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A walker that steps at random along a grid comes back to where it started for certain on a line and for certain on a plane — but on a plane 28.25% are still out after 4,000 steps. In three dimensions it comes back only 34.05% of the time, ever. One walk drawn in each, the chance of being home by any step, and why the third dimension is the one where the walker escapes.

A test that is right about 95 in every 100 sick people and 95 in every 100 well ones, used where one person in a hundred is sick, says yes to 590 people in 10,000 — and only 95 of them are sick, so a positive is right 16.1% of the time. A second independent positive takes that to 78.5%. Ten thousand people drawn one square each, with the prevalence, the sensitivity and the specificity on dials.

A fair coin pays 50% on what you stake and takes 40% of it, so staking everything is worth 5% a round on average — and after a hundred rounds the average player holds 131 times the stake while the player in the middle holds one 194th of it. Thirteen players in a hundred end ahead, one in a hundred beats the average, and that one per cent holds 98% of all the money there is. The stake that grows a run fastest is a quarter, break-even is a half, and the second is exactly twice the first for any coin of this kind.

Fill an album of 640 stickers by buying them one at a time and it takes 4,505 stickers on average — but the first half of the album costs 443 of those and the last tenth costs 3,036, because the k-th new sticker takes n over n minus k plus one tries to find. Six faces of a die take 14.7 rolls on average and 13 more often than any other number, and 57 collectors in a hundred finish before the average, at every album size this page draws.

Every colour rule on this wall compares an ink with the ground under it, which is a ratio of brightnesses. None of them had ever compared two inks with each other — so from this wall's first figures onward it drew its two main series colours, an amber and a green, at 1.01:1: the same brightness, differing only in hue. Most readers see two plainly different colours; a protanope sees them 7.1 apart, which is one colour, and so does anybody at all in print. This page puts three kinds of colour blindness on the wall's own palette, and sweeps the 496 colours that sit at the old green's brightness and that anybody would call green to show that none of them would have helped: the repair had to move the brightness, not the hue.

Riffle a deck of 52 cards seven times and it is close to random by the measure Bayer and Diaconis made famous: after seven, the best single bet about the order beats a fair deck by 33 points, and after six by 61. This page works their table out exactly rather than quoting it. Ask instead that every order get at least half its fair chance and it takes eleven, because the deck turned upside down cannot come up at all before the sixth riffle. And eight perfect shuffles put the deck back exactly where it started.

Jugglers write a pattern as the list of how long each throw stays up, in beats. The average of the list is always the number of balls, which is why 441 is a three-ball trick — and the average is not enough, because 432 averages three too and sends all three balls to the same beat, while 423, the same throws in another order, juggles. Every list with a whole-number average has some order that juggles; the page checks that against all 1,546 such lists of up to six throws. And the patterns can be counted exactly: there are 175 lists of four throws that juggle three balls, which is 4⁴ − 3⁴.

Five towns share a council in proportion to their populations, and the fairest-looking rule there is — give every town the whole part of its quota, then hand the leftovers to the largest fractions — takes a seat away from a town three times as the council grows from five seats to sixty, at 29, 34 and 51, with nobody moving house. Seven people split 3, 3, 1 is the smallest population that can do it, and two towns can never do it at all. Hold the council still and let the people move instead and it fails the other way round: nine people becoming thirteen is enough for a town that gains somebody to lose a seat to a town that gains nobody. The rules that cannot do either buy that by handing a town more seats than its quota allows — Balinski and Young proved in 1982 that you have to give up one or the other.

Three ordinary six-sided dice: amber 4 4 4 4 4 9, blue 2 2 2 7 7 7, green 0 5 5 5 5 5. Amber beats blue 7 in 12, blue beats green 7 in 12, and green beats amber 25 in 36 — so there is no best die, and whoever chooses second can always answer the one you chose. Now roll two of each and add. Every one of those three verdicts turns round: blue beats amber 85 in 144, green beats blue 85 in 144, and amber beats green 671 in 1,296. The ring is still a ring and it runs the other way. From three dice each onwards it stops being a ring at all and settles into the order the averages were in the whole time. Two of the three chances are not merely close but the same fraction exactly, at every length; and the two totals can only be equal at all when the number of dice is a multiple of five.

Two people toss a fair coin, a point each time, and you would expect the lead to change hands often and the score to sit near level. Over twenty tosses the likeliest thing that can happen is the opposite: one player leads for every single toss, 35.24 per cent of games, against 6.06 per cent for the even split of ten each. A longer game does not make the lead fairer, it only stops depending on the length. Over ten thousand tosses one player is in front for at least 96 per cent of them in 25.7 games in a hundred, the share stays between 45 and 55 per cent in 6.4, and the lead changes hands about 39 times in the whole game — with no change at all the single likeliest outcome, at any length whatever.

