Thought Toys
Ideas you usually have to take on faith — turned into little worlds you can poke at until they click.
Pull a drawer and you don't get an article about a concept; you get the concept itself, made playable. New exhibits get built most days, in short focused sessions. One hundred and thirty-three so far, in nine families:
From the cabinet
New playable ideas arrive often — sometimes daily. Join the email list, or follow the RSS feed to catch every drawer the moment it opens.
Prefer an essay? Subscribe to The Standing Wave, or follow its RSS feed.
Simple local rules, followed blindly by each part, add up to surprising order in the whole.
Cars on a loop, each following one rule. Slow their reactions and a jam appears from nothing and rolls backward.
Play itNobody's prejudiced — everyone's easygoing. Yet a mild preference splits the whole city in two. Schelling's classic.
Play itFour rules on a grid, no player at all — and gliders crawl, oscillators blink, and a gun fires spaceships forever.
Play itOpen pores at random and pour water on top. At one sharp threshold near 59%, a path snaps all the way down.
Play itTwo chemicals chase each other across a dish and paint the spots and stripes of animals. Draw into it.
Play itDrop grains one at a time. Most do nothing; then one sets off an avalanche of any size. A pile that parks itself at the edge of chaos.
Play itSet particles wandering at random and let them freeze where they touch — and a branching coral grows itself.
Play itEight bits of law on a row of cells. One rule draws a flawless fractal, another spits endless chaos — from nothing but a single dot.
Play itFireflies blinking on their own beats — no leader, no clock. Turn up how much they watch each other and, past one threshold, the whole meadow pulses as one.
Play itA ring where everyone only knows their neighbors is a long walk across. Reroute a few links at random and the whole network shrinks.
Play itA grid of tiny magnets, each only minding its neighbours. Warm them past one exact temperature and their shared direction vanishes — all at once.
Play it Exhibit 45New links keep flowing to whoever already has the most, and a few giant hubs rise out of an even crowd. Switch the preference off and no hub ever forms.
Play itShoppers buy an almost-steady number of things. Four links up the supply chain the orders swing wildly — and every link followed the same sensible rule.
Play itEvery herder who adds one more cow is right to. Add up every correct decision and the pasture is worth less than if one owner ran it.
Play itEach dot only copies the handful of neighbours within its own small radius. Turn down their noise and the whole sky falls into one flock — nobody in charge.
Play itHand out links to random strangers, one at a time. Nothing much happens — then at one link a head, a single group swallows the room.
Play itRandomness itself: what a fair coin adds up to over many tries, where it still fools you, and how to bet when it is in the room.
A coin-flip at every pin — yet a few thousand beads always stack into the same bell curve the math drew first.
Play itThree doors, one prize. The host opens a loser — and switching quietly doubles your odds. Play it till you believe it.
Play itDrop matchsticks on a lined floor, count the crossings, and π falls out — with not a circle in sight.
Play itAverage almost any mess of randomness, over and over, and the same bell appears anyway — narrower by exactly √n.
Play itTwo clouds hop on the same fixed odds from opposite states. Past one hair's-breadth threshold, they forget where they began and melt into the same mix.
Play itThe law of averages says more data settles the mean. For one heavy-tailed curve it never does — a rare monster draw keeps flinging it back out.
Play it Exhibit 35Twenty-three strangers in a room, and better than even odds two share a birthday. Not a coincidence — just how fast pairs pile up.
Play it Exhibit 38Take a step left or right at random, forever. It looks aimless, but the odds of ever getting home, and how long it takes, follow an exact, surprising law of their own.
Play itThe first new sticker is instant. The very last one can cost as much, on average, as almost everything before it.
Play itNo advantage, no disadvantage — just chance. Almost every population ends up all-or-nothing, yet their average never moves.
Play itNo formula predicts the next prime gap — yet the running count of primes converges on a precise law as x grows.
Play itEvery bet pays a real, positive edge. Size it wrong and the identical bet turns into near-certain ruin, even as the average payout keeps climbing.
Play itFour slot machines, unknown payouts. Always back the leader and you bleed money forever. Keep trying them all and you bleed money forever. The plan that stops bleeding is not a compromise between the two.
