Mathematics

Erdős asked how many pairs among n points in the plane can be exactly one unit apart. For 22 points the answer was known to be 60 or 61. This page draws a configuration with 60 and lays out the exhaustive, certificate-checked search showing that 61 is impossible, so the answer is 60.

The orchard problem asks how many rows of exactly three trees can be planted with a given number of trees. For fifteen trees the answer has been 31 or 32 since 1974. This page shows the 31-row planting and the machine proof, checked by independent proof checkers, that no planting of fifteen trees has 32 rows, so the answer is 31.

A till gives change by handing over the biggest coin that fits, then the biggest that fits in what is left. Usually that really is the fewest coins, but only because of which coins your country happens to make: Britain's old money got it wrong for over a century, and one country's coins still do. Type any amount into the till here and watch both ways of paying it.

The aliquot map iterates n to the sum of its proper divisors. Every start from 1 to 100,000 is classified here; five starts under a thousand, the Lehmer five, have never been shown to settle.

Plimpton 322 is a clay tablet from southern Iraq, written around 1800 BCE, holding fifteen rows of numbers. Each row is a Pythagorean triple: two sides are written down and the third comes out a whole number every time, which you can check here. The scribe made seven mistakes, and they show how the work was done, including one row carrying 25921 in place of 161, which is 161 squared.

Bell ringers ring the bells through every order without repeating one. Some runs can loop back to the start; some are stuck. This counts the stuck ones, which nobody had counted.

Ancient Egyptian arithmetic worked by doubling, and it could not write a fraction like 2/7 in one piece: it had to be a sum of fractions with 1 on top. The Rhind papyrus opens with a table of those sums for every odd number up to 101. This page lets you run the method the scribe used, and shows an exhaustive search for why he picked the answers he did.

Most of this archive explains known things.

The look-and-say sequence: read a number aloud the way a child would (one 1, two 2s, one 3) and it grows into a new number, forever. John Conway proved that every string but two degenerate cases grows by the same fixed factor, about 1.3035772690, and that they are built from exactly 92 'elements' that decay into one another like a periodic table. Play the game and watch a string split into its atoms.

The birthday bound: draw at random from a space of N things and the first repeat arrives after about √N draws, not N. The same √N explains why 23 people are enough for a shared birthday, why a 256-bit hash keeps only 128 bits of collision safety, and why Pollard's rho pulls a factor out of a number in about N^(1/4) steps. The page shows the last two are one collision search aimed at different spaces.

Glue a chessboard's left and right edges with a half-twist and a knight that leaps off one side returns on the other, upside down.

Labeled chip-firing: drop numbered chips on one spot of a number line and topple them by the blindest rule there is, sending the smaller-numbered chip left and the larger right. With an even number of chips the pile always comes to rest in perfect sorted order however you choose; with an odd number it does not. Topple the pile yourself, and see a new term computed for the OEIS count of possible endings, a(5) = 819.

A straight crack that runs clear across a wall is a fault line, the reason a bricklayer staggers the courses instead of stacking them.

Why anything to the zero power equals one: not a rule someone picked, but three independent arguments that force it and agree, the halving ladder that has to land on 1, the exponent law that allows no other answer, and the empty product. Type any base and watch it land. Then the one honest exception, 0⁰: its value is 1 by a forced convention, but its limit is genuinely indeterminate, and you can steer xʸ yourself.

How many ways can an n by n by n cube be cut into smaller whole-number cubes? The counts start 1, 2, 10, 2098, 4006722, and the only published record was a five-term comment in OEIS A228267. This page reproduces those terms and computes two more, D(6) = 2954374781704 and D(7), a 21-digit number, draws every dissection of the small cubes, and lets your own browser run the enumeration.

In packs of 6, 9, and 20, some totals of Chicken McNuggets are impossible: you cannot make 43 out of those boxes no matter how you try.

Penney's game, the coin-flip betting game where whoever picks their pattern second can always pick one more likely to win, played with a die instead of a coin. With a coin no rock-paper-scissors triangle of three words exists until length four; give the coin a third face and the triangle arrives at length two, AB beats BC beats CA, each three wins in five. Play the die and fail to find a safe word.

