The mathematics of perfect shuffles

Abstract

There are two ways to perfectly shuffle a deck of 2 n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle O leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tricks are reviewed. The main result is the determination of the group 〈 I, O 〉 generated by the two shuffles, for all n. If 2 n is not a power of 2, and if 2 n ≠ 12,24, then 〈 I, O 〉 has index 1, 2, or 4 in the Weyl group B n (the group of all 2 n n! signed n × n permutation matrices). If 2 n = 2 k , then 〈 I, O 〉 is isomorphic to a semi-direct product of Z 2 k and Z k . When 2 n = 24, 〈 I, O 〉 is isomorphic to a semi-direct product of Z 2 11 and M 12, the Mathieu group of degree 12. When 2 n = 12, 〈 I, O 〉 is isomorphic to a semi-direct product of Z 2 6 and the group PGL(2,5) of all linear fractional transformations over GF(5).