arithmetic

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.

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.