Small cases lie — page 4
Analysis
An error with an unknown in it
Taylor's theorem does not say a partial sum is close to anything. It says the error is one more derivative evaluated somewhere nobody can name, and everything the theorem is worth comes from what happens when that somewhere is replaced by the worst case.
Analysis
The points that ruin the fit
A polynomial through eleven points of a gentle curve should be a good approximation to it, and adding more points should make it better. On evenly spaced points it makes it worse, without limit, and the reason is not the polynomial but where the points were put.
Applied
Two equilibria and no way to choose
A game can have two states nobody wants to leave, one paying more than the other, and the definition of an equilibrium has nothing to say about which happens. The two standard tie-breakers disagree, and the one that wins is usually the worse.
Dynamics
A carpet with two dimensions
For every self-similar set met so far the two definitions of dimension agree, and the agreement is a theorem about sets built from copies of themselves scaled equally. Stretch one direction more than the other and the two numbers come apart, by an amount that can be computed exactly.
Probability
Giving up on the best
The secretary rule treats landing the second-best exactly as badly as landing the worst, which is a strange thing to want. Ask instead for the smallest average rank and the answer is about the fourth-best candidate — whatever the size of the field, and whether it is ten or ten million.
Analysis
A series that converges nowhere
Expand Euler's integral in powers of x and the coefficients are the factorials, so the series converges at no x but nought. Stopped at its smallest term it still computes the integral to within about e^(−1/x) — and every term added after that makes the answer worse.
Algebra
A plane disguised as an arrow
The cross product of two arrows is an arrow perpendicular to both, as long as the area of their parallelogram. Reflect everything in a mirror and it points the wrong way, because it was never an arrow: it is a plane, written as the one direction a plane in three dimensions leaves over.
Discrete
When several pairs share the roads
For one pair of places, the most that can travel between them equals the cheapest cut that separates them. Give three pairs a hub of three roads to share and the two numbers come apart: the pairs can send 3/2 between them, while separating every pair costs 2. Two pairs still meet their cut, but only by splitting units in half.
Number
The exponent that is smaller than Euler's
Euler's theorem raises every unit to the count of the units and gets one. The smallest exponent that works for all of them at once is often much smaller — and a composite is invisible to Fermat's test exactly when that smaller number divides n − 1.
Analysis
Which curves have a length at all
A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.
Analysis
The length belongs to the journey
Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.
Dynamics
A rotation in different coordinates
At an unlocked parameter the map is not merely like a rigid rotation; it is one, after a change of coordinates built out of a single orbit. The theorem needs the map to be smooth enough, and the map that shows why is one whose orbit leaves a hole.
Computation
A quintic a sliding mark reaches
The eleven-sided polygon needs a number of degree five, and five is not a product of twos and threes — so no conic and no angle trisector reaches it. A ruler with two scratches does, which places the marked ruler strictly above the conics and leaves its exact reach unknown.
Applied
The reading that is almost right
Every account of simultaneous choice so far has assumed the payoffs are known to both choosers and known to be known. Replace that with each chooser seeing a private reading off by a little, and a band of equilibria closes to a single point — so the assumption nobody states decides the answer.
Number
Almost no number is one
Sums of two squares look common — a sixth of all numbers up to a million are one. The fraction is falling to nothing, at a rate so slow that no computation will ever make it obvious, and the constant in front of it has been computed to fifty places and identified with nothing.
Probability
The one that hardly ever comes up
Make the kinds unequally likely and the tidy decomposition into stages fails, because a stage's rate now depends on which kinds turned up rather than on how many. What replaces it is an alternating sum over every subset — and the rarest kind turns out to be nearly the whole answer.
Probability
Two patterns, one chance, different waits
HTH and HTT are equally likely in any given window of three tosses. Waiting for HTH takes ten tosses on average and waiting for HTT takes eight, and the difference is not about probability at all — it is about what a failed attempt leaves behind.
Geometry
The slice that has to match
The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.
Number
The sum of the parts, taken again
Replace a number by the sum of its proper divisors and do it again. Most numbers fall to 1, a few land on a perfect number or a cycle, and some climb for hundreds of steps. Below a thousand there are twelve whose fate nobody knows — and the thing that keeps them climbing is a perfect number hiding in their factorisation.
Geometry
The fewest ordinary lines a polygon allows
Take the corners of a regular polygon and add the points at infinity where its parallel chords meet. Every chord then carries three points, the line at infinity carries all the new ones, and the only lines left with exactly two points are the tangents at the corners — half as many as there are points. Dirac guessed in 1951 that nothing does better, and Green and Tao proved it in 2013.
Probability
Two children and the sentence about one of them
A family has two children and at least one is a boy. The chance that both are boys is one in three — or one in two, or anything from one in three to certainty — and every one of those answers is right for some way the sentence could have come to be said. There is no host and no door, and the protocol is still the whole problem.
Analysis
A coin in front of every term
Put all plus signs in front of 1, 1/2, 1/3, … and the sum runs off to infinity; alternate them and it settles on log 2. Toss a fair coin for each sign instead, and the sum settles — every time, on a different number. Where it tends to settle has a smooth, flat-topped shape, and at the value 2 that shape takes a height that agrees with one eighth to forty-two decimal places and is not one eighth.
Computation
One cell short of a transversal
A transversal of a Latin square picks one cell in every row and every column with every symbol different. The cyclic squares of even order have none, and that was settled by a parity argument centuries old. Whether every square of odd order has one is a conjecture from 1967 that nobody has proved; whether every square comes within one cell of having one was settled only in 2023, and only for squares large enough.
Number
A factorisation that hides its primes
Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.