View the question sheet / student exercises at the bottom of this article.When Persi Diaconis was a kid, his favorite hangout was the magic store. He and his friends goofed around, practiced their tricks and longed for the books and tools. Then one day, in walked the world’s greatest living magician. Diaconis, a thirteen-year-old whippersnapper, decided to show off his card tricks. The great magician, Dai Vernon, was so impressed that he decided to teach the teenager a few new things. Each time the pair met in the shop, Vernon taught Diaconis a bit more. And within a year, Vernon offered ...

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Here’s the rule: To assure cards get sufficiently mixed up, shuffle a deck about seven times. Mathematician, magician and card shark Persi Diaconis of Stanford University, along with David Bayer of Columbia 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 and his colleagues are issuing an update. When dealing many gambling games, like blackjack, about 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 al...

New techniques are beginning to unravel the mysteries of knots, revealing a great mathematical superstructure in the process.

In the 1980s, an inkling emerged among some scientists that very disparate phenomena might on some deep level be related. The weather, protein folding, computers, evolution, the stock market, the immune system … each shows complex behavior arising from fairly simple interactions among its parts. For the past 20 years, researchers have labored to understand how these and other “complex systems” work. But there’s still no agreement about even the most basic of questions: What is a complex system? The frustration of this enduring question has led one researcher to a ...