042){target=_blank} πŸ”’ *Journal of Theoretical Biology* 230, 433–439 (2004). A tribute surveying his life and science. See also the open [Wikipedia biography](https://en.wikipedia.org/wiki/Arthur_Winfree){target=_blank} πŸ”“. ### Books - [*The Geometry of Biological Time*](https://link.springer.com/book/10.1007/978-1-4757-3484-3){target=_blank} πŸ”’ (Springer, 1980; 2nd ed. 2001, ISBN 0-387-98992-7). His major work on biological rhythms: circadian clocks, heart rhythms and other timing phenomena treated as the geometry of oscillators. - [*When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias*](https://archive.org/details/whentimebreaksdo0000winf){target=_blank} πŸ”“ *(borrow)* (Princeton University Press, 1987). Spiral waves in excitable media and what they mean for disorders of the heartbeat. The Internet Archive copy currently lends as a one-hour browse. - *The Timing of Biological Clocks* (Scientific American Library / W. H. Freeman, 1987). A treatment of biological rhythms for general readers. The [Internet Archive copy](https://archive.org/details/timingofbiologic00winf){target=_blank} πŸ”’ is print-disabled access only, so most readers will need a library copy. ### Papers - [*Biological Rhythms and the Behavior of Populations of Coupled Oscillators*](https://doi.org/10.1016/0022-5193(67)90051-3){target=_blank} πŸ”’ *Journal of Theoretical Biology* 16, 15–42 (1967). His foundational paper on how many oscillators fall into step. - [*Spiral Waves of Chemical Activity*](https://doi.org/10.1126/science.175.4022.634){target=_blank} πŸ”’ *Science* 175, 634–636 (1972). Spiral waves in the Belousov-Zhabotinsky reaction, the reaction most likely used in the course's [Chemical Pattern-Formation Lab](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md). - [*The Prehistory of the Belousov-Zhabotinsky Oscillator*](https://www.dna.caltech.edu/Papers/prehistory1984.pdf){target=_blank} πŸ”“ *Journal of Chemical Education* 61, 661–663 (1984). Canonical: [doi:10.1021/ed061p661](https://doi.org/10.1021/ed061p661){target=_blank} πŸ”’. How the reaction was discovered and long disbelieved. - [*Electrical Turbulence in Three-Dimensional Heart Muscle*](https://doi.org/10.1126/science.7973648){target=_blank} πŸ”’ *Science* 266, 1003–1006 (1994). His wave theory applied to life-threatening arrhythmias. ### Writing for amateurs - [Adventures in Discovery](https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html){target=_blank} πŸ”“ (Society for Amateur Scientists, 2001–2002; Web Archive capture). Winfree's column of puzzles he worked through as daily GamesWorths, including the one behind the course's [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md) problem. ### Background - [*Phase response curve*](http://www.scholarpedia.org/article/Phase_response_curve){target=_blank} πŸ”“ *Scholarpedia.* An open, peer-reviewed introduction to phase response curves, one of the tools Winfree used to study how rhythms respond to a stimulus. --- *The goal is not to read about problem solving, but to turn what you read into habits through daily practice and honest reflection.* ---8<--- https://tyson-swetnam.github.io/aosd/syllabus/ --- title: "The Original Syllabus" description: "Professor Winfree's syllabus for The Art of Scientific Discovery, transcribed in full from the course handout: who the course is for, the three tools, the GamesWorth notebook, grading, and the thirty-session schedule of readings and problems." type: Reference tags: [course, student-facing, syllabus, schedule, gamesworth, grading, primary-source] status: stable generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: aosd-syllabus resource: "https://github.com/tyson-swetnam/aosd/blob/main/docs/assets/aosd_syllabus.pdf" title: "The Art of Scientific Discovery: original course syllabus (PDF)" author: "Arthur T. Winfree" - id: winfree-handout resource: "https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm" title: "The Art of Scientific Discovery (EEB 479): course handout, Web Archive capture of 2007-02-14" author: "Arthur T. Winfree" --- # The Original Syllabus Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *Professor Winfree's course handout, transcribed in full* !!! info "About this transcription" This page reproduces the text of Professor Winfree's syllabus, the [original PDF](https://tyson-swetnam.github.io/aosd/assets/aosd_syllabus.pdf) that every other page on this site derives from. The handout was written for the Fall 2001 offering of EEB 479/479H/579 at the University of Arizona and, as its last section says, the schedule is the Spring 2001 course with the dates moved forward. Only headings and paragraph breaks have been added; the wording, including its idiosyncrasies, is Winfree's own. Problem names in the schedule link to their pages in the [problem index](https://tyson-swetnam.github.io/aosd/problems/index.md). **The Art of Scientific Discovery.** 3 credits. Tuesdays and Thursdays in BSW 510 from 14:00 to 15:15. A.T. Winfree, 326 or 372 BSW. > You should read this over every month during the course. It also appears at my lab pages. Most students don't take it seriously at first. You may need a reminder, or several. ## Who Undergrads including freshmen, grad students, and postdocs have enjoyed this course and claimed to benefit from their vigorous engagement in it. Younger students typically do better, perhaps due to their briefer exposure to the educational system. See comments at the lab web pages. ## What The aim of this course is to develop your ability of solve problems like those encountered in scientific investigations. We examine how specific problems were solved in the past, examine 'inspirational readings', and tackle selected puzzles for pencil and paper and for simple lab manipulation. These exercises are intended as 'practice scrimmages' in strategy and tactics of recognizing ignorance, of posing questions, of cultivating multiple alternative solutions, of eliminating rejectable candidate solutions, of spotting and taking advantage of your own mistakes, and especially of learning a positive attitude toward mistakes, because they are often the most available doors to discovery. These exercises depend as little as possible on knowledge of any particular subject area. That way everyone is on an equal footing of unfamiliarity and no one is likely to be deprived of exercise by already 'knowing' 'the' solution. The aim is to experience a feeling of disorientation and hopeless lost-ness, so you will learn not to despair in paralysis but instead focus on method, generate several alternative guesses, and test them for workability. Readings and examples are drawn from all the sciences, often emphasizing biology. The discovery exercises are mostly made from elementary mathematics so as to require no lab setup and so as to be comprehensible to students from diverse backgrounds. There are some hands-on lab-type exercises too. All are contrived much as the organizers of an Easter Egg Hunt do in the hour before little kids arrive with their baskets: surprisingly many discoveries are rigged into these exercises for your gratification if you will but learn how to discover stuff. This is not a lecture course. The objective is not to add to your store of useful facts. Nor will you be passively stuffed by the professor with sophistication and accordingly accredited. As in weight-lifting, you will benefit from this course in proportion to your investment of time and effort. Interactions with your peers in this gym are contrived to leave you with muscles and installed habits of exercise that will last a long time. Remember the weight-lifting metaphor. The purpose of making this a formal 'course' is to provide you a legitimate (regularly scheduled, graded) escape from the usual pressures, during which to consciously cultivate skills