MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

May 28, 2007

Up the Magician's Sleeve

If you're good at keeping track of cards, here's a fairly simple but nearly foolproof mind-reading trick you can try out on your friends.

Ask a friend to shuffle a standard deck of 52 playing cards, then have her secretly pick a number between 1 and 10. Tell your friend to slowly and steadily deal out the cards, one by one and face up, to form a pile. As she does so, she is to count them silently, following these rules.

Suppose her secret number is 6. The sixth card that she deals becomes a "key" card, and its face value tells her how many more cards she must deal out to get to the next key card. For example, if the key card happens to be 3, she counts from 1 to 3 to find the next key card. She silently repeats this procedure—without pausing, because pauses would tip you off—until she has dealt out all 52 cards. An ace counts as 1, and a king, queen, or jack counts as 5.

At some point, your friend will run out of cards and won't be able to complete the count. Her final key card becomes her secret "chosen" card. Your task as the magician is to read your friend's mind and identify that card.

Here's what you do while your friend is dealing out the deck. You pick your own secret number, then watching the cards, count your way to the end at the same time as your friend. You'll end up with your own "chosen" card. Amazingly, no matter what secret number you pick, you're likely to end up at the same card as your friend.

One way to see what's going on is to deal out a shuffled deck so the cards are in long rows. You can then use different coins, colored poker chips, or other markers to identify the key cards associated with each of the 10 possible starting points.

Suppose your secret number is 1, and the first card is a 10. The first chip goes on the 10. The second chip goes on the 11th card in the row. If that card is an ace, the next chip would go on the next card in line, and so on, until you reach the final key card. Do the same for the other secret numbers, laying down a trail of colored chips for each one.

You'll see that, for nearly all arrangements of cards, every starting point leads to the same "chosen" card. Somewhere along the way, two separate trails of chips meet on the same card, then coincide from that point on. Here's an example.

This prediction trick is known as the Kruskal count, named for mathematician Martin Kruskal of Rutgers University, who discovered it. Mathematicians have studied the trick and have worked out the chances that a magician will guess the "chosen" card correctly. They put the odds of being successful at five times out of six. It helps a little bit of the magician chooses a lower secret number (say, 2 instead of 5) or simply starts with the first card.

The Kruskal count shows how seemingly unconnected or unrelated chains of events can lock together after a while. Such counterintuitive processes underlie many amazing coincidences and startling predictions. They're what the magician has up his sleeve.

Muse, October 2003, p. 42-43.