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Perfect Shuffles through Dynamical Systems
Daniel J. Scully
To perform a perfect shuffle on a deck of 2n cards you split the deck into two packets of n cards each and interleave the cards in the two packets together perfectly. Magicians do perform perfect shuffles. They appeal to magicians because they appear random but are not. In this article we present a new way to model perfect shuffles that uses the well-known doubling function from dynamical systems. This model shows the intimate relationship between the orbits of the cards under the perfect-shuffle permutations and the binary expansions of certain rational numbers from the unit interval. We exploit that relationship to show connections between the various perfect shuffles on the various sized decks, and we use it to link deck sizes to orbit lengths. The model generalizes nicely to k-handed perfect shuffles, and we use the model to describe more general (less-than-perfect) riffle shuffles.