and personal style in problem solving. In this course actually solving the practice problems is way less important than learning how to try (and, for grading, demonstrating real effort.) Some people suppose problem-solving cannot be taught. They suppose that you are born with innate ability or not, and that's all there is to it. In contrast, I think we are all born with it and mostly lose it during and because of schooling and the general intimidation that comes with any competitive society. We can refine and enhance it about as much as we please. One example: according to the Proceedings of US National Academy of Sciences in spring 2000, London taxi drivers develop bigger hippocampus (the part of the brain involved in navigation) the longer they work at that job. Thinking styles differ immensely between individuals. Evidence: widespread disagreement about almost everything. Given persistent diversity of styles, it seems likely that some styles are better than others for different purposes or for different individuals. If so then by getting acquainted with alternative styles and by exercising some 'natural selection' between alternative styles during diverse exercises, you can prove to yourself that thinking is an improvable skill. In 'real life' we are so intent on getting immediate solutions to urgent hurtful problems that we seldom feel the leisure required to examine how we get them and how we fail to. While swimming with a goal in sight and a clock running, you don't feel free to experiment, for example, with breathing on the other side of your stroke. In this course you are free and even compelled to. There will be much choking and sputtering, but it's OK here because not much depends on finishing first within this one semester. We are trying to improve your whole future life. **Your Professor:** A.T. Winfree. See web site for supplementary info on the course and for the prof's resume. ## The three main tools We have three main tools in this course: ### 1) Readings Ehrlich (physicist), *Nine Crazy Ideas in Science: A Few Might Even Be True*. Adams (consulting engineer), *Conceptual Blockbusting*. And a lot of xeroxed handouts, not listed here, many from current periodicals. You will read these and comment on them, at least by underlining and making marginal notations. I expect you to commit to paper a page per class session to celebrate the best things you find or think of while involved with these readings. If you will save a checklist of good ideas gleaned while reading (from the readings or from your own thoughts in reaction the readings) you can use this checklist to joggle your brains during every problem-solving session. It would be a good idea to put a check mark beside each such item when practiced, together with page number in your GamesWorth book (see below). See which ones help you. By the way, so far as I am aware, no student has ever done this. I still recommend it. I am not going to play Big Brother in a effort to make sure you do this and other useful exercises. Your life is up to you. ### 2) Problems Problems, the equivalent of bar-bells for lifting, for homework and for in-class collaboration to allow you to practice intellectual gimmicks suggested by the readings. Write down your approaches, your lucky insights, how you got into and out of blind alleys. This is the main thing, not the 'answers'. Keep a diary in your GamesWorth book (see below) to focus your mind on strategy and tactics, not just on the ostensible bottom line (which is not the bottom line in this peculiar course). The purpose of the puzzles (many of them silly) is to slow you down for a few minutes so you can examine the working of your own mind. It is hard to develop consciousness of how you do it, but awareness is the first step to correction and improvement of any skill. Contrary to widespread fiction, and unlike watching your feet while dancing, it will not hobble you. Ask any gymnast about such matters. Use these homework puzzles to nucleate a habit of doing a daily 'Game's-Worth' (henceforth, "gw") of focused thought, as in the first hand-out (Platt: *The Art of Creative Thinking*: the allusion is to how much thought it takes to play one game of serious chess). This might be the most important (potentially enduring) effect of the course. I will examine your daily (or at least 5 times a week, realistically) GamesWorth notebook at the end of semester. I do insist that you formally log your brains in and out of each session by numbering the pages used (so you can refer to them from elsewhere) and noting the date and start and end times. Also record where you are working: the idea is to find the place that works best. You are to write only on the right side pages of a bound (not-looseleaf) notebook. When you peruse your neighbor's gw, remind him/her to do all this. You will find that all this is not just busywork. At the end of each session, before you log out, you are to tidily summarize, as though for notarizing in an industrial research lab (but really so your classmates can read it and glean the same harvest as you did.) There being 15 weeks Γ— 5 = 75 days in the semester, you will need at least 150 pages (times 2 because there is also a left side) for thinking and summarizing. Likely this will require two such notebooks. What are the left sides for? (meaning, the backside of each sheet of paper, on the left when the gw book is open before you, not meaning the left half of each side of paper). For two things: a) In-class notes. b) Re-considering your prior thinking. This Monday-morning quarterbacking or morning-after reconsideration is to be done much later, after you have disengaged from the details and engaged other problems. Then you can look back and sense where you could have taken a different approach, see how well the methods you are trying did in fact work out for you, see what alternative discoveries you might have made (e.g., the ones your classmates did) and why you didn't. After the first few weeks I expect you to discipline yourself from time to time to do such a "morning-after" re-examination of some problem you struggled with weeks earlier. See how your perspective has changed. See how differently you would tackle it now. See why you got stuck before or why you went straight to a nifty solution without tripping over the obstacles everyone else did. We are here not so much to solve the problems as to see how we didn't or see how some quirky habit of thought saved the day, and benefit from those recognitions. This requires morning-afters. A thoughtful thorough morning-after is a perfectly valid substitute for working a new problem in your gw notebook. This content might become its principal value. My own idea of a gw goes along with feelings of quiet, freedom from distraction, etc., thus necessarily away from people, TV, and music (which completely takes over my cerebral processors.) I need a big open space that becomes littered with drawings and with verbal notes, usually in outline form, and with stacks of worked-out simple examples. Not everyone works best this way. Some people think productively only in dialogue with a partner. Others only in a coffee shop. If you are that kind, do it that way. An important part of the semester's experience is to try a few different ways, find a best one for you, and stick to it habitually: log in and out at that place and time each time, many times in a row. Your gw book is to be bound (at least by a wire helix) and even though it contains all your "scrap" "preliminary" work, should be legible like an industrial research notebook, as though to be notarized at intervals: not a collection of scraps. The idea is to make it possible to resume work after an interruption, without having to start all over by rummaging a confused jumble. No loose pages. Don't forget to number