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Shuffling a Deck of Cards

It takes just seven ordinary, imperfect shuffles to mix a deck of cards thoroughly, researchers have found. Fewer are not enough, and more do not significantly improve the mixing. The mathematical proof, discovered after elaborate computer calculations, was complicated because of the immense number of ways the cards in a deck can be arranged; any of 52 could be first in the deck, any of 51 second, and so on. Multiplied out, the number of possible permutations is exactly 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505, 440,883,277,824, 000,000,000,000. Persi Diaconis, a Harvard University mathematician who is co-author of the study with Dave Bayer, a mathematician and computer scientist at Columbia University, says, “Most people shuffle cards three or four times. Five times is considered excessive.

Reader’s Digest, 1990