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Card tricks and mathematics: applying the magician's trade to numerical dilemmas

Instead of reaching for a pencil, Stanford mathematician Persi Diaconis reaches for a pack of cards.

"They are easier for my mind to think about than sets of differential equations and stuff like that," says Diaconis, who spent 10 years as a traveling magician. He explains that cards can be used as simple mathematical models to describe complex real-life systems.

Diaconis, the Mary V. Sunseri Professor of Mathematics and Statistics at Stanford, will perform a card trick at the annual meeting of the American Association for the Advancement of Science (AAAS) in San Francisco. His presentation is part of symposium entitled, "Juggling, Magic, Sport and Combinatorics," to be held Friday, Feb. 16, at 9 a.m. PT. Thomas M. Cover, the Kwoh-Ting Li Professor in the School of Engineering and professor of statistics and electrical engineering at Stanford, will speak in the same session about the mathematical aspects of sports.

During the AAAS meeting, Diaconis will explain how he uses a magician's skill to solve long-standing problems in physics, biology and engineering. Lately he has been using cards to work out problems in statistical mechanics a branch of physics that uses statistics to predict and explain how particles interact.

First, he spreads a deck of cards on a table. Each card represents an individual atom, he says, and each has a unique position relative to the others. Also, every card appears either face-up or facedown, which Diaconis equates to individual atoms that can move either forward or backward.

He then starts shifting cards around and flipping them over, just as atoms move and change direction. His goal is to predict all the different combinations possible for the entire system the group of atoms or pack of cards.

"In my talk, I'll try to explain just what arrangements are possible, as an illustration of how a grown-up mathematician could apply grown-up math ideas to a very concrete problem," he says. "The card trick helps because it's simple and neat enough that I can do the math completely."

Of course, a small group of cards is a very small sample compared to the number of atoms in the world. But that is Diaconis's forte: applying what he learns from a small, defined system to the larger, complex reality. It's as if he is looking at a small square of mesh screen in order to learn something about the whole screen door.

"It's sort of like a magic trick, really," says Diaconis. "You have these particles bouncing around, obeying Newton's Laws, and somehow, instead of having to clock how they bounce around, we're able to say that if you know one or two things about them, we can answer all kinds of other questions about them."

During the AAAS symposium, Diaconis will describe how he can apply his card trick to more than atoms. For example, each card could represent a protein, a drop of moving fluid or a person carrying HIV.

Diaconis often chats with biochemists about protein folding, listens to chemists puzzle over fluid mechanics and helps computer scientists design computers that play solitaire. "One nice thing about statisticians and applied mathematicians is that we get to talk to all kinds of people if we want," he says. The downside is that he usually ends up working in a field he knows nothing about. "I often think I get paid for my ability to tolerate feeling stupid," he says.

Diaconis says that his magician's tricks can be valuable tools for solving tangible scientific dilemmas. But that may not be his ultimate motivation. "I'm actually happy to just have a good card trick," he says with a smile.


By Louisa Dalton

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