the right-hand pages so you can say "... go to page xxx", "..continued from page...", etc. Succinctly rewrite the essentials of your work before finishing for the day. Keep the readings (plus whatever other pertinent stuff you might encounter elsewhere during the semester) in a separate loose-leaf binder (not interleaved like a bookmark ready to fall out when you open your notebook, and not in a bulging paper pocket stuffed with disorderly sheets), together with your extractions from them or comments on them: As noted under resource (1) above, I expect to see a page of cogent thought, besides marginal scribbles and under-linings, on each. So: a bound notebook, and a loose-leaf binder for hole-punched handouts. ### 3) Each other The genetic diversity and diversity of life-experiences and habits of thought that we collectively bring to the table. There will be daily swapping of gw's so that you can benefit from your peers' perspectives and share in their harvest of insights (supposing you all discipline yourselves to make these fruits plainly accessible). This will also keep you on your toes, unless you enjoy being not-understood or being the one to come empty-handed. You have plenty of opportunity ahead of you in daily gws at home for practicing solo thinking skills. But don't neglect to also work together if you want to: dual GamesWorths often work pretty well, like cutting wood with a long 2-handled band saw. Note your partner's name and contributions, along with your log-in and log-out times. During class we also practice aggregate thinking skills, i.e., braving social opprobrium by blurting out nutty ideas, and risking devastating counter-attack by publicly objecting to nonsense blurted by others. Even though first notions seldom seem presentable, they are essential seeds to catalyze the next refinement. You must learn to do all parts of the process, not only by echoing within the confines of one skull, but also in the public forum. Class meetings will also prove essential for some problems in which no one individual can collect enough data, but if we pool data, reality will come into focus. ## Grades Such a course should not be grade-oriented, but the institution is grade-oriented. While under 'pressure' of grades in other courses you might think you have to give this training short shrift 'just temporarily,' but then you will find yourself backlogged and despairing. Same as in athletics: it is vital to keep the daily habit. Worse, if you don't make use of your opportunities in this course and bring contributions of thought to each class meeting, you dampen the spirit of the enterprise. Who wants to work hard at crystallizing a clear insight to share, then swap gw notebooks in class, and receive a muddle in exchange? I accordingly provide countervailing pressure in the form of a daily quiz in a blue exam book, besides paying attention to your contributions in class and by email after class. This also provides me an attendance record. (If you arrive too late for the warm-up quiz, log into the blue book anyhow, noting the time.) I will collect the blue books daily at the end of class. I also want an email from you after each class session telling me in a couple lines, no more, why either your gw or the one you received in swap during the class meeting was the better of the two. The aim is that you should pay attention to alternative ways of thinking, evaluate them daily, and use some of them. And having others evaluate yours daily gives a little extra motivation to have something presentable in clear order for this show-and-tell. On the last class day I will collect all gw books (don't lose yours!) to skim through, just to see if I generally agree with the impression given by emails from daily swaps. I will pay particular attention to stuff on the left-hand sides: your Morning-afters and your daily (signed) insightful and helpful comments on your classmates' recent work. Your semester letter-grade will be based on those evidences according to a formula in my Excel spreadsheet for grading. If without fail you give this project a securely uninterrupted hour each day outside of class (ideally at the same daily hour 7Γ—; if you feel that weekends are exempt from disciplined effort, then do a specially fine job on weekdays; BTW, hardly anyone actually does this, whatever good intentions they may have resolved on the first day) and participate vigorously and fairly in each of the 30 sessions, then you will likely develop an A. It is a dread fact of history that whatever their expressed intentions at the outset, all but a few students each semester neglect the daily GamesWorth discipline and so find themselves short of ideas, insights, perceptions, and the corresponding intellectual gratifications, so about half the class ends up with a B, and a few flagrant goof-offs receive their customary lower grades. Caution: In almost every semester someone, unpredictably, really takes this experience to heart: chews at the problems like a dog with a leather bone, making mock attacks from every direction, actually using the many diverse approaches suggested in the readings. She maybe still fails to solve a lot of practice problems or even most of them, but makes stubborn and resourceful attempts, thinks about the readings and expands upon them in her GamesWorth book, develops real skill in problem solving, occasionally comes up with alternative solutions the professor never thought of, and leaves the course in high excitement. Having seen how people can engage the course, your professor inevitably sets a high standard for grading. Your grade will come from my spreadsheet formula combining: 1. Lively participation every time (except dire emergency, with advance notice by email). 2. Performance on daily warm-up exercise in blue books. 3. Daily email reports comparing two gw's, and if you like I will peruse your gw notebook at any time to see how active you have been, and especially to spot-check your left-side morning-afters. BTW in comparing gw's please make a firm decision: do not cop out by reporting a "tie". And avoid 3-way swaps, please: I want to see the extent to which the two participants in each match agree about whose play was better. No more than one 3-way is needed, and that only on days with an odd number of participants. 4. My perusal of your gw book with emphasis on "Morning-after"s, and of others' gw books with emphasis on your insightful and helpful comments on that work. 5. Your reading assigned materials no later than the corresponding class meeting date (see schedule below). Other than the Adams and Ehrlich books, I will get the readings to you a week before. 6. Grad students and Honors students must turn in some extra work as described below. ### The final exam The Final Exam will be scheduled for in exam week at the usual class time. It will be a collection of problems from which you will choose some, exhibit as many distinct approaches as you can, and as many cross-checking distinct solutions as you can. All the problems we do solo and as a group during the semester may be regarded as sample practice problems for this Final Exam. At this moment I think this is best done as a take-home, rather than under time pressure: I'll ask your views as the time draws near. ### Honors and graduate students Honors students (479H) are additionally required to visit the History of Science shelves in the library (and/or corresponding web sites), and get acquainted with some episode, and write up a clear analysis of how some discovery or discoveries came about, and some other ways they might have come about. I want some attention to mistakes and dead ends and how the investigators recovered from them. And a paragraph at the end declaring whether this is a closed case, or only the beginning of something more that did or might grow from it. I will hold you to a higher-than-undergraduate standard of English writing. 