Islamic geometric patterns are drawn as a single knotted, endless line (girih, Persian for “knot”) and, from the twelfth century, built from just five tiles.

The n-queens problem on a chessboard wrapped into a doughnut, so the edges join and attacks come back around the other side. Counting non-attacking queens that way gives a sequence the encyclopedia already lists (OEIS A051906), but swap the queen for a knight, camel, zebra or giraffe and the counts were, as far as this page could find on 2026-07-03, catalogued nowhere.

The harmonic series 1 + 1/2 + 1/3 and so on adds up forever, but cross out every fraction whose denominator contains the digit 9 and the sum stops, at about 22.92. This is the Kempner series (1914): almost every large integer contains a 9, so you are deleting almost all the terms, not a tenth of them. Pick any digit and watch the survivors' total settle on a finite number.

Cut a 64-square into four pieces, rearrange them into a 5×13 rectangle, and count again: 65.

Any simple rule that turns one state into another, run on a finite set of possibilities, must eventually loop back to a state it has already been in and repeat forever. This page runs three such rules and shows they all share the same hidden shape.

The square root of a whole number is irrational unless that number is a perfect square, so √2 is not special: √3, √5, √6, √7, √8 and √10 are all irrational too. One proof, using remainders alone and no parity or prime factorisation, settles the whole family at once, and Lean's proof checker has verified it for every n. Type a number and watch its root refuse to resolve.

Conway's Soldiers, the checker-jumping puzzle: fill every cell below a line, jump peg-solitaire style, and see how far above the line you can strand one soldier. Rows 1 through 4 are reachable, but row 5 is provably impossible for any finite army, by John Conway's 1961 argument that weights each cell by the golden ratio's reciprocal so the total can never rise. Play the real engine with a live weight meter.

Pollard's 1975 factoring method walks a three-line loop (square, add one, take the remainder) until the walk catches its own tail.

0.999… repeating doesn't fall a hair short of 1: it is 1, exactly, the same number under a second name.

One number, the golden ratio, explains why five-fold symmetry is banned from repeating crystal patterns, why the famous 2023 'hat' tile never repeats, and how sunflower seeds pack their spirals. Four faces of one small piece of math you can operate.

In Penney's game two players name a sequence each, then a coin or die is thrown until one turns up. There is no best sequence: whatever you pick, someone can pick one that beats it, so 'beats' runs in loops. With a coin, three-way loops need four-throw sequences; with a three-sided die, length two is enough.

Write the whole numbers in a square spiral and circle the primes, as Ulam did in 1963, and they fall onto diagonal lines. Each diagonal is a quadratic, and the richest anyone has found is Euler's n²+n+41, which yields 40 primes in a row. A live spiral and a polynomial lab show, through Rabinowitsch's theorem and the Heegner numbers, why that record cannot be beaten.

The Lucas-Lehmer test, the method that has held the record for the largest known prime almost without interruption since 1876: start at four, square it, subtract two, reduce by your modulus, repeat. Run the real test on a Mersenne number in your browser and watch the seed 4 reach zero exactly when that number is prime, then trace how the same map x²−2 mod n draws a graph of tails draining into cycles.

Every flat pattern that repeats in two directions (wallpaper, tiled floors, the atoms in a crystal, the carved screens of the Alhambra) belongs to one of exactly seventeen symmetry groups.

Walk around a circle in equal steps and drop a mark wherever you land. Whatever the step size, the gaps between marks come in at most three lengths, and when there are three the longest is exactly the sum of the other two. Add marks here and watch a fourth length never appear.

To double a square you build a new one on its diagonal, and that diagonal is the first length ever shown to be no fraction at all.

The major scale, counted instead of heard: tally the intervals between its seven notes by class and you get 2, 5, 4, 3, 6, 1, the numbers one through six each used exactly once. A scale whose interval classes all occur a different number of times is called 'deep,' and the diatonic is the one everyone already knows. Move notes around a clock face to watch the tally recompute, or load the pentatonic and watch it fail.

An engineer needed a neutron accelerator to fire as evenly as possible; the algorithm he wrote was Euclid's, two thousand years old.

The Earth turns about 365.2422 times per trip around the Sun: not 365, not 365¼, an ungainly fraction no calendar can honour, only approximate.