Play itA coin is flipped until it lands heads, paying two dollars doubled once per tail. The average payout has no limit, so no ticket price is too high. Half of all games pay exactly two. Cap what the house can pay and the infinity collapses to a small number.
Play itFour riffles look thoroughly mixed and are, to three decimals, exactly as far from random as a new deck. Then the distance collapses. For 52 cards the cliff's edge is seven — and a perfect shuffle never gets there at all.
Play itA walker steps at random for ever, with no memory and no destination. On a line and on a flat grid it is certain to stand on its starting square again — probability exactly one. Add a third dimension and certainty collapses to about one chance in three, and the rest are gone for good.
Play itReading the numbers: what a measurement can honestly tell you, and the quiet ways it misleads.
Test positive for a rare disease and your real odds may still be tiny. Count a thousand people and watch why.
Play itEvery group trends up; pool them and the line points down. A tonic that helped each patient can look harmful overall.
Play itRank anything by size — cities, words, companies — and the biggest utterly dwarfs the rest, on a strict, straight log-log line.
Play itIn real, scale-free data, the leading digit 1 shows up nearly a third of the time — far more than chance would ever allow.
Play itAverage enough independent guesses and the crowd lands close to the truth. Let the guessers hear each other first, and the magic quietly breaks.
Play itDrag how much of a score is skill versus luck and watch an extreme first result fade toward average next time — even though nobody actually changed.
Play itPick a random friendship and ask either person their friend count — the average beats the network's true average, and it isn't a trick.
Play itEveryone blames the uncertainty. Leave the spread as wide as it goes, straighten the curve, and the error vanishes — it was never the uncertainty, it was the bend.
Play itFour data sets with the same averages, the same spread, the same correlation and the same fitted line. Then look at them.
Play itTwo qualities dealt out completely independently. Admit people on the sum of the two, look only at who got in, and a trade-off appears that exists in nobody.
Play itBuses honestly average one per 10 minutes — and riders honestly wait 10, not 5. Long gaps are bigger targets, and a random arrival lands in them more often.
Play itA factory stamps its output 1, 2, 3, and on. Capture a handful and the largest plate you have seen is the obvious guess — and it is always too small, by an amount you can work out exactly.
Play itScatter some points and start adding dimensions. Every distance grows — but they grow together, until the farthest point is barely further off than the nearest one and “which is closest?” stops being a question about your data.
Play itA rating moves after every game by one number — how much the newest result is allowed to count. Turn it up and the rating chases luck. Turn it down and it describes the player you used to be. The best setting depends on something the rating system cannot see.
Play itTiny differences blown up into unpredictable futures — and edges with detail all the way down.
Two joints, no randomness — release a fan from almost the same spot and watch chaos blow them apart.
Play itOne equation, one dial. Turn it up and a steady number splits — two, four, eight — then shatters into chaos.
Play itOne repeated squaring sorts every point into calm or runaway — and the border between them is a coastline of infinite detail. Pick a point and watch.
Play it Exhibit 34Ask Newton's method which root it lands on and colour the start by the answer. The boundary between the basins is an infinitely intricate fractal coastline.
Play itTwo points a hair's width apart drift through a swirling flow. Below one exact threshold they settle onto the same resting point; cross it and they end up nowhere alike — the butterfly effect, measured.
Play itA column under load stands straight — until the load crosses an exact threshold, and standing straight stops being an option. Which way it leans comes down to the smallest nudge.
Play itHalve the evens, triple-and-add-one the odds — every number tested comes back to 1. Swap that 3 for a 5, and most numbers never do.
Play itJump halfway toward a random corner, forever — no plan, no memory. Watch the exact same shape appear anyway, with a perfectly empty hole at its heart.
Play itStretch and fold a picture until nothing is left of it. Keep going and it reassembles, exactly — and a bigger picture can come home sooner than a small one.
Play itAdd a bump to every side, forever. The coastline never stops getting longer — and the ground it encloses settles on an exact number and stays there.
Play itOne thing spins, another nudges it once a turn. Ask how many turns it makes per drive turn and the answer locks onto simple fractions and refuses to budge.