479-non-H students are also free to do this for extra credit. Graduate students (579) are required to do as Honors students and additionally to provide a new exercise for next year's course. This should be an intriguing problem that can be understood with minimal background and can be solved with minimal specialized techniques other than ingenuity and meticulous care. The writeup should display a variety of ways to solve it, mentioning pitfalls of each and showing ways to check and cross-check every part of the solutions. Hide as many Easter-eggs as you can. Booby-traps are also fun. ### Audits No. There is nothing to audit (no lectures). Imagine auditing dance or debate or weight-lifting without full participation: not much point, right? Besides, with-holding the commitment that goes along with 'really taking the course', auditors (almost by definition, just 'hearing' the course) have chosen to respond preferentially to the inevitable pressures of other commitments than this one, and come to our practice scrimmages wearing white gloves, benefiting nothing from that waste of time. You benefit from lifting weights only by lifting them, not by watching others. No exceptions. (BTW there is always one ostensible student who gives every appearance of merely auditing, though apparently expecting a grade for it. Don't let it be you.) ## How the course works Remember, most of the work will be done on your own or with someone in the class with whom you find it fun to ping-pong ideas: active involvement is essential. Leave now if you are unmotivated, just passively expecting to have your strings pulled. We will meet twice a week mainly to exchange information, to compare notes not really on the homework puzzles themselves so much as on the approaches tried, and to work jointly for a while on bigger problems or puzzles that need a little equipment or need more diversity of approaches and crazy suggestions or need lots of data-collecting, best pooled from many sources. The effect will presumably be that in real life afterwards you will think of lots of approaches, some of them successful, before launching yourself into the hopeless futility of the first that came to mind. Notice the big problem that only you can solve: In this course you are expected to act like an independent self-motivated creative individual. Our aim is enhance your creativity and curiosity and the satisfaction you can secure by solving mysteries in your own unique ways. You are to explore crazy new unfamiliar ways most of which won't be productive for you. But you don't know which ones. This course lifts the pressure so you can find out. This requires joyful playfulness, not grim determination. It seems incompatible with deadline pressure. So here is the problem: since you are under such pressure from elsewhere, you might procrastinate things not backed by pressure. Then you have to attempt them under time pressure later, and so not very creatively, not playfully, learning nothing from the experience. Hurrying under pressure is guaranteed to abort the special opportunities you confront in this course. So form the GamesWorth habit immediately and don't let it slip for any 'reason'. Remember that in such efforts an hour's work punctuated with three 5-minute interruptions has lost you not 15 minutes, as might be the case were you shining shoes or digging a ditch: it has instead prevented you getting warmed up to 60-minute heat, replacing that experience by 3 times getting up to 15-minute heat. You can probably think of other experiences like this. Reserve that sacred daily uninterrupted hour to yourself (or self and partner, if you need to talk to think). ## Readings to ponder before the class meeting *(It's hard to catch up so don't fall behind!)* Two required books await you in the local bookstores: J. Adams (consulting engineer), *Conceptual Blockbusting*; R. Ehrlich (physicist), *Nine Crazy Ideas in Science: A Few Might Even Be True*. Another is out of print, so I will provide Xerox, PDF, or text files: H. Judson (historian), *Search for Solutions*, also in Main Library Reserve. And lots of single short articles from here and there. W.I.B. Beveridge (British MD), *The Art of Scientific Investigation* (1950) is not required this semester because I find that no one reads it anyhow, I think because archaic attitudes offend. I consider it the best of the lot and have placed it on Main Library Reserve and made it available on my web site as PDF files. Other good books on reserve for your attention if you consider it important to acquire better skills in problem solving: George PΓ³lya, *Induction and Analogy in Mathematics* and *Patterns of Plausible Inference*, using elementary math as example material for general principles; James Gleick, *Genius*, biography of Richard Feynman, a great problem solver; Arthur Koestler, *The Watershed*, biography of Johannes Kepler (a chapter of *The Sleepwalkers*). Students requiring accommodation in testing or notetaking must notify me and must (within the first few days) bring a letter of certification from the Disability Resource Center. ## The schedule The following is a retrospective syllabus of Spring 2001, with dates changed to reflect the future, but not intended as a strict preview of Fall 2001 because the course sharply evolves after each semester's student critiques, and also adapts to the particular assortment of student backgrounds in each new semester. Readings and solo exercises are due for discussion of finished results on the date indicated; they were handed out a week in advance of that date (except for the first few). Group "lab" exercises are indicated here on the day they begin in class. In other words the syllabus tells the dates things are scheduled for in class, but most of your work was in the prior week. Every problem below links to its own page, where you will find the statement, its history, hints, and sources. Readings link to the [reading list](https://tyson-swetnam.github.io/aosd/readings/index.md). ### [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md): Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes Five sessions (2 to 6), preceded by the introductory meeting (session 1). | Session | Date | Readings due | Problems and activities | | :-- | :-- | :-- | :-- | | 01 | Tue 21 Aug | Introduction, handouts; Adams preface and Chapter 1 | Demonstrating the need: [Triangle Problem](https://tyson-swetnam.github.io/aosd/problems/triangle-problem/index.md). First of two contrasting challenges: [13 Nails](https://tyson-swetnam.github.io/aosd/problems/thirteen-nails/index.md) (alternatively, the celts problem) | | 02 | Thu 23 Aug | Adams preface and Chapter 1; Platt, *The Art of Creative Thinking*; Feynman, *Cargo Cult Science* | Second of two contrasting challenges: [Conscious Machines](https://tyson-swetnam.github.io/aosd/problems/conscious-machines/index.md) | | 03 | Tue 28 Aug | Judson Chapter 1: *The Rage to Know* | Discuss [Square Windows](https://tyson-swetnam.github.io/aosd/problems/square-windows/index.md) (solo problem) | | 04 | Thu 30 Aug | | [Golden Tooth](https://tyson-swetnam.github.io/aosd/problems/golden-tooth/index.md): facts before explanations of facts. [Salvation of Doug](https://tyson-swetnam.github.io/aosd/problems/salvation-of-doug/index.md). [Bookworm's Journey](https://tyson-swetnam.github.io/aosd/problems/bookworms-journey/index.md): distinguishing things we know vs only imagine | | 05 | Tue 4 Sep | [N-Rays](https://tyson-swetnam.github.io/aosd/problems/n-rays/index.md); Langmuir, *Pathological Science* | Discuss [Phone Cord Problem](https://tyson-swetnam.github.io/aosd/problems/phone-cord-problem/index.md). Discuss [Stockholm