A hundred candidates arrive one at a time, you can only compare them to the ones you have already seen, and every rejection is final. The famous answer is to let 37 go by and then take the first one better than all of them, which lands the very best of the hundred 37 per cent of the time. It also finishes twentieth on average, because in the rounds where no record turns up — 37.4 per cent of them, a different number from the one above and close to it for the same reason — it is left holding whoever walked in last. A different rule, which relaxes what it will accept as the queue runs out, takes the best only 26 per cent of the time and averages rank 3.6 — and never worse than 3.8695 however long the queue.

We each name a run of three coin flips, then flip a fair coin until one of the runs appears. You name yours first — and that is the losing move. Whatever you pick there is an answer that beats it: HHH loses to THH seven times in eight, and the four safest openings still lose two times in three. This page works out all fifty-six ordered pairs in exact fractions by Conway's leading numbers, solves the same fifty-six again by clearing an absorbing Markov chain over rationals, plays two thousand seeded rounds against every opening, and draws the four-sequence cycle that shows why no best sequence exists.

Berkeley's six largest graduate departments, autumn 1973: 2,691 men applied and 44.5% got in; 1,835 women applied and 30.4% got in. Fourteen points. Yet four of the six departments admitted women at a higher rate than men, and the two that did not favoured men by 2.9 and 3.8 points. Put the men through the departments the women applied to, at the rates the men themselves were admitted at, and they come out at 30.25% against the women's 30.35% — a tenth of a point the other way. This page works the whole table out in exact fractions, splits the fourteen points into the part that is where they applied and the part that is how they were treated, and lets you move either group's application pattern onto the other's and watch the gap cross zero.

Four thousand drivers, two routes, sixty-five minutes each. Open a new road between the two halves that takes no time at all to drive, and every one of them takes eighty. Nobody is behaving badly and nothing is congested by accident: the eighty-minute outcome is the only stable one. This page works the whole thing out in exact fractions, shows that the harm exists only between three thousand drivers and nine thousand, peaks at exactly 22.5 minutes, and lands on the same four-thirds that bounds every network of this kind.

In February 1944 the Allies estimated German tank production two ways. The spies said 1,400 a month. The serial numbers on captured gearboxes said 270. The German records, found after the war, said 276. The whole method is one line — the highest number you have seen, plus one more average gap — and this page shows that it is exactly unbiased, that the obvious alternative is more than twice as wide, and how many serial numbers you need before the answer is worth printing.

Sample a wave and you lose the difference between it and every wave a multiple of the sampling rate away — not approximately, exactly: the numbers you record are the same numbers. A wheel turning 23 times a second filmed at 24 frames looks like one slow turn backwards, and a 30 kHz tone recorded at 44.1 kHz comes back as an audible 14.1 kHz. The fold is at half the rate and there is nothing on the far side of it.

Take the powers of two and look only at the leading digit. It is a 1 three times in ten and a 9 once in twenty-two, and that is not a quirk of two: it follows from log base ten of two being irrational, and it fails exactly when that number is not. The powers of ten start with a 1 every single time, and the powers of the square root of ten use two digits and no others.

A hanging chain is a catenary and a suspension bridge cable is a parabola, and they are not the same curve. Over a hundred-metre span sagging ten metres they part by 32 millimetres — a hand's width, at one place a little past a third of the way out from the middle, and nowhere else. The difference is the fourth term of a series, which is why every drawing of a chain you have ever seen was near enough right.

A hundred people each open half the boxes and every one of them must find their own name. Guessing gives one chance in 2^100. Following the loops gives 31.2%, and the number barely moves as the hundred becomes a million: it settles on 1 minus the natural log of two. The argument works because a cycle longer than half the boxes is the only one of its kind — and this page shows exactly where that stops being true.

A comparison answers one bit, so k of them tell apart at most 2 to the k orders — and five things can be in 120 orders, which six questions cannot separate. The bound is log2(n!), it is reached exactly up to eleven things, and at twelve it stops being reachable: the counting says 29 comparisons and 30 are needed.

Look at the previous letter and the Gettysburg Address falls from 4.12 bits a character to 2.99. Look at the previous four and it falls to 0.35, which sounds like a triumph and is a model that has memorised the text: the table you would have to send costs 11,466 bits, and the address is 11,664 bits as plain bytes. The same letters shuffled tell you which part of the fall was structure and which was memory.

