The Mathematics of Perfect Shuffles

There are two ways to perfectly shuffle a deck of 2n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle 0 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, 0) generated by the two shuffles, for all n. I f 2 n is not a power of 2, and if 2n * 12,24, then (I, 0) has index 1,2, or 4 in the Weyl group B, (the group of all 2”n! signed n x n permutation matrices). I f 2n = 2“, then (I, 0) is isomorphic to a semi-direct product of Zi and Z,. When 2 n = 24, (I, 0) is isomorphic to a semi-direct product of 2-j’ and M,,, the Mathieu group of degree 12. When 2n = 12, (I, 0) is isomorphic to a semi-direct product of Zi and the group PGL(2,5) of all linear fractional transformations over GF(5).