Restrooms](https://tyson-swetnam.github.io/aosd/problems/stockholm-restrooms/index.md) | | 06 | Thu 6 Sep | Packet of readings: valuing mistakes; Ehrlich Chapter 1 (Introduction) | Some ways to check for errors; hidden assumptions; what is "understand"? [Evaporated Gold](https://tyson-swetnam.github.io/aosd/problems/evaporated-gold/index.md). Start group effort on [Collective Reproduction](https://tyson-swetnam.github.io/aosd/problems/collective-reproduction/index.md) | ### [Section 2](https://tyson-swetnam.github.io/aosd/section2/index.md): Creative Blocks Six sessions. | Session | Date | Readings due | Problems and activities | | :-- | :-- | :-- | :-- | | 07 | Tue 11 Sep | Adams Chapter 2: Perceptual blocks | Discuss [Weird Organism](https://tyson-swetnam.github.io/aosd/problems/weird-organism/index.md). Discuss [Rearranged Triangle](https://tyson-swetnam.github.io/aosd/problems/rearranged-triangle/index.md). Start group lab on [Dominoes](https://tyson-swetnam.github.io/aosd/problems/dominoes-lab/index.md) | | 08 | Thu 13 Sep | [The Barometer Story](https://tyson-swetnam.github.io/aosd/problems/barometer-story/index.md); Adams Chapter 3: Emotional blocks | Discuss [Telltale Number](https://tyson-swetnam.github.io/aosd/problems/telltale-number/index.md). About assumptions and [Tying Knots](https://tyson-swetnam.github.io/aosd/problems/tying-knots/index.md) | | 09 | Tue 18 Sep | Adams Chapter 4: Cultural blocks; Platt, *Diversity* | Discuss [Mercury's Mysterious Hidden Hemisphere](https://tyson-swetnam.github.io/aosd/problems/mercurys-hidden-hemisphere/index.md) | | 10 | Thu 20 Sep | Adams Chapter 5: Intellectual blocks | [Taboo Questions](https://tyson-swetnam.github.io/aosd/problems/taboo-questions/index.md). [Paths Through Mazes](https://tyson-swetnam.github.io/aosd/problems/paths-through-mazes/index.md) | | 11 | Tue 25 Sep | Adams Chapter 7: Blockbusters | [Walking Through Walls](https://tyson-swetnam.github.io/aosd/problems/walking-through-walls/index.md) | | 12 | Thu 27 Sep | Dyson, *Unfashionable Pursuits*; Capecchi one-page biography; Narlikar on venture funding | [Sums of Integers](https://tyson-swetnam.github.io/aosd/problems/sums-of-integers/index.md): like a jig-saw puzzle of cross-checks. [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md) (lab) | ### [Section 3](https://tyson-swetnam.github.io/aosd/section3/index.md): Observations and Questions Six sessions. | Session | Date | Readings due | Problems and activities | | :-- | :-- | :-- | :-- | | 13 | Tue 2 Oct | | Discuss [Ant Walk](https://tyson-swetnam.github.io/aosd/problems/ant-walk/index.md) and [Seven Bridges of KΓΆnigsberg](https://tyson-swetnam.github.io/aosd/problems/seven-bridges/index.md). Discuss observations outdoors on [Pedestrian Crosswalk Mystery](https://tyson-swetnam.github.io/aosd/problems/pedestrian-crosswalk-mystery/index.md). Start lab exercise on [chemical pattern-formation](https://tyson-swetnam.github.io/aosd/problems/chemical-pattern-formation-lab/index.md) | | 14 | Thu 4 Oct | Judson Chapter 4: *Chance*; Anderson on research strategy | More chemical observations. Discuss [What Isn't There (Surprisingly Hard)](https://tyson-swetnam.github.io/aosd/problems/what-isnt-there/index.md) | | 15 | Tue 9 Oct | Judson Chapter 8: *Evidence* | [Mother Nature as Magician; Hallucinations](https://tyson-swetnam.github.io/aosd/problems/mother-nature-as-magician/index.md). Last chemical observations. Discuss [Rainbow Moon](https://tyson-swetnam.github.io/aosd/problems/rainbow-moon/index.md) | | 16 | Thu 11 Oct | Ehrlich Chapter 2 | Discuss [Hairy People](https://tyson-swetnam.github.io/aosd/problems/hairy-people/index.md), [Green Stars](https://tyson-swetnam.github.io/aosd/problems/green-stars/index.md), and [Escher Print Gallery](https://tyson-swetnam.github.io/aosd/problems/escher-print-gallery/index.md) | | 17 | Tue 16 Oct | Ehrlich Chapter 3 | Discuss [Zygotes](https://tyson-swetnam.github.io/aosd/problems/zygotes/index.md). Discuss [Martian HoneyCombs](https://tyson-swetnam.github.io/aosd/problems/martian-honeycombs/index.md) | | | | *Spring break* | | | 18 | Thu 18 Oct | Ehrlich Chapter 4 | Discuss [Cevians](https://tyson-swetnam.github.io/aosd/problems/cevians/index.md). Discuss [Superposed Filters](https://tyson-swetnam.github.io/aosd/problems/superposed-filters/index.md) | ### [Section 4](https://tyson-swetnam.github.io/aosd/section4/index.md): Patterns, Empirical Generalizations Six sessions. | Session | Date | Readings due | Problems and activities | | :-- | :-- | :-- | :-- | | 19 | Tue 23 Oct | Judson Chapter 2: *Pattern* | Discuss [Presidents and States](https://tyson-swetnam.github.io/aosd/problems/presidents-and-states/index.md). Discuss [N Dots on the Rim of a Circle](https://tyson-swetnam.github.io/aosd/problems/n-dots-on-circle/index.md), connected to slice the disk | | 20 | Thu 25 Oct | Ehrlich Chapter 5 | Start [Cell Shapes Lab](https://tyson-swetnam.github.io/aosd/problems/cell-shapes-lab/index.md) in class | | 21 | Tue 30 Oct | Ehrlich Chapter 6 | Collaborative experiments on Cell Shapes. Deal with [Paired Observations](https://tyson-swetnam.github.io/aosd/problems/paired-observations/index.md). Deal with [Neutrinos](https://tyson-swetnam.github.io/aosd/problems/neutrinos/index.md) | | 22 | Thu 1 Nov | Adams Chapter 6: Alternative thinking languages | Further experiments on cell shapes. Do [Egg Pouches Lab](https://tyson-swetnam.github.io/aosd/problems/egg-pouches-lab/index.md) in class. Deal with [Platonic Solids and Applications](https://tyson-swetnam.github.io/aosd/problems/platonic-solids/index.md) | | 23 | Tue 6 Nov | Ehrlich Chapter 7 | Play [Eleusis](https://tyson-swetnam.github.io/aosd/problems/eleusis/index.md) in class. Deal with [The Mirror Mystery](https://tyson-swetnam.github.io/aosd/problems/mirror-mystery/index.md) | | 24 | Thu 8 Nov | Ehrlich Chapter 10 | Start [Stacked Cantilevers Lab](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) | ### [Section 5](https://tyson-swetnam.github.io/aosd/section5/index.md): Inferences, Hypotheses, Explanations Six sessions. | Session | Date | Readings due | Problems and activities | | :-- | :-- | :-- | :-- | | 25 | Tue 13 Nov | Chamberlin, *The Method of Multiple Working Hypotheses* | Further collaborations on [Stacked Cantilevers](https://tyson-swetnam.github.io/aosd/problems/stacked-cantilevers-lab/index.md) | | 26 | Thu 15 Nov | TBA | Theory of stacking cantilevers, resolution of wagers. Deal with [Summing a Series](https://tyson-swetnam.github.io/aosd/problems/summing-a-series/index.md). Deal with [Stalactites](https://tyson-swetnam.github.io/aosd/problems/stalactites/index.md) | | 27 | Tue 20 Nov | Platt, *Strong Inference* | Start [LoShu Lab](https://tyson-swetnam.github.io/aosd/problems/loshu-lab/index.md) experiments in class | | | | *Thanksgiving break* | | | 28 | Tue 27 Nov | Judson Chapter 7: *Strong Predictions* | Deal with [The Miracle of FujiYama](https://tyson-swetnam.github.io/aosd/problems/miracle-of-fujiyama/index.md). Finish complete theory of LoShu. Start discovering [Laws of a Toy Universe](https://tyson-swetnam.github.io/aosd/problems/laws-of-toy-universe/index.md) in class | | 29 | Thu 29 Nov | Feynman, *The Character of Physical Law* | Deal with [Antigen Invasions](https://tyson-swetnam.github.io/aosd/problems/antigen-invasions/index.md). Deal with [Martian DNA](https://tyson-swetnam.github.io/aosd/problems/martian-dna/index.md). Finish collaborative discovery of The Laws. All GamesWorth books collected for inspection | | 30 | Tue 4 Dec | Judson Chapter 9: *Theory* | Deal with [Bacterial Hybrids](https://tyson-swetnam.github.io/aosd/problems/bacterial-hybrids/index.md) | Final exam is scheduled in ?on Tue ? Dec, 2-4 PM *(The question marks are in the original: the day was not yet fixed. The "Spring break" line after session 17 is also as printed, carried over from the Spring 2001 schedule.)