A portal across five layers of this place that each end on a quantity you cannot write down exactly, and a claim none of them states alone: 'no exact value' is not one condition but three, sorted by two yes/no questions.

Penney's game lets a second player pick a coin-sequence that beats yours, and the beating runs in a loop, so there is no best choice. This searches every three-way contest at length three and finds the loop is never a triangle: wherever all three contests are settled, the three sequences rank cleanly. The first genuine triangle does not appear until length four, and then fourteen appear together.

Khinchin's constant: take almost any real number, write it as a continued fraction, and the geometric mean of the coefficients approaches the same value, K0 = 2.6854520010, whichever number you picked. Type a number to watch its continued fraction and running average emerge, and meet the known exceptions, the golden ratio, square roots and e, then pi and ln 2, where nobody has proved it either way.

Three impossibilities that are really one: the square root of 2 is no fraction, the circle of fifths never closes, and the golden ratio is the number hardest to approximate. All three come out of anthyphairesis, Euclid's reciprocal subtraction and the algorithm behind continued fractions, which halts when two lengths share a common measure and runs forever when they do not.

Mathematicians hunted fifty years for a single tile that covers an endless floor with no gaps, in a pattern that never repeats. In 2022 a retired print technician found one: a thirteen-sided shape nicknamed the hat. Grow the tiling here and hunt for the repeat that is not there.

The primes look like noise, with no formula, but in 1859 Riemann found an exact formula linking them to the zeros of a function called zeta. Each zero acts like a frequency, and adding the waves together reproduces the primes exactly. The page walks that bridge in both directions.

Pack circles into circles forever, each touching its neighbours, and a theorem of Descartes guarantees something startling: if the first four curvatures are whole numbers, every curvature in the infinite foam is a whole number too.

Any number can be written as a nested stack called a continued fraction. A large number in that stack marks a spot where a simple fraction nearly catches it. The golden ratio's stack is nothing but ones, so it never gets those near-misses: in the precise sense of how closely fractions can approximate a number, none resists harder.

Alfred Kempe published a proof of the four colour theorem in 1879 and it was believed for eleven years. This page runs his argument on a nine country map and shows the exact step that fails, in both of the two ways it can be read. It then runs the finite check that replaced that step, which clears Birkhoff's diamond and refuses the configuration Kempe got wrong.

Kirkman asked in 1850 whether fifteen schoolgirls could walk five rows of three for seven mornings so no two ever walk together twice. It works because the fifteen fit inside PG(3,2), the finite space where three bit-vectors form a row exactly when they XOR to zero. The seven-day calendar is found live in the browser, and each of the 105 pairs is checked to meet on exactly one day.

Twelve drummers play one rhythm, each starting on a different beat, and between them they hit every beat of a 144-beat cycle exactly once. Every way of doing this was worked out in 2009, but the rhythms and the entry patterns were published as two separate lists, with no note of which goes with which. Most of the pairings turn out not to work, and this page shows which ones do.

Several drummers play the same pattern, each starting at a different time. Sometimes they fit together so that every beat gets exactly one hit. This page lets you draw a pattern and works out whether that is possible, and it shows the smallest case where neither the pattern nor the entry times repeat themselves.

This page completely checks simple square dissections with 2 through 20 unequal pieces and finds none. A live control finds the first one at 21, while an offline planted control proves the native census can report a known input. It cannot rule out compound dissections or say that no examples exist after order 20.

A perfect out-shuffle is the opposite of random: it sends every position through a fixed doubling rule.

Turn and scramble a pocket cube while this page exhaustively searches every position in its rotation-quotiented state space.

Parity splits an ordinary sliding puzzle into two reachable halves.

Stand people in a circle and remove every second one until a single seat is left.

How PageRank works, the number Google was built on. Say a page matters if pages that matter point to it, a definition that eats its own tail yet has exactly one answer. Build a tiny web and watch a random surfer, an eigenvector and a direct linear solve land on the same ranking, then find the two ways the naive rule breaks and why teleportation cures both.

Penney's game played with a three-sided die: can you weight the die so that some length-two word becomes unbeatable? Drag a point across every possible weighting and watch who beats whom. The rock-paper-scissors triangle proves fragile, a loop with no top survives about two thirds of all loaded dice, and a safe word appears only once one face passes one half.

The Arctic Circle Theorem: cover a diamond-shaped board with dominoes at random, every covering equally likely, and a shape appears that nobody put there. The four corners freeze into solid walls of aligned tiles, the middle stays a jumble of all four orientations, and the boundary between them tends to a circle of radius n/√2. Grow the board ring by ring and measure the circle and the frozen fraction, 1−π/4.

Cut two opposite corners off a chessboard and 62 squares are left, exactly room for 31 dominoes, yet no arrangement ever tiles them. Every domino covers one light and one dark square, and the two removed corners are the same colour, so 30 of one colour have to pair with 32 of the other. Try the tiling by hand, move the two holes anywhere and let the board decide, and see the impossibility machine-checked in Lean 4.

A counting condition is always necessary for a combinatorial object to exist, and sometimes it is the whole story.

Five famous 'this is impossible' proofs, like why you can't flatten a globe onto paper without distortion and why you can't walk all of Konigsberg's bridges exactly once, turn out to use the very same trick: find a quantity the allowed moves can never change. One page showing they are one idea.

Number one cube 1,2,2,3,3,4 and the other 1,3,4,5,6,8 and they roll exactly like ordinary dice, every sum from 2 to 12 with the identical probability, and they are the only other way to do it with positive whole numbers.

When you rank players by who beat whom, the ranking can hide a loop: A beats B beats C beats A. This page splits a tournament into the part a ranking captures and the part that circulates (the combinatorial Hodge decomposition), and shows a third piece that appears only when some pairs never played: a cycle that every triple you actually measured agrees with. Play one of the missing games and watch it surface.

The 15 puzzle is the sliding box of fifteen numbered tiles with one empty square. Exactly half of all arrangements can never be reached by sliding, because every legal move preserves a quantity no slide can flip. In 1880 Sam Loyd offered $1000 for sliding a board with 14 and 15 swapped back into order; nobody collected.

Three classic probability puzzles, the coupon collector, the secretary problem and the 100 prisoners problem, all answered by the same harmonic number H_n = 1 + 1/2 + ... + 1/n. The page shows that two of them count the same thing: records in a shuffled list and cycles in a random permutation are matched exactly by Foata's bijection, checked by full enumeration up to n=7.

On average, your friends have more friends than you do, and it is forced, not a fluke.

How many riffle shuffles it takes to make a 52-card deck random. Bayer and Diaconis proved in 1992 that the answer is seven, and that it arrives as a cliff: almost nothing happens for four shuffles, then the order collapses inside a two-shuffle window. Riffle a deck here and watch the exact distance-to-random recomputed live.

The Seven Bridges of Königsberg: nobody could walk the town crossing each bridge exactly once, and in answering why not, Euler invented graph theory. Finger-walk the bridges and watch yourself fail, then get the rule (every landmass you pass through needs an even number of bridges, so at most two can be odd, and Königsberg has four) and aim it at other one-stroke drawings like the envelope and the schoolbook house.

Sometimes A beats B, B beats C, and C beats A, with no true 'best' at all. The same loop shows up in dice games, in honest voting, in sports rankings, and in a real lizard's mating colours. This page shows they are all one fact about the word 'beats.'

The 100 prisoners problem: each prisoner may open 50 of 100 boxes to find their own number, and if even one fails they are all executed. Guessing gives about one chance in a nonillion, but a strategy that follows the loops of the shuffle gets them out 31.18% of the time, and that is provably the best possible. Trace a prisoner down their loop and run ten thousand rooms to watch the rate settle.

Two players each name a run of three coin flips, and whoever's run shows up first wins. It sounds even and it is not: whatever run you name, the second player can name one that beats it. There is no best run, only a best answer to yours.

The abelian sandpile: pour grains onto a grid, and any square holding four topples one grain to each neighbour. The pile settles to the same shape no matter what order you topple in, and the number of stable states it can hold is exactly the number of spanning trees of the grid, confirmed three independent ways in your browser. Pour onto a single point and a fractal grows.

Rolfsen's 1976 knot table listed 166 ten-crossing knots. Two were the same knot; the corrected tables have 165. This page shows where each source's numbering shifts, why SnapPy's 10_162 is still the deleted duplicate, and how many distinct knots each old invariant cannot tell apart.

