# 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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [Seven Bridges of Königsberg](https://tyson-swetnam.github.io/aosd/problems/seven-bridges/index.md) (puzzle) - Can a walker cross all seven bridges of Königsberg exactly once? Euler's answer threw away the map and kept only the connections — the founding move of graph theory.
* Session 14 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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 — [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.