The randomization of a deck of Cards by repeated interleaving. More generally, a shuffle is a rearrangement of the elements in an ordered list. Shuffling by exactly interleaving two halves of a deck is called a Riffle Shuffle. Normal shuffling leaves gaps of different lengths between the two layers of cards and so randomizes the order of the cards.

A deck of 52 Cards must be shuffled seven times for it to be randomized (Aldous and Diaconis 1986, Bayer and Diaconis 1992). This is intermediate between too few shuffles and the decreasing effectiveness of many shuffles. One of Bayer and Diaconis's randomness Criteria, however, gives shuffles for a -card deck, yielding 11-12 shuffles for 52 Cards. Keller (1995) shows that roughly shuffles are needed just to randomize the bottom card.

**References**

Aldous, D. and Diaconis, P. ``Shuffling Cards and Stopping Times.'' *Amer. Math. Monthly* **93**, 333-348, 1986.

Bayer, D. and Diaconis, P. ``Trailing the Dovetail Shuffle to Its Lair.'' *Ann. Appl. Probability* **2**, 294-313, 1992.

Keller, J. B. ``How Many Shuffles to Mix a Deck?'' *SIAM Review* **37**, 88-89, 1995.

Morris, S. B. ``Practitioner's Commentary: Card Shuffling.'' *UMAP J.* **15**, 333-338, 1994.

Rosenthal, J. W. ``Card Shuffling.'' *Math. Mag.* **54**, 64-67, 1981.

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1999-05-26