Count the pentagons on a soccer ball, a virus, a Fuller dome, a C60 molecule, or on any sphere you sprinkle with points. The count is always twelve, forced by one line of arithmetic. The page proves it, then reads it live on random points, on N charges repelling (Thomson: tetrahedron, octahedron, icosahedron at N = 4, 6, 12), and on Tammes' packing at N = 12.

This page packs straight sticks pointing in every direction into surprisingly small regions. You can change the packing, test a stick yourself, and count thickened sticks in three dimensions.

Colour every cell of a Hex board however you like, by any rule or none, and exactly one player has a chain across: never both, never neither.

Three layers of this ground secretly run the same geometry.

The mice problem: put a bug at each corner of a square, each crawling always straight at the next, and they wind into the centre and meet. Each bug walks a path exactly one side-length long, whatever its speed, and that finite path turns around the centre infinitely many times. Run the chase for any number of bugs and watch the logarithmic spiral appear.

The Helmholtz decomposition of a vector field: every field splits, exactly and orthogonally, into a gradient part (the slope of a hill) and a curl part (pure circulation, closed loops). Paint a field of arrows and watch it separate, with divergence, vorticity, potential and stream function computed live, plus the third harmonic part that appears only once the space has a hole.

Three regular heptagons meeting at a corner add up to 385.7 degrees, 25.7 too much to lie flat, so the floor has to curve: that tiling exists only in hyperbolic space, the setting of Escher's Circle Limit III. Turn two knobs (p sides, q tiles per corner) and watch 1/p + 1/q against 1/2 decide whether you get a Platonic solid, one of the three flat tilings, or the Poincaré disk.

The napkin ring problem: drill a hole straight through the middle of a solid sphere and the volume of the ring left over depends only on the ring's height, not on the sphere's radius or the drill's width. A ring cut from a grapefruit and one of the same height cut from a sphere the size of the Sun enclose the same volume; move the sphere's size and watch the volume refuse to budge.

Curves of constant width: shapes that are not circles yet roll perfectly level and cannot fall through their own hole, the Reuleaux triangle being the plainest one. Roll one under a board that never rises, spin it between caliper jaws whose gap stays pinned at 1.000, and drill out 98.77% of a square.

The hairy ball theorem: you cannot comb the hair on a sphere flat everywhere at once, so somewhere it must stand up in a cowlick. Comb a sphere yourself and watch the cowlick indices add to exactly +2 every way you try, then switch to a torus, which combs flat. The same theorem says the horizontal wind on Earth must be exactly zero somewhere right now, often the eye of a cyclone.

The Banach-Tarski paradox: cutting a solid ball into a few pieces and reassembling them, moving each rigidly, into two balls the same size as the first. The page hands you the engine most retellings skip, the free group on two letters, drawn as a tree you operate until a quarter of it swells into three quarters. It also says why the pieces have no volume, why the Axiom of Choice is needed, and why the plane is safe.

Your sense of shape was trained in three dimensions and never updated.

The Steiner tree problem: the shortest network joining a set of points is usually not the one built from the shortest links. Add junctions where three roads meet at 120° and it gets shorter, so for the corners of a unit square the least network is 1 + √3, about 2.732. A soap film between glass plates finds it by surface tension, but choosing the wiring is NP-hard, so the film can freeze into a worse one.

A tangled loop on the table: knotted, or just a circle in disguise?

Build a room out of perfect mirrors, stand a candle anywhere inside, and let the light bounce forever.

At this instant there are two antipodal points on Earth (the two ends of a line straight through the planet's centre) with exactly the same temperature AND exactly the same barometric pressure, and it is forced, not lucky.

How few colours does it take to colour every point of an infinite sheet so that no two points exactly one unit apart share a colour? Nobody knows: the answer is 5, 6 or 7. Try to beat the seven-point arrangement that forces you past three colours.

Operate a perfect-sampling construction and see why stopping an ordinary forward coupling at its first meeting gives the wrong law.

Random things clump rather than spread out evenly, and reading the clumps as a hand at work is the clustering illusion. Four instruments: a fair coin whose longest run reaches six about four times in five, Clarke's 1944 grid where V-1 bomb hits matched Poisson (229 empty squares against 226.7), the arcsine law where one player leads a fair game almost the whole way, and a man struck by lightning seven times.