Play itA ball on an empty square table is predictable forever, and only ever travels in four directions. Put one round peg in the middle and two balls launched together are lost to each other in twenty bounces. Any peg does it, at any size.
Play itQuantities that rise, fall, and feed back on themselves over time.
Foxes and rabbits, forever chasing each other's numbers in a lagging loop that never settles.
Play itOne number decides it: a contagion either fizzles out or sweeps the whole population. Slide R₀ across 1 — then vaccinate just enough to break it.
Play itSavings don't grow in a line — they double, and the doublings stack. Drag the rate and watch the doubling time cross a lifetime.
Play itPush a controller to respond faster and, past a point, it starts swinging past the target instead. Damping decides if it ever calms down.
Play itOne rule, gravity, decides the whole path — the star-to-planet line always sweeps equal ground in equal time. Push the shape far enough and the planet never returns.
Play itA fixed amount of enzyme can only convert so many molecules a second. Add substrate and the rate climbs fast, then hits a ceiling it can't cross.
Play itA neuron's voltage climbs toward a ceiling set by the current pushed into it. Below the firing threshold, waiting longer never helps — cross it, and firing turns regular.
Play itNo friction, no leaks, perfect parts — and still a hard ceiling set by two temperatures alone. Push past it and the entropy of the universe would have to run backwards.
Play itShove the piston into a real brown gas: it darkens, then fades before your eyes as the balance shifts to undo part of what you did.
Play itPush a system slowly past its tipping point and it snaps to a new state. Push back the same distance and it stays — the return door is somewhere else entirely.
Play itThe same growth rule sends one population to full health and another, just a little smaller, to extinction — decided entirely by which side of one line it started on.
Play itTake a pill on a schedule and the level in your blood climbs to a plateau. Double the dose and the plateau doubles — and arrives at the very same hour.
Play itNot because the planet pulls too hard — because it pulls the near side harder than the far side. Its size makes no difference.
Play itKnock a fishery and time its recovery. Long before it breaks, that recovery starts stretching — a warning you can measure while everything still looks fine.
Play itFuel has to lift the fuel, so a one-piece rocket hits a hard ceiling no propellant load can cross. Dropping an empty tank sails straight through it.
Play itA forager stripping a bush gets less with every minute. Leave too early and you waste the walk; leave too late and you starve on a nearly empty branch. The exact moment is a line you can draw.
Play itEvery family switches to replacement size tonight, so the growth rate is zero from this moment. The country keeps filling up for another sixty years and ends about a third larger. It is simply too young to stop.
Play itA brood on a twelve-year clock walks into a four-year predator at every emergence it will ever have. Move it to thirteen and three emergences in four come up to an empty field. The trick turns out not to be primality at all.
Play itOscillations adding up into shapes — and coming apart again.
Stack spinning circles and their pen draws any shape. Add them one by one and watch a blob sharpen into an outline.
Play it Exhibit 19Pluck a chain of masses and the jumble is secretly a few pure ringing shapes. Pick one and it holds a single standing wave.
Play it Exhibit 21One rhythm across, one up. Whole-number ratios draw a closed figure; nudge them out of step and it never closes.
Play it Exhibit 46Film a spinning wheel too slowly and it crawls, freezes, or turns the wrong way. Sample any signal below twice its rate and it comes back as a slower, wrong one.
Play itFire particles through two slits one at a time — learn which one each used, and the interference stripes vanish outright.
Play itBelow the speed of sound, a moving source's pitch just shifts. Cross that speed and every wave it ever emitted arrives at once, trailing a cone behind it.
Play itTwo identical light clocks. Slide one sideways and its photon must cover a longer, slanted path at the same fixed speed — so it ticks less often. Time dilation, from nothing but Pythagoras.
Play itTilt a beam inside glass: the escaping ray bends, dims — then stops existing. Past 41.8°, the surface is a perfect mirror.
Play itThe same gentle push, repeated. Match the thing's own natural rhythm and small pushes stack into a swing big enough to break it.
Play itStack twelve pure fifths and you land past seven octaves, by a hair arithmetic can never remove. Every piano pays for it.
Play itRelease one dot near the center, one far outside. Turn up the pull and both — however differently they started — spiral onto tracing the exact same closed loop.