Every number on this wall is a double: 52 bits of fraction, and a gap that doubles with the magnitude. Between 1 and 2 there are 4,503,599,627,370,496 of them and between 1024 and 2048 there are exactly as many, which is why half of all doubles lie between minus one and one — and why a clock holding the seconds since the Big Bang can only tell you the time to the nearest minute.

Every display is sold with two numbers: 16.7 million colours, and a percentage of the visible spectrum. This page rebuilds both from the 1931 colour-matching functions. The eye reports three numbers about a spectrum with seventy-eight dimensions of invisible freedom; no three primaries, however perfect, can ever cover more than 73.9% of the chromaticity diagram; and the count of distinguishable colours is an arbitrary threshold cubed.

Equilibrium theory says the tide is a bulge 36 cm high that follows the Moon. The Bay of Fundy has 16 metres, Tahiti has almost none, and no bulge could keep up with the Moon anyway. This page derives the 36 cm from Newton, derives the tide's whole clock from five astronomical periods, then runs a rotating sea until amphidromic points appear — and bills the difference to the length of the day.

The Gregorian calendar repeats its dates every 400 years, but Easter takes 5,700,000 — and the seven-day week is the only reason it is not 39,900,000. This page derives the number from the two correction rates, counts all 35 possible Easters over the whole cycle, and then asks the sky, which disagrees eight times a century.

A peal of bells is 5000 changes because the extent on seven bells is 7! = 5040, and half a tonne of swinging bronze can only move one place at a time. This page counts the legal changes (a Fibonacci number), rings the extent, and finds the half of it that bobs alone can never reach.

Every geodesic dome, every football and every fullerene has exactly twelve pentagons, whatever its size. The count is not a design decision — it is 720 degrees divided by 60. This page builds the domes, counts them, and hands the strut lengths to a table printed for builders in the 1970s.

Before a GPS satellite leaves the ground its clock is ground to the wrong frequency on purpose — 10.22999999543 MHz instead of 10.23. That offset, the altitude at which it would have been zero, and a constant printed in the system's public specification all fall out of one expression in GM, c and the potential of the sea.

A sail has one number. It sets how close to the wind you can point, how many times the wind speed you can go, the best angle to tack downwind — and the top speed of a propeller cart running dead downwind, which turns out to be exactly the same number by a completely different route.

Nobody picked the angle of the rainbow. It falls out of one law of refraction and one number measured in a laboratory — and so does the dark band above it, the reversed order of the second bow, the two bows around the sun that nobody sees, and the exact size of raindrop at which the extra fringes stop being visible.

Every famous approximation — 22/7, the leap year, the twelve-note octave, the Metonic cycle — is a rung on one ladder, and the ladder is the same algorithm every time. The page climbs it, then checks every rung against a brute-force search over every denominator up to two hundred thousand, and finds the good fraction nobody quotes.

A sunflower has to put every new seed where there is still room, and the angle that never runs out of room is the number that fractions approximate worst. Turn the dial and watch the packing fail everywhere else. The page measures the packing, verifies Steinhaus's three-gap theorem on every turn of the dial, and checks its own arithmetic against Markov's equation of 1879.

Throw a book spinning and it comes back the way it left — unless you spin it about the middle of its three axes, in which case it flips over, and keeps flipping. The page throws it, checks its own arithmetic against the closed form in elliptic functions, and measures the one number that says how long the trick can be postponed.

A sentence is sent down a wire that flips bits at random, in six different encodings, and comes back either perfect or ruined — never nearly right. The page measures where each code's cliff is, checks it against the bounds that say it cannot be anywhere else, and then against Shannon, who says none of them is good enough.

A sphere cannot be flattened, so every flat map is wrong about angle or about area or about both. Twelve of them are drawn here, and the page measures how wrong each one is — from the map itself, not from what its author claimed.

A browser breaks a paragraph one line at a time and never looks back. TeX looks at the whole paragraph at once. Both run here on the same words, and the better one is checked against every arrangement there is.

Twelve pure fifths do not make seven octaves, and the gap is a sound rather than an argument. Six temperaments, played from arithmetic, each one measuring its own beats.

Where the sun and the moon actually are, for any place and any day, from arithmetic alone. Nothing is fetched, nothing is stored, and it will still be right in fifty years.

A chess engine and a board in one file, with nothing behind it. It castles, takes en passant, knows a draw from a win, and will show you its own proof of correctness.