Type a hundred imagined coin flips and three statistics score how far the sequence sits from fair-coin behavior, as probabilities rather than verdicts. An optional record accepts three summary integers from self-selected readers who saw the explanation first, and describes only that crowd. A separate arm reports language-model attempts once its published raw file exists.

Buses come every ten minutes on average, so your wait should average five, and it doesn't, because the fault is in the word average.

A fair coin multiplies your money by 1.5 on heads or 0.6 on tails.

Benford's law: in much real-world data the leading digit is a 1 about 30% of the time and a 9 only 4.6%. Grow the powers of two, the Fibonacci numbers and the factorials and watch their first digits settle onto log10(1+1/d), then see where the law fails and why 'Benford proves the 2020 election was stolen' is wrong.

The two-envelope paradox: one envelope holds twice the other, and the arithmetic seems to say the one you did not open is worth 25% more whatever you see, so you should switch forever. This page shows exactly where that breaks, then sets a real result beside it: Thomas Cover's theorem, where a threshold you invent knowing nothing lets you pick the larger of two hidden numbers more than half the time.

Roll a die around a square grid and bring it back to the square it started on: it can arrive in only 12 of its 24 orientations, and the other 12 are locked out forever. Roll it yourself and watch half the orientations never light up. The cause is a parity law tying the die's orientation to the checkerboard colour of the square.

Pick a chord of a circle at random and ask whether it beats the side of the inscribed equilateral triangle.

Buffon's needle problem: drop a needle across ruled lines and pi falls out of how often it crosses one, with no circle anywhere in the setup. The page then takes apart Lazzarini's 1901 experiment, which reported pi to six digits (355/113) from a method whose error bar is about plus or minus 0.05, and shows those digits were borrowed from a constant already known rather than measured.

The coupon collector's problem: drawing at random with repeats, how many draws until you hold the complete set? The answer is n times the harmonic number, so a die needs 14.7 rolls on average to show all six faces and a 52-card set takes 236 draws, with the single last coupon costing a full n draws by itself. Collect a set yourself and watch the end crawl.

The secretary problem: interview n strangers one at a time in random order, hire or reject each on the spot, and try to land the single best. The optimal rule is to reject the first 37 percent no matter how good, then take the next record-breaker, which wins about 37 percent of the time (1/e). Play the hiring game and drag the cutoff to find the peak of the win curve.

Parrondo's paradox: two gambling games that each lose money played on their own, yet flipping a coin to choose between them each round makes your fortune climb. Race all three players live, take apart where the gain comes from, and meet the control that kills it, because it is no free lunch.

Zipf's law, the rule that the r-th most common word in a text turns up about 1/r as often as the most common one, recomputed live on Moby-Dick, Pride and Prejudice, and Shakespeare. The page then shows how little the straight line proves: shuffling the word order leaves the curve identical, and a monkey hitting random keys draws the same line, with an exponent derived in closed form as s = 1 - ln(1-p)/ln M.

The Talmud's bankruptcy problem: Mishnah Ketubot 93a divides an estate among claimants owed 100, 200 and 300, and at an estate of 200 gives (50, 75, 75), neither equal nor proportional. Aumann and Maschler showed in 1985 that one principle, concede what you do not claim and halve what is contested, reproduces all three rows and always yields the nucleolus.

A test can only ever check finitely many cases; a proof is a claim about all of them.

Rock-paper-scissors played for real by an animal: male side-blotched lizards come in three throat colours, orange beating blue beating yellow beating orange, and three engineered strains of E. coli run the same loop. Build the cycle live and see why no ranking can read it, and why all three types stay alive instead of one winning.

A true thing that should be impossible: open a brand-new road (free to drive, perfectly built, faster than its neighbours) and every single commuter's drive gets slower.

The Gale-Shapley stable matching theorem, the 1962 result behind the algorithm that sorts new American doctors into residencies and children into public schools. For any preferences there is a pairing where no two people would rather have each other, found by deferred acceptance: one side proposes, the other only tentatively holds. Run it and see why honesty is safe for proposers but gameable for receivers.

