Juggling Siteswap Checker

Digits, or numbers separated by spaces for throws above nine. Letters work too — a is ten, z is thirty-five.

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Beat by beat

What the numbers mean

• A siteswap is a list of numbers saying how many beats later each throw lands. Most strings of digits are not patterns at all, because two balls would have to come down on the same beat.
• The test for that is exact and is the definition rather than a rule of thumb: work out (position + throw) for every throw, take it modulo the length, and the pattern is valid exactly when those are all different. Nothing about hands, height or gravity comes into it.
• The number of balls is the AVERAGE of the throws. "531" averages three and so does "441" and so does plain "3" — all three-ball patterns. A sequence whose digits do not average to a whole number cannot possibly be valid.
• A whole average is not enough on its own, though, which is why the collision test is the one that decides. Both theorems are checked here by enumerating every sequence up to a given length rather than by being quoted.
• Rotations of a pattern are the same juggling started at a different moment, so counting distinct patterns means counting rotations once. That is why there are fewer real patterns than the raw enumeration suggests.
• Which hand throws which beat is convention, not arithmetic — the numbers do not know about hands. Odd throws cross and even throws do not, and that follows from the hands alternating rather than from the siteswap itself.