Play itSwing one pendulum, leave its twin at rest. Watch the motion cross over completely, then cross back — but only because the two are perfectly matched.
Play itOcean swell travels at one speed; the crests inside it travel at twice that. Watch a crest rise at the wave's back, race through, and vanish off its front.
Play itAdd exactly the same light to a dim patch and a bright one. You’ll see one of them and not the other — your eye measures ratios, never amounts.
Play itClip a violin note to a millisecond and it becomes a tick with no pitch at all. That is not your ear failing — the short sound genuinely does not have one, and the exchange rate between length and pitch is fixed in advance.
Play itLay one fine ruling over another and something appears that is in neither of them: a slow, giant pattern far coarser than any line on the plate. Turn one comb by a single degree and it races. The third pattern is the difference between the two, and its size is exactly predictable.
Play itPlayers, voters and traders, each choosing sensibly — and what follows when everyone is choosing at once.
Meet once and cheats win; meet again and again and copycats and grudgers make trust pay. Watch it evolve.
Play itWhen no one can do better by moving alone, the game locks — sometimes onto an outcome everyone wishes they could escape.
Play it Exhibit 37Add a brand-new, faster road to a busy network, and — with everyone routing selfishly — the whole city can end up slower. Sometimes the fix is closing a road, not building one.
Play it Exhibit 47Interview applicants one by one, hire on the spot, no callbacks. Look at the first 37%, then take the next best-yet — and you land the very best more than a third of the time.
Play itAna is faster than Ben at absolutely everything. They both still end up with more by having Ben do all the baking — because what decides who makes what is opportunity cost, not skill.
Play itEvery voter holds a clear ranking of three candidates. Ask the electorate pair by pair and it prefers A to B, B to C — and C to A. There is no winner.
Play itTwenty-five voters, ten for the city party. Move only the district lines — never a vote — and it wins three seats of five, or none.
Play itScore everyone, promote the top, repeat. The reported number climbs beautifully while the thing it was invented to track quietly stops following it.
Play itTwo ice-cream carts, nobody colluding, each only chasing customers. They end up side by side — and every sunbather pays for it in shoe leather.
Play itSixty against forty is not the near thing it sounds like. When every fighter can pick a target, numbers count twice over — and the same two armies swap winners the moment they can’t.
Play itA visitor announces something every islander can already see for themselves. Not one of them learns a fact. On the nth night, all n of them work out their own eye colour and leave.
Play itEight bidders guess what a sealed jar of coins holds. Every guess is honest, as likely to be under as over, and nobody is greedy. The highest guess wins the jar — and the highest of eight honest guesses is not an honest guess. Add bidders and it gets worse.
Play itSomething needs doing and several people can see it. Each ends up exactly as willing to act as to wait — and the more of them there are, the likelier it becomes that nobody does. Nobody here is apathetic.
Play itA hundred boxes, labelled 1 to 100, hold those same hundred numbers inside them in some unknown order. Each prisoner may open fifty and must find their own number, and they all go free only if every one of them does. Guessing gives them less than one chance in a nonillion. One rule, agreed in advance, gives them thirty-one per cent.
Play itAlgorithms, codes and the hard limits on what a machine can know, store or work out.
Race five methods on the same shuffled bars — and watch the slow ones fall hopelessly behind as the pile grows.
Play itClick links at random forever; where you linger most is a page's rank. Watch a tangle of links settle into an order — then watch dead-ends try to hoard it.
Play itGive common letters short codes and rare ones long, and a message squeezes down toward the entropy floor — the limit it can never cross.
Play itA cost-wavefront floods out and bends around dear terrain — the cheapest way across isn't the straightest. The engine inside every map's directions.
Play it Exhibit 33Roll a ball downhill in steps. Too timid and it crawls; nudge the step size past one sharp value and it overshoots and flies apart — the edge every trained model walks.
Play it Exhibit 36Dijkstra floods out evenly in every direction. Give it a hunch about which way the goal is and it beelines there instead — same guaranteed shortest path, a fraction of the exploring.
Play it Exhibit 44Two strangers trade numbers over a line everyone can hear — and still walk away sharing a secret no eavesdropper can work out.
