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Small cases lie — page 1

Patterns that hold for every example anyone would check by hand, and then stop. The cases within reach are not a sample of the cases.
Ulam's spiral to 900. The integers up to 900 laid out in a square spiral, with the primes marked; they crowd onto diagonal lines. Discrete

The primes on a spiral, and a pattern nobody ordered

Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.

 The whirling squares. Squares with Fibonacci sides 1, 1, 2, 3, 5, 8, 13, each attached to the long side of what came before. They fill a 13 by 21 rectangle exactly. Geometry

The rectangle that eats itself

Cut a square off a golden rectangle and what is left is a golden rectangle. That single property is the whole of the golden ratio, and it explains both what the number really does and most of what is wrongly claimed for it.

 Terms that vanish, a total that does not. The first 24 terms of the harmonic series as bars, with the running total above them. The last bar is 0.042 tall and the total has reached 3.776. Analysis

A sum whose terms vanish and whose total does not

Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.

 Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere. Analysis

One point's worth of information

A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.

 When a shared birthday becomes likely. The chance that some pair in a group shares a birthday, against group size. It passes a half at 23 people, where the probability is 50.7%. Probability

Twenty-three people

A room needs 253 people before someone probably shares a birthday with you. It needs 23 before two of them probably share one with each other. The gap between those numbers is the whole problem.

 Bayes' theorem as two rectangles. A unit square split by how common the condition is (1.0%) and then by how the test behaves. Of everyone who tests positive, the fraction who have it is 16.7%. Probability

Bayes' theorem is a picture of a square

A test that is 99% accurate returns a positive result. The chance it is right can easily be under one in five, and the reason is visible the moment the population is drawn as a square rather than described as a formula.

 The same terms, with the signs alternating. The partial sums of 1 - 1/2 + 1/3 - 1/4 + …, out to 24 terms. They close on 0.69315 from both sides at once, and the gap between consecutive sums is the next term, so the answer is trapped. Analysis

The same terms, in a different order, adding to whatever is asked

Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.

 Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way. Discrete

Numbers that wrap

A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.

 Six people, and the trio that cannot be avoided. The fifteen pairs among six people, coloured at random. Whatever the colouring, three people are all mutual acquaintances or all mutual strangers — here 1, 2, 5. Discrete

Six people at a party

Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.

 Nine walks, and the square root. 9 independent walks of 400 steps, each step one place left or right. The dashed curves are ±√n: the walks stay near them, spill past them, and come back — which is what a typical distance means as opposed to a limit. Probability

A walk that always comes home, until it does not

Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.

 Three doors, as areas. Staying wins 33.3% of the time and switching wins 66.7%, because the host's choice is constrained by what the host can see, so opening a door rules a region out without moving any boundary. Probability

The door that was not opened

Three doors, one prize, a host who opens a losing door and offers a swap. Switching wins two times in three, and the reason is not about doors — it is about what the host was allowed to do.

 The sieve of Eratosthenes below 100. A grid of the whole numbers with the composites struck out by the prime that removes them. Number

The primes are what is left over

Eratosthenes' sieve does not find the primes. It removes everything else, one prime at a time, and whatever survives is prime by default — which is a strange way to reach the most studied objects in arithmetic.

 Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over. Number

There is no last prime

Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.

 Two factor trees of 360. The same number split two different ways, both ending in the same primes. Number

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

 28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime. Number

Numbers that are their own parts

Six is one plus two plus three. Twenty-eight is one plus two plus four plus seven plus fourteen. Euclid explained where such numbers come from; Euler proved there are no others of that kind; and whether an odd one exists has been open for two thousand years.

 The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points. Dynamics

The road paved with doublings

Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.

 Where the period doubles, and by how much the gaps shrink. The parameters at which the period doubles, with the ratio of consecutive gaps beside them. Dynamics

A constant that does not care which map

The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.

 The basins of Newton's method on z³ = 1. The complex plane coloured by which cube root of one Newton's method converges to from each starting point. Dynamics

Where Newton's method goes instead

An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.

 The 256 syllogistic forms, and the 24 that work. A grid with one cell per syllogistic form, marked according to whether it is valid and what it needs to be valid. Logic

Twenty-four out of two hundred and fifty-six

Aristotle's syllogisms are four sentence forms in four arrangements, which makes 256 patterns of argument. Fifteen of them are valid. Nine more become valid if you assume the things being talked about exist, and the gap between those numbers is a two-thousand-year-old disagreement.

 The Goodstein sequence from 4, with the ordinal beside each term. A table of the Goodstein sequence with each term's hereditary representation and the ordinal obtained by replacing the base with omega. Logic

A sequence that explodes and still stops

Goodstein's sequence starting at 4 climbs past any number you care to name and reaches zero after about ten to the hundred and twenty million steps. The proof that it stops is a second sequence, running alongside it, that goes down.

 A degree-2 polynomial over GF(11), and the 7 values sent. A grid of the finite field with the polynomial's value at each point marked, and the transmitted symbols picked out. Computation

A polynomial through the gaps

Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.

 The arithmetic of GF(4), and of the integers mod 4. Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size. Computation

The field with four elements

The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.

 Transversals of the cyclic square of order 6. A cyclic Latin square with a transversal marked if it has one, beside a count of transversals at neighbouring orders. Computation

The thirty-six officers

Six regiments send six officers each, one of every rank. Arrange all thirty-six in a square so that each row and each column holds every rank once and every regiment once. Euler could not, guessed why, and was wrong about the reason.

 A majority cycle over 3 candidates, and how often 3 voters produce one. The majority tournament as a directed polygon with each arc's margin, beside one cell for every profile of the stated size, filled where no Condorcet winner exists. Applied

The majority that goes in a circle

Every voter hands in a ranking, and a ranking is transitive by construction. Compare the candidates two at a time and let the majority decide each pair, and the verdicts need not fit together into a ranking at all.

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