Randomization and eventual reordering: a number theoretic approach Available to Purchase

Shuffling a deck of cards is normally used for randomization. An imperfect shuffle would not produce the desired randomization, since there would be residual correlation with the original order. On the other hand, from the classical card magic literature it is known that eight successive perfect riffle shuffles returns the deck to the original order. The question addressed here is whether this observation is in fact unusual and surprising. Although a general closed‐form analytical solution does not appear to be possible, a simple program could be written to determine deck sizes and numbers of shuffles for which eventual reordering occurs. This computational approach correctly predicts the original observation of eight shuffles for a deck of 52 cards; in fact if the trivial solutions of integral multiples of eight shuffles are discarded, eight shuffles appears to be the unique solution for a 52 card deck.