Juggling Patterns

Throwaway remark by Freeman Dyson:

There was also a performing juggler who happens to be a professor of mathematics. He stood on the stage, simultaneously juggling five balls in the air and proving elegant theorems about the combinatorics of juggling. His theorems explain why serious jugglers always juggle with an odd number of balls, usually five or seven rather than four or six.

Intuitively the idea is that there are some patterns of throws that only work with odd numbers and some that only work with even numbers -- e.g. the cascade (odd) and the fountain (even), of which the cascade is easier. There are also more complicated patterns, such as the Mills mess and Rubinstein's Revenge, that work only with three balls and aren't easily extended.

For a more scientific take, there's a SciAm article here. There's also, apparently, a seminal paper by Claude Shannon that appears in his collected works, but I can't find a copy online.