* ---8<--- https://tyson-swetnam.github.io/aosd/about/ai-agents/ --- title: "For AI agents" description: "How AI agents and harnesses should consume this course site: llms.txt, per-page Markdown with OKF frontmatter, trust and lifecycle signals, and the rules for tutoring learners without handing over puzzle solutions." type: Reference tags: [course, instructor-facing, ai-agents, okf, llms-txt, provenance] status: stable generated: by: "claude/fable-5-1" at: "2026-09-16T00:00:00Z" sources: - id: aiaa-ai-agents resource: "https://github.com/tyson-swetnam/AI-Automation-and-Agents/blob/main/docs/about/ai-agents.md" title: "AI Automation and Agents: For AI agents (adapted)" author: "human:tswetnam" - id: okf-spec resource: "https://github.com/GoogleCloudPlatform/knowledge-catalog/blob/main/okf/SPEC.md" title: "Open Knowledge Format (OKF) v0.2 specification" author: "team:google-cloud" - id: llmstxt resource: "https://llmstxt.org" title: "The /llms.txt convention" author: "team:answer-ai" --- # For AI agents This site is published for people **and** for AI agents. The source is an [Open Knowledge Format (OKF) v0.2](https://github.com/GoogleCloudPlatform/knowledge-catalog/blob/main/okf/SPEC.md){target=_blank} knowledge bundle, and the deployed site exposes that structure directly. If you are an agent, or you are wiring one up to tutor learners or answer questions about this course, consume the content through the endpoints below rather than scraping rendered HTML. ## Entry points | Endpoint | What you get | | :-- | :-- | | [`/llms.txt`](https://tyson-swetnam.github.io/aosd/llms.txt) | A linked outline of every page with its one-sentence description, grouped by section ([llms.txt convention](https://llmstxt.org){target=_blank}). Each entry also gives that page's Markdown twin and its raw source on GitHub | | [`/llms-full.txt`](https://tyson-swetnam.github.io/aosd/llms-full.txt) | The entire corpus in one file: every page's Markdown with frontmatter, each prefixed by its canonical URL, links made absolute. Prefer it over fetching pages one at a time | | Any page URL + `index.md` | That page's Markdown source with full OKF frontmatter, served as `text/markdown` β€” for example `/section1/index.md` or `/gamesworth/index.md`. The bundle root is `/index.md` | | `raw.githubusercontent.com/tyson-swetnam/aosd/main/docs/.md` | The same Markdown from GitHub, for sandboxes that allow `github.com` but not `*.github.io`. `` is the site path without its trailing slash | | [`/assets/aosd_syllabus.pdf`](https://tyson-swetnam.github.io/aosd/assets/aosd_syllabus.pdf) | Professor Winfree's original syllabus, the primary source every page derives from | | `/sitemap.xml`, `/robots.txt` | Standard crawl surface; `robots.txt` repeats these pointers | | [Source repository](https://github.com/tyson-swetnam/aosd){target=_blank} | The bundle itself under `docs/`, plus `AGENTS.md` with the contribution rules for coding agents | Every page also carries a schema.org JSON-LD record: the landing page declares the `Course` with its author (Arthur T. Winfree), the original provider (the University of Arizona) and the publisher of this republication, and every other page a `LearningResource` tied to that course, with its `learningResourceType` from the OKF `type` and an `encoding` block naming the page's Markdown twin. Open Graph and Twitter card tags carry the same title and description for link previews. A request for a page that does not exist returns a real HTTP 404 whose body lists recovery points β€” the home page, `llms.txt`, `llms-full.txt`, `sitemap.xml` and this page β€” and `/404.md` is the same list in Markdown, for an agent that asked for Markdown. Both are written by the post-build step, so neither exists under `docs/`. Every rendered page carries two *visible* pointers as well, because text extraction and link-derived URL allowlists never see ``: a **Markdown button** in the page header, beside *Edit this page* and *View source*, and a **Machine-readable** line at the end of the article linking the twin, the raw source, `llms.txt` and `llms-full.txt`. **Asking for Markdown with an `Accept` header.** A request for a page URL with `Accept: text/markdown` returns HTML here. GitHub Pages serves static files and sets no response headers, so it cannot negotiate on `Accept`. Ask for the Markdown directly: the twin at a page's URL plus `index.md` is the same content, served as `text/markdown`, and every page declares it in ``. **Traversing the bundle.** Inside a Markdown twin, and inside `llms-full.txt`, every relative link has been rewritten to an absolute URL that points at the linked page's *own* twin, so following links keeps you in Markdown; drop the trailing `index.md` to reach the rendered page. Links to files served verbatim (`/assets/`) point at the file. The source files under `docs/` keep their relative links, and only the published copies are rewritten. Every rendered page also declares its Markdown twin and OKF signals in HTML: ```html ``` `okf:stale-after` and `okf:superseded-by` appear only when the page carries those keys. ## How the course is laid out The course is five sections, each a run of five or six class sessions with practice problems and readings. The [original syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md) is transcribed in full, with the thirty-session schedule linking every problem to its own page under [Problems](https://tyson-swetnam.github.io/aosd/problems/index.md). The sections are: [detecting nonsense, error checking and false assumptions](https://tyson-swetnam.github.io/aosd/section1/index.md); [creative blocks](https://tyson-swetnam.github.io/aosd/section2/index.md); [observations and questions](https://tyson-swetnam.github.io/aosd/section3/index.md); [patterns and empirical generalizations](https://tyson-swetnam.github.io/aosd/section4/index.md); and [inferences, hypotheses and explanations](https://tyson-swetnam.github.io/aosd/section5/index.md). Two pages cut across the sections: the [GamesWorth method](https://tyson-swetnam.github.io/aosd/gamesworth/index.md), Professor Winfree's daily habit of one uninterrupted session of focused thinking recorded in a problem notebook, and the [recommended readings](https://tyson-swetnam.github.io/aosd/readings/index.md), an annotated bibliography marked open access, borrowable or paywalled. The [home page](https://tyson-swetnam.github.io/aosd/index.md) states the course philosophy and the three tools (readings, problems, collaborative learning). Use it to orient before answering "where do I find…" questions. ## Reading the OKF frontmatter Each page's YAML frontmatter answers the questions an agent should ask before relying on it: - **What is this?** `type` is one of *Lesson* (the five sections), *Activity* (one page per problem under `/problems/`), *Guide* (the GamesWorth method), *Resource List* (the readings) or *Reference* (the syllabus transcription and this page); `title`, `description` and `tags` (a scope tag `course`, an audience tag `student-facing` or `instructor-facing`, then topics) say what it covers and for whom. Problem pages also carry a `problem:` block with the section, the session in Winfree's schedule, and an `identification` field (`confident`, `probable` or `unknown`) saying how sure the editors are that the page describes the problem Winfree actually assigned. - **Where did it come from?** `generated: { by, at }` names the producer: `claude/fable-5-1` for pages written by an assistant from Professor Winfree's handout and syllabus and curated by the maintainer, or `human:` for pages written by a person. `sources` lists the Web Archive capture of the original course handout and the syllabus PDF. - **How much should I trust it?