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MatheMUSEments
Up the Magician's Sleeve
By Ivars Peterson
Muse, October 2003, p. 42-43.
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. |
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