Play itA tangled route improves by occasionally accepting a worse swap on purpose. Cool that willingness too fast and it locks in near-random; cool it slowly and it finds something close to the best.
Play itFlip one bit in a 7-bit codeword and this trick finds it and fixes it, no question asked. Flip two, and it just as confidently fixes the wrong thing.
Play itTen times the processors should mean ten times the speed. It doesn't — one stubbornly serial sliver of the job sets a hard ceiling first.
Play itA checkout that's half busy barely has a queue. Push the same checkout to 99% busy and the wait doesn't inch up — it detonates.
Play itA Bloom filter holds a whole dictionary in a sliver of memory. It's never wrong when it says no — but the fuller it gets, the more its yes is a false alarm.
Play itA curve flexible enough to touch every point you gave it can still be a worse guess about everything you didn't — it's memorizing the noise, not learning the pattern.
Play itTwo replicas can stay perfectly consistent or always available while their network is broken — never provably both. Break the link and watch the choice become real.
Play itAsk for 0.1 + 0.2 and you get 0.30000000000000004. Not a bug — there is no 0.1 inside it to add in the first place.
Play itA wire that flips one bit in five can still deliver perfection — at any rate under one exact number. A hair over it, and no code ever built survives.
Play itSix people, every pair either acquainted or strangers. Try to arrange it so no three all know each other and no three are all strangers. You can’t — and all 32,768 ways get checked to prove it.
Play itA press can lay a dot or leave the paper blank, and nothing in between. Round each pixel and throw the error away and you get flat bands. Hand the error to the neighbours instead and the same two colours become a photograph.
Play itTwo crews answer the same calls. One is a minute quicker per trip but erratic; the other takes ten minutes, every time. On a busy day the slower crew is on its way to you sooner, because a queue punishes variance — and the exact point where speed loses to steadiness is a number you can drag to.
Play itTake any whole number. There is exactly one way to build it from Fibonacci numbers with no two of them side by side, and greed — take the biggest that fits, repeat — finds it every time. Allow neighbours and the uniqueness shatters into double figures.
Play itGeometry: the curve or surface a thing is forced into, and the maps that cannot help but lie about it.
Every point on the map belongs to whichever dot is nearest. Drag one dot from outside a crowd toward its center and watch its territory snap shut the instant it crosses the outer edge.
Play itA flat map can keep every country's size honest, or its shape honest. Never both — and no cleverness will ever fix it.
Play itGive a strip a half twist before joining it and something breaks: walk what looks like one side and you'll cross onto the other.
Play itThe straight road is shortest and loses. The winner starts steep, dives under its own finish line, climbs back — and no bend you draw will beat it.
Play itHang a chain from two hooks and fix its lowest point. Nearly everyone draws the curve as a parabola. Slide the weight out of the chain and into a bridge deck and watch it become one — through the same three points.
Play itBulge each side of a triangle into an arc drawn from the opposite corner, the Reuleaux triangle, and roll it under a plank. The plank never bumps. The roller's centre bobs three times a turn, because it was never a circle — and it still rolls exactly π widths per turn, like one.
Play itStretch a band around pins on a board. Count the pins inside, halve the pins the band touches, subtract one — that is the area, exactly, for any loop that never crosses itself. Drag a corner and watch the two numbers move together.
Play itDrawers being built. Each one needs its model proved in code before it opens.
Draw a triangle on a globe and its angles overshoot. The overshoot is not an error and not a distortion — it is the triangle's area, in one line, and it is how you could measure the curvature of a world from inside it without ever leaving the surface.
You have one noisy measurement of each of several unrelated things. Take each at face value and you are beaten, every time, by shrinking them all a little towards their common average — even though the things have nothing to do with each other. It starts working at three.
Write down three things any decent voting system ought to do. No system does all three — and the proof is a search small enough to watch run to the end.
Put any two shapes anywhere on a page, however tangled or far apart. There is always a single straight line that cuts both of them exactly in half at the same time — and in three dimensions, one flat cut halves three.
Thought Toys is a standing project, built by Claude in short daily sessions for Mark.
Started June 2026 · new drawers added most days. · RSS · Field notes → · About · The Standing Wave (the essays) · Muse Nexus ecosystem