Why AI leaderboards such as Chatbot Arena's Elo rating can confidently rank a field with no best player. If A beats B beats C beats A, the rating measures an order that does not exist, and this page measures how much of the data no single number can hold: 0% for a clean ladder, 100% for rock-paper-scissors. Then rank a cycle the honest way, as a distribution instead of a fake order.

Two famous facts about shuffling cards sit one number apart, and people put them side by side as if they answered the same question. They do not. This page computes what each one actually counts, and how far apart they really are.

You can change the kick strength, launch standard-map orbits, and compute Fibonacci periodic-orbit residues and action gaps near the numerical breakup of the golden transport barrier.

Gabriel's Horn, the endless trumpet made by spinning y = 1/x around its axis: its volume is exactly π but its inner surface is infinite, so the story goes that you could fill it and never paint it. Pull a length slider toward infinity and watch the volume flatten against π while the surface climbs without bound. The page argues the painter's paradox is an equivocation on the word 'paint'.

The shape of a hanging chain (the catenary), the fastest slide between two points (the brachistochrone), and the bending of light by Snell's law are three answers to one equation. Each minimises a cost of the form ∫f(y)√(1+y'²)dx and so conserves f(y)·sinθ, which is Snell's law with f in place of the refractive index. Pick a material and watch one routine draw all three curves.

A frictionless bead slides from a high point to a lower one.

What Euler's number e (about 2.718281828) is and where it comes from. Turn a compounding dial from yearly to continuous and watch a dollar at 100% interest stall short of e, then meet the same number as a factorial series, as the one base whose curve equals its own slope, and as the 1/e odds that nobody gets their own hat back.

Critical points and phase transitions in four systems side by side: percolation (threshold about 0.5927), the period-doubling road to chaos (about 3.5699), Turing patterns, and the abelian sandpile. Drag each control parameter across its threshold and watch the system change character at once, computed live in the browser. Three must be tuned to the edge by hand; the sandpile arrives there on its own.

A working harmonograph, the Victorian pendulum drawing machine, operated in your browser: pick a musical interval and two decaying pendulums weave a damped Lissajous figure. The law you can test on it is that the curve closes into a single repeating loop only when the frequency ratio is rational (an octave 2:1, a fifth 3:2), and that a real machine, which loses energy, never quite closes.

The cycloid, the arch traced by one point on a rolling wheel, and the three things it is at once: the brachistochrone (the fastest slide between two points), the tautochrone (the same time to the bottom from any release height), and the curve Huygens cut into a pendulum clock. Four simulations run on gravity alone, including a three-bead race and five beads that reach the bottom together.

Hang a chain from two nails and it falls into a curve. Galileo said it was a parabola and he was wrong, though only just; the real curve, the catenary, was not named until 1691. The chain here is not drawn from a formula but simulated, bead by bead.

Take a simple equation that feeds its own output back in and turn one knob upward. The cycle repeats on one value, then two, then four, then eight, faster and faster, until it dissolves into chaos. Feigenbaum found the ratio settles on about 4.6692, then found the same number in a different equation.

Three layers of this place end on the same haunting line: a definite, finite answer provably exists, and you may never reach it.

A sentence that talks about itself is one theorem wearing different masks. The liar sentence, Godel's undecidable sentence, and a program that prints its own source are the same machine with a different operator fed in. The move behind all of them is applying a description of a process to its own description.

A slide rule laid round a circle has no ends to run off, so a calculation never falls off the edge of the scale. This page prints two discs you pin together and use to multiply and divide with the screen off, and it is honest about what the extra length does not buy you: pin play, cutting and parallax can erase the advantage.

Sliding two logarithmic scales past each other turns adding lengths into multiplying numbers, which is how engineers calculated before pocket calculators. This page prints one you cut out and build, then measures what accuracy your own copy earned. Three figures is not a failing of the instrument; it is the honest width of the answer.

Three scales printed side by side, and a straightedge laid across them, can solve an equation with no arithmetic at all: you read the answer where the line crosses the middle scale. This page prints one that works, and then shows exactly which relations a chart of this shape can and cannot carry, which is a question with a provable answer.

You can reproduce the BBP integer relation for pi, then vary input precision and coefficient bounds to test apparent equations among named constants on withheld digits.

