Card Shuffling, Card Tricks, and Mathematics

Professor Peter Diaconis of Harvard has discovered that seven shuffles are required to mix a deck of 52 cards thoroughly. Rather surprisingly, after eight perfect shuffles, the order of the cards in a deck is identical to the original order. Let us see why this is so.

By a perfect shuffle we mean one in which the deck is first cut into two equal halves (each containing 26 cards) and then the two halves are meshed together so that, if the original sequence of cards was c0, c1, c2, . . ., c50, c51, then the new sequence of cards is c0, c26, c1, c27, c2, c28, . . ., c25, c51. Notice that cards c0 and c51 remain in their original positions. Therefore, we need only keep track of cards c1, c2, . . ., c50. If we refer to the original location of card c1 as position 1, that of card c2 as position 2, and so on, we see that the following changes take place: