Mathematical Daydreams: Exploring Mixed Passing

by Phyllis Chinn and Danny Sleator

At a mathematics meeting in Kyto, Japan, we began exploring the mathematics of passing three balls and three clubs on an "every other" count. We noted that the progression of who had various numbers of balls and clubs seemed to depend on the starting configuration. We analyzed the possibilities theoretically, then checked our predictions by juggling.

Let us think of the pattern of six objects as forming a circle, with objects moving around the circle clockwise as they are passed. With three balls and three clubs, there are only four basic distributions of the objects.                   

With every other passing, in the first three configurations each juggler always has two clubs and a ball or two balls and a club. In pattern A, the number of each type changes with each exchange. In Band C each juggler has two of one type for three exchanges in a row, then switches to an excess of the other for three exchanges. In D, each juggler goes through as regular pattern of - 3 clubs, 2 clubs, I club, 0 clubs, ] club, 2 clubs, and back to 3.

Throwing a right-to-right double in the pattern causes similarly interesting results. Given the following configuration of objects:

The following table can be constructed showing the resulting patterns of a right­to-right double given the starting pattern indicated:

By the way, at two math meetings where we juggled, in Kyoto and Hakone, many Japanese people gathered around, interested in watching and anxious to learn. We're sure they look forward to their next juggling encounter!