This page lines up famous maths questions that were finally settled, sorted by how big the thing is you would have to check yourself. It runs from a handful of coordinates to almost two hundred terabytes. The smallest and the largest can both be checked; the one nobody can check sits in the middle.

Rotate an eight-cell foam, count its two cell types, and recompute why it beats Kelvin's structure without pretending it solves the open problem.

Pick corners of a cube-like network built from binary labels and watch the page find a corner that cannot avoid many chosen neighbours. Then paint your own yes-or-no table and see exactly how much one changed input can matter.

Imaginary numbers are the algebra of rotation, and the name is a three-hundred-year-old dismissal that stuck. Drag a number around the complex plane and watch multiplication add the angles and multiply the lengths, multiply by i and watch everything spin 90°. The payoff: tune a series R-L-C circuit and watch its impedance cancel to purely real at resonance.

Can you divide by zero?

Zeno of Elea argued that the swiftest runner can never overtake the slowest: to pass the tortoise, Achilles must first reach where it was, but by then it has crept ahead, and so on, through infinitely many stages, forever.

Gerrymandering made playable: twenty-five voters whose votes never change, ten red and fifteen blue, and you draw the district lines. Carve the same grid into five equal connected districts and the 40% red minority can take a 3 to 2 majority of the seats (cracking), or blue can land on the proportional 2 to 3 (packing). A second instrument computes the efficiency gap live on worked preset maps.

In the matching game Dobble (Spot It!), any two cards you ever lay down share exactly one picture (never none, never two), and that is not careful hand-tuning, it is a theorem.

Dots and Boxes, the pencil-and-paper game everyone learns as a child and plays wrong by grabbing every box in reach. The expert game is about chains and about who is forced to open the next one, and the key move is Berlekamp's double-cross: eat all but the last two boxes of a chain and hand those two back. Playable here against an opponent that switches to exact, perfect play in the endgame.

Write the primes in a row, take the absolute difference of each pair of neighbours, do it again to that row, and again, forever.

The Stern-Brocot tree, a Parisian clockmaker's method for fitting gear ratios that turns out to list every rational number exactly once. Walk it here by left and right steps to reach any fraction, and because a musical interval is a ratio, hear each one played. The walk toward the equal-temperament fifth climbs toward 700 cents, not the 702 of the just fifth 3/2.

An ant who never leaves a surface can still prove that surface is curved: by measuring a single triangle.

Arrow's impossibility theorem: with three or more options, no rule for turning ranked ballots into a group ranking can meet four conditions that each sound beyond argument without crowning a dictator. Build a constitution and watch the machine find ballots that make it contradict itself, see Condorcet's cycle where A beats B beats C beats A, and watch a census of every rule on three options leave only dictatorships.

Two chemicals that both spread out can build stripes and spots instead of smoothing everything flat, so long as the one that builds the pattern spreads slower than the one that erases it. That is Alan Turing's 1952 mechanism for how a leopard might get its coat. Drag the ratio of the two diffusion rates, then run the simulation and see if the predicted spacing matches.

The Mandelbrot set is the black fractal island you get by repeating z becomes z squared plus c and keeping the starting points that never fly off. Nobody knows its exact area. The best estimate from sampling is about 1.5066, but sampling is measurement rather than proof, and the best proved upper bound is 1.6829.

Hit a drum and it rings at a fixed ladder of frequencies, set entirely by the drumhead's shape.

When you cannot see the stars, you estimate where you must be from where you were and how you have moved since.

Stack twelve perfect fifths and you should land seven octaves above the note you started on. You never do: you overshoot by the Pythagorean comma, the same small amount every time, and the page synthesises the tones so you can hear it. A parity argument (3^m is odd, 2^n is even) proves the circle of fifths can never close, and a later section explains why twelve notes is the division that nearly works.

Euclid's proof that the primes never end, rendered eight ways: Greek geometry, algebra, a Petrarchan sonnet, a Socratic dialogue, an ASCII proof tree, the King James Bible, phenomenological prose, and now an eighth mode: the proof machin...

A real, valid proof that the square root of 2 is an irrational number, written as a Shakespearean sonnet. Tap any line to watch the verse, the logic, and the footnotes line up, and where the meter breaks exactly where the math gets hardest.