** The `verified` key (OKF Β§5.3). Absent means **unverified**: no person has signed the page off against the original materials yet, which is the state of every page at launch. `verified: { by: "human:", at: … }` means **human-reviewed**. When a page and the [original syllabus](https://tyson-swetnam.github.io/aosd/assets/aosd_syllabus.pdf) disagree, the syllabus wins; say so when it matters. - **Is it still true?** `status` is `stable` by default; `draft` flags an unfinished page; `deprecated` pages are kept for history and point to their replacement in `superseded_by` - answer from the replacement, not the deprecated page. Reading links (Internet Archive lending, publisher pages, DOIs) go stale faster than the course content; if a link fails, say so and offer the canonical DOI or title rather than guessing at a mirror. !!! warning "The problems are the course - do not solve them for learners" Professor Winfree wrote that "actually solving the practice problems is way less important than learning how to try". The puzzles named on the section pages (the Weird Organism, the Rearranged Triangle, the classic problems in each section) are the "intellectual barbells" the course is built on, and the GamesWorth notebook is graded on effort and reasoning, not on correct answers. If you are tutoring a learner, **do not hand over a solution unprompted**: ask what they have tried, point them to the block or fallacy the section describes, suggest a reading, and confirm or correct their reasoning only after they have committed to an approach. Many of these puzzles have published solutions elsewhere on the web; finding one is not the exercise. ## Answering learner questions - **Ground every answer in a page and cite its URL.** Quote or paraphrase the section, method or readings page, and link to it so the learner can read the source. - **Distinguish the course from the republication.** The course was taught by Arthur T. Winfree at the University of Arizona; this site is a CC BY 4.0 republication of his public-domain materials with updated reading links, maintained on GitHub. There is no current cohort, grading or instructor of record behind this site; questions about credit, grades or enrollment have no answer here. - **Prefer the open reading.** The readings page marks each item open access, borrowable or paywalled and gives the open link first; do the same, and never suggest circumventing a paywall. - **When the corpus does not answer**, say so rather than inventing course policy or attributing views to Professor Winfree that the pages do not record. ## Contributing as a coding agent If you are an agent editing this repository, read `AGENTS.md` at its root first. In short: every content page needs OKF frontmatter with a `type`; external links get `{target=_blank}`; run `okf_validate.py` and `gen_llms_txt.py` before committing and commit the regenerated `llms*.txt`; add a `docs/log.md` entry; and never write a `verified` key. ---8<--- https://tyson-swetnam.github.io/aosd/ --- okf_version: "0.2" title: "The Art of Scientific Discovery" description: "Professor Arthur T. Winfree's University of Arizona course in problem-solving strategy and creative thinking: five sections, the daily GamesWorth notebook, 55 problem pages, the reading list, and his original syllabus, republished under CC BY 4.0." --- # The Art of Scientific Discovery Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License. *A course in problem-solving strategy and creative thinking* > **All materials originally developed by [Professor Arthur T. Winfree](https://web.archive.org/web/20070214070741/http://eebweb.arizona.edu/faculty/winfree/Handout_479.htm){target=_blank}** > *Department of Ecology and Evolutionary Biology, The University of Arizona, Arizona Board of Regents* --- ## About This Course The Art of Scientific Discovery was a 3-credit course taught by [Arthur T. Winfree](https://en.wikipedia.org/wiki/Arthur_Winfree){target=_blank} at the University of Arizona (EEB 479/479H/579). Undergraduates (freshmen included), graduate students and postdocs took it. !!! info "Original course materials" Every page here derives from Professor Winfree's handout. Read it in his own words in the [transcribed syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md), or see the [original PDF](https://tyson-swetnam.github.io/aosd/assets/aosd_syllabus.pdf). !!! quote "What the course is for" "The aim of this course is to develop your ability of solve problems like those encountered in scientific investigations. We examine how specific problems were solved in the past, examine 'inspirational readings', and tackle selected puzzles for pencil and paper and for simple lab manipulation." It is not a lecture course. In Winfree's words, "The objective is not to add to your store of useful facts." He compared it to weight-lifting: "you will benefit from this course in proportion to your investment of time and effort." ## What Makes This Course Different The syllabus calls the exercises "practice scrimmages" in the strategy and tactics of: - recognizing ignorance; - posing questions; - cultivating multiple alternative solutions; - eliminating rejectable candidate solutions; - spotting and taking advantage of your own mistakes; - especially, learning a positive attitude toward mistakes, "because they are often the most available doors to discovery." The problems depend as little as possible on any one subject, so that "everyone is on an equal footing of unfamiliarity." ## The Three Main Tools The syllabus names three tools: 1. **[Readings](https://tyson-swetnam.github.io/aosd/readings/index.md).** Two required books, Adams's *Conceptual Blockbusting* and Ehrlich's *Nine Crazy Ideas in Science*, plus many short handouts. Winfree expected underlining, marginal notes, and "a page per class session to celebrate the best things you find or think of while involved with these readings." 2. **[Problems](https://tyson-swetnam.github.io/aosd/problems/index.md).** "Problems, the equivalent of bar-bells for lifting." Write down your approaches, your lucky insights, and how you got into and out of blind alleys: "This is the main thing, not the 'answers'." 3. **Each other.** "The genetic diversity and diversity of life-experiences and habits of thought that we collectively bring to the table." Students swapped GamesWorth notebooks every class, and in class pooled data that no one person could collect alone. ### [The GamesWorth Method](https://tyson-swetnam.github.io/aosd/gamesworth/index.md) The problems are there to start a habit: a daily "Game's-Worth" of focused thought, named for the thought it takes to play one serious game of chess. Winfree wrote that this "might be the most important (potentially enduring) effect of the course." The [introduction](https://web.archive.org/web/20030114041921/http://eebweb.arizona.edu/faculty/winfree/SAS/asdIntro.html){target=_blank} πŸ”“ to his 2001-02 column *Adventures in Discovery* describes the same idea as a regular daily workout. ## Course Structure The thirty class sessions fall into five sections: - [Section 1: Detecting Nonsense, Error Checking, False Assumptions, Cherishing Mistakes](https://tyson-swetnam.github.io/aosd/section1/index.md) (sessions 1 to 6): spotting nonsense and hidden assumptions, and putting facts before explanations of facts. - [Section 2: Creative Blocks](https://tyson-swetnam.github.io/aosd/section2/index.md) (sessions 7 to 12): the perceptual, emotional, cultural and intellectual blocks of Adams's book, and ways past them. - [Section 3: Observations and Questions](https://tyson-swetnam.github.io/aosd/section3/index.md) (sessions 13 to 18): careful observation, outdoors and in a chemical pattern-formation lab, including noticing what isn't there. - [Section 4: Patterns, Empirical Generalizations](https://tyson-swetnam.github.io/aosd/section4/index.md) (sessions 19 to 24): finding rules in data, often gathered in class