The Mathematics of Juggling

Juggling patterns can be described by Siteswap notation – a sequence of numbers indicating how high each throw goes. A '3' means a three‑beat throw, a '5' a five‑beat throw, etc. The average of the digits must be an integer (the number of balls). For example, '531' averages to 3 – a three‑ball pattern. Siteswap was discovered by mathematicians in the 1980s. It turns juggling into a branch of combinatorics: any valid siteswap corresponds to a permutation of throw heights modulo the period.