Monday, November 10, 2008

Shuffling The Cards: Math Does The Trick

Science News: "Here’s the rule: To assure cards get sufficiently mixed up, shuffle a deck seven times. Mathematician, magician and card shark Persi Diaconis of Stanford University created shock waves in Las Vegas when he figured that out back in 1992. Most dealers had been shuffling much less.
But now Diaconis is issuing an update. When dealing many gambling games, like blackjack, four shuffles are enough. The reason for the lower number is that many games require randomness for only a few specific aspects of the cards, not all. In blackjack, for example, suits don’t matter. Diaconis and his collaborators extended the earlier analysis to account for these variations.
Gamblers and casinos aren’t the only ones who will benefit. One the most useful tools for applied mathematicians — the Monte Carlo simulation — was inspired by the games of chance that are main attractions in Monte Carlo, Monaco. The new card-shuffling results apply directly to this method, promising to save mathematicians computer time.
Shuffling starts by cutting the deck roughly in half. During the shuffling, a few cards fall from one side, then a few from the other. Diaconis, Sami Assaf of the Massachusetts Institute of Technology and K. Soundararajan of Stanford University made the same assumption Diaconis and his collaborator Dave Bayer made back in 1992, that the cards are more likely to fall from the larger stack — an assumption borne out in real life.
Assaf started by using a very small deck, just four cards, and played with it a lot. Then she tried five, then six. From her experiments, she guessed a formula for how mixed the cards were, for whatever property she cared about. Then she worked out a proof.
The formulas she generated, though, were a mess. “We couldn’t actually calculate them,” she says. “We would have had to run the computer for 64 years or something like that.”
So she took each messy, complicated formula to Diaconis and Soundarajan, and for each they found a simple, easy-to-compute equation that approximated it. “We found a beautiful simple pattern,” Diaconis says. “There’s no reason this problem should have a nice answer. I’m not a religious person, but this is as close as I get.”"

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