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Pigeonhole — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 13 into 12. 13 items spread as evenly as 12 boxes allow. Even at their most even, some box holds 2, because 13 is more than 12 × 1.

    More things than boxes

    If there are more objects than containers, some container holds two. That is the entire principle, it is impossible to disagree with, and it settles questions that look nothing like it.

    part 1 · discrete
  2. 8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.

    How close a fraction can get

    Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.

    part 2 · number
  3. Rotating by √2 − 1 of a turn, 40 times. Points on a circle produced by repeatedly turning through the same angle.

    The orbit that must come back

    A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

    part 3 · dynamics
  4. Langford pairings of order 3, 4 and 7. Rows of boxes with arcs joining equal numbers: order 3: 3 1 2 1 3 2; order 4: 4 1 3 1 2 4 3 2; order 7: 7 3 6 2 5 3 2 4 7 6 5 1 4 1.

    A sum that forbids half the pairings

    Put two 1s, two 2s, …, two ns in a row so that between the two ks there are exactly k other numbers. For n = 3 there is one way, 312132; for n = 4 one way; for n = 5 and 6 none at all, and the reason is a single sum — adding up the positions of every entry in two ways forces n(3n − 1)/2 to be even. A search confirms the count forbids nothing that exists and finds 26 arrangements at n = 7 and 108,144 at n = 12, but the sum is all anyone knows about why.

    part 4 · discrete
  5. Thirty-seven points in thirty-six boxes. n=6: pair q=8,15; difference 7: ‖q√2‖=0.100505, ‖q√3‖=0.124356.

    Two numbers approximated at once

    Thirty-seven points in thirty-six boxes force a multiple of √2 and a multiple of √3 to land near whole numbers together, and the same count in more boxes makes q times the two distances less than one infinitely often. Littlewood guessed that the product can be made as small as anyone likes. Each number alone has a floor; the product finds a new record eight times below twenty million, and nobody can prove it will keep going.

    part 5 · discrete
  6. Alone on the track. Speeds 1, 2, 3, 4 relative to a still runner: the best time 0.2000 leaves every other runner at least 0.2000 away, exactly 1/5.

    Alone on the track

    Runners set off together at different constant speeds. Is each of them, at some moment, far from all the others? A pigeonhole on time gives half the conjectured distance in two lines, and sixty years of work have not closed the other half. The speeds 1, 2, …, k are exactly at the limit, and so, surprisingly, are 1, 3, 4, 7 and 1, 4, 5, 6, 7, 11, 13.

    part 6 · discrete

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