7 is the Magic Number
Playing a friendly card game (i.e. poker), you shuffle the card deck five times and start to deal...."Wait!" Your mathematician friend interjects loudly..."That is not enough. You need to shuffle some more."
Persi Diaconis (Harvard University mathematician and magician) and Dave Bayer (Columbia University computer scientist) have shown that it takes exactly "seven ordinary, imperfect shuffles to mix a deck of cards thoroughly.... Fewer are not enough, and more do not significantly improve the mixing."
And in case anyone asks, a card deck can be arranged in 80,658,175,170,943,878,571,660, 636,856,403,766,975,289,505,440, 883,277,824,000,000,000,000 different ways. But, how many of these are considered a "thorough" or random mixing?
A final concern is in Las Vegas, where decks of cards are shuffled from four to seven times, at the discretion of the casino owners...but Diaconis said "he almost never sees a dealer
shuffle seven times. He said his research also shows that when dealers shuffle several decks at once, they need to shuffle more. Two decks should be shuffled nine times, he said, and six decks should be shuffled 12 times, which is unheard of in the casinos."
Seems that there is a pattern lurking here...7, 9, 12, ....
Source: New York Time, January 9, 1990