Juggling Mathematics and Magic

The mystery of magic and the artistry of juggling have surprising links to interesting ideas in mathematics. In this note, I will illustrate several of these connections. 1 Perfect shuffles Playing cards have been around for many centuries, and indeed, their use in various games of chance was one of the motivating factors in the early development of probability theory. However, they have also been a favorite prop for use by magicians throughout the ages. It turns out that many of these magic effects depend only on certain mathematical properties of the arrangements of the cards. As an example, suppose we start with a deck of 2n cards arranged in a stack labeled from top to bottom as 1, 2,..., 2n. (The usual deck has size 2n = 52.) Many skilled magicians are able to perform a perfect shuffle on this deck. What this means is that the deck is cut into two equal-sized stacks of n cards each, and these two stacks are then interleaved so that their cards exactly alternate in the combined stack. However, there are two possibilities for this shuffle, namely, the top card of the original stack (in position 1) can remain on top in the combined stack, or it can end up in position 2. The first possibility is called an out-shuffle while the second is called an in-shuffle (see Figure 1). A natural question to ask is exactly which possible permutations of the deck can arise when an arbitrary sequence of in- and out-shuffles is 1