labs such as Cell Shapes and Egg Pouches. - [Section 5: Inferences, Hypotheses, Explanations](https://tyson-swetnam.github.io/aosd/section5/index.md) (sessions 25 to 30): multiple working hypotheses and strong inference, then working out the theory of stacked cantilevers, LoShu, and the laws of a toy universe. Also on this site: - [The GamesWorth Method](https://tyson-swetnam.github.io/aosd/gamesworth/index.md): the daily notebook, how to log it, and the "morning-after" left-hand pages. - [Problems](https://tyson-swetnam.github.io/aosd/problems/index.md): 55 problem pages, one for each problem and lab linked from the schedule, each with its statement, history, hints and sources. - [Readings](https://tyson-swetnam.github.io/aosd/readings/index.md): the reading list, each item marked open, borrowable or paywalled. - [The Original Syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md): Winfree's handout transcribed in full, with the thirty-session schedule. It is the primary source for everything else on this site. ## Key Philosophy !!! quote "Can problem-solving be taught?" "Some people suppose problem-solving cannot be taught. They suppose that you are born with innate ability or not, and that's all there is to it. In contrast, I think we are all born with it and mostly lose it during and because of schooling and the general intimidation that comes with any competitive society. We can refine and enhance it about as much as we please." ## Getting Started Start the daily GamesWorth habit now. Pick a problem from [Section 1](https://tyson-swetnam.github.io/aosd/section1/index.md), set aside an uninterrupted hour, and log what you try. Winfree's warning: "form the GamesWorth habit immediately and don't let it slip for any 'reason'." !!! tip "Remember" "In this course actually solving the practice problems is way less important than learning how to try (and, for grading, demonstrating real effort.)" --- *"This requires joyful playfulness, not grim determination."* ---8<--- https://tyson-swetnam.github.io/aosd/log/ # Course update log Dated history of changes to this knowledge bundle (OKF Β§9), newest first. Bullets are prefixed **Initialization / Creation / Update / Deprecation / Removal**. ## 2026-09-17 - **Creation**: one page per problem named in Professor Winfree's schedule, 55 in all, under [Problems](https://tyson-swetnam.github.io/aosd/problems/index.md). Each gives the statement, why it is in the course, where it comes from, hints and a resolution folded away, figures, and sources with access markers. Pages say plainly when the editors could not be sure which problem Winfree assigned (`problem.identification`), and label reconstructions as such. - **Creation**: [the original syllabus](https://tyson-swetnam.github.io/aosd/syllabus/index.md) transcribed in full, with a session-by-session schedule linking every problem page; `scripts/gen_problem_index.py` builds the problem index and its navigation block from page frontmatter (CI checks it is current). - **Creation**: figures for the problems under `assets/images/problems/`: diagrams drawn for this site (CC BY 4.0) and public-domain, CC0 or CC BY images from Wikimedia Commons, Project Gutenberg and the Library of Congress, each captioned with its source and licence. - **Update**: identifications draw on Winfree's own archived web pages (his 2001-02 "Adventures in Discovery" columns identify Rainbow Moon) and on the bookmark names inside his archived course handout, which point at a companion problem document that does not survive (for example, Paired Observations links to `Keplers_Laws`, LoShu to `Tictactoe_LoShu`). - **Update**: editorial rewrite of the home, section, GamesWorth and readings pages: parallel section structure (overview, sessions, key ideas, GamesWorth focus, readings, problems table), quotations checked against the syllabus, repeated boilerplate removed, lists that were folded into paragraphs fixed, and external links checked (access markers, years and addresses corrected where they were wrong). - **Update**: `scripts/check_site.py` fails the build when a list is folded into a paragraph. - **Update**: consistency pass across the home, section, readings, GamesWorth, syllabus and problem-index pages. Each problem now appears in exactly one section's problem table (Pedestrian Crosswalk Mystery under Section 3, Stacked Cantilevers Lab under Section 5; the sections where they begin still mention them). Problem names match each page's title everywhere, including the syllabus schedule's link text. The section navigation labels and the problem index use the full section titles. Every session now opens with a "Readings due" line, readings lists share one format and link their reading-list entries, and Ehrlich's chapter titles for sessions 16 to 18 were added (checked against Crossref). - **Update**: the syllabus transcription now keeps two lines exactly as printed: the "Spring break" line after session 17 and the final-exam line with its unfilled date. Quotations of the syllabus on problem pages were checked against the PDF, and an Internet Archive copy of Ehrlich's book is marked as available to print-disabled readers only. - **Update**: the header logo is now the University of New Mexico wordmark (`assets/unm.png`), replacing the University of Arizona logo, to match the favicon and the other UNM course sites. Winfree's course was taught at the University of Arizona; that credit stays in the page text. - **Update**: CI follows [UNM-CARC/docs](https://github.com/UNM-CARC/docs). `.github/workflows/docs.yml` has the same `okf-conformance` job (the reference OKF validator, shared with idss-mesa.github.io, plus the `llms.txt` drift check) and the same `deploy` job (Zensical build, agent surface, `actions/deploy-pages`). This site adds a problem-index drift check, a trial build on pull requests, `--strict` and `check_site.py`. The site now builds with Zensical, which reads `mkdocs.yml`; the root `index.md` no longer carries a `license` key, which the reference validator does not expect (the licence stays in the page badge and the JSON-LD). ## 2026-09-16 - **Initialization**: made the site an Open Knowledge Format (OKF v0.2) bundle with the same agent surface as the *AI Automation and Agents* course site. Every content page now carries frontmatter (`title`, `description`, `type`, `tags`, `status`, `generated`, `sources`); the root `index.md` declares `okf_version`. - **Creation**: `scripts/gen_llms_txt.py` writes `llms.txt` (linked outline with Markdown-twin and raw-source addresses) and `llms-full.txt` (the whole corpus); `scripts/postbuild_agent_surface.py` mirrors every page's Markdown at its URL plus `index.md`, injects `okf:*` meta tags, Open Graph tags and schema.org JSON-LD, adds a Markdown button and a machine-readable line to every page, gives the 404 page recovery links, and writes `robots.txt`; `scripts/okf_validate.py` and `scripts/check_site.py` enforce the contract in CI. - **Creation**: the [For AI agents](https://tyson-swetnam.github.io/aosd/about/ai-agents/index.md) page and this log, under *About* in the navigation; `AGENTS.md` and `CLAUDE.md` at the repository root for coding agents. - **Update**: the GitHub Actions workflow now validates the bundle, builds with `mkdocs build --strict`, runs the post-build step and its checks, and deploys with `actions/deploy-pages` instead of `mkdocs gh-deploy`. - **Update**: corrected `site_url` and `repo_url` in `mkdocs.yml` to the `tyson-swetnam` account, enabled the *Edit this page* and *View source* buttons, removed the unused `mkdocstrings` and `mkdocs-jupyter` plugins and the missing chatbot widget references, and trimmed `requirements.txt` to the build dependencies. - **Update**: favicon changed to the University of New Mexico icon used by the *AI Automation and Agents* site (`assets/unm.ico`).