Alex Stone is the author of Fooling Houdini: Magicians, Mentalists, Math Geeks and the Hidden Powers of the Mind. His writing has appeared in DISCOVER, Harper’s, Science, The New York Times, and The Wall Street Journal.
There was a time when people thought of playing cards as cosmic instruments. Fortunes were told, fortunes were lost, and the secrets of the universe unveiled themselves at the turn of a card. These days we know better. And yet, a look at the mathematics of card shuffling reveals some startling insights.
Consider, for instance, the perfect, or “faro” shuffle—whereby the cards are divided exactly in half (top and bottom) and then interleaved so that they alternate exactly. Most people think shuffling tends to mix up a deck of cards, and usually that’s true, because a typical shuffle is sloppy. But a perfect shuffle isn’t random at all. Eight consecutive perfect shuffles will bring a 52-card deck back to its original order, with every card in the pack having cycled through a series of predictable permutations back to its starting place. This holds true for any deck, regardless of its size, although eight isn’t always the magic number. If you have 25 cards, it takes 20 shuffles, whereas for 32 cards it only takes 5; for 53 cards, 52 shuffles are needed. You can derive a formula for the relationship between the number of cards in the deck and the number of faro shuffles in one full cycle.
There’s a neat connection between the faro shuffle and the binary number system, the language of modern computing. The connection involves a formula for moving the top card to any position in the deck using a sequence of perfect shuffles. To understand how the formula works, you first have to understand that there are two types of faro shuffles. You can either weave the cards together so that the top and bottom cards stay in place—this is called an out-faro—or you can do what is known as an in-faro, in which the top and bottom cards each move inward (toward the middle) by one card. (Remember that the shuffle includes both separating the deck into two halves and then recombining it.) To see the difference between the two faros, braid your hands together prayer-style. This can be done in one of two ways, with either your left or right index finger closest to you. Just as there are two distinct ways to plait your fingers, there are two distinct ways to do a faro.
Now let’s say that the ace of spades is on top, and you want to position it 20 cards down. To do this you need to move 19 cards above the ace. The sequence of in- and out-faros required to bring about this arrangement is found by writing the number 19 in binary notation, which looks like this: 1 0 0 1 1. For each 1 you do an in-faro, and for each 0 you perform an out-faro, going from left to right. So in this case, you would do an in-faro (1), followed by two outs (0 0) and, lastly, two more ins (1 1).
Many astounding features of the faro shuffle can be expressed and generalized by a class of mathematics known as group theory. (In math patois, a faro is an element of a symmetric group. So is a Rubik’s cube.) Roughly speaking, group theory is the mathematical language of symmetry. Nature is a big fan of symmetry. In physics, group theory is the organizing principle behind the Standard Model, the overarching schema of fundamental particles and forces—the periodic table for physics. The Standard Model is the crowning achievement of modern physics, and arguably—for its sheer generality, prognostic power (it led to the prediction of the Higgs boson 48 years before it was actually discovered), and funny terminology (to wit: the “Particle physics gives me a hadron” t-shirt)—the greatest scientific achievement of all time. One can therefore state, without doing too much violence to the truth, that an ordinary deck of playing cards and the deepest laws of nature share a common mathematical language. So in a sense, the secrets of the universe really are contained within a deck of playing cards.
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