juggling

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.