The magic of mathematics: amaze your friends

Maths isn’t always deadly serious. To prove it, here’s a card trick to amaze your friends in the pub. It was invented by Arthur Benjamin, a mathematician and magician at Harvey Mudd College in California.

Take a pack of cards and deals the top 16 face down in four rows of four. Turn four of these cards face up. Call for a volunteer from the audience, who will repeatedly ‘fold up’ the square of cards, like folding a sheet of stamps along the perforations, until it ends up as a single pile of 16 cards.

The audience will decide where the folds occur. For instance, the first fold can be along any of the three horizontal lines between the cards, or the three vertical lines.

When the cards have been folded into one pile, the volunteer spreads them out on the table. Either 12 cards are face down and 4 face up, or 4 cards are face down and 12 face up. In the first case, the face-up cards miraculously turn out to be the four aces. In the second case, the volunteer takes the four face-down cards, and turnsthem over to reveal... the four aces.

Magic!

To prepare the pack, the magician arranges the four aces in positions 1, 6, 11, and 16 from the top down. After dealing out the four rows of cards, the aces lie along the diagonal from top left to bottom right. But they’re face down, so the audience doesn’t see that. He also has to turn over the right four cards, to make the squarelook like this:

The trick then works automatically, no matter which sequence of folds the audience chooses. Magic again!

Why does the trick work?

Imagine turning the cards along the diagonal the other way up. Then the square has a pattern like a chessboard, and the aces run along the diagonal:

However you fold the square, the cards that end up in a given position will all face the same way: either all up or all down. For instance, suppose you fold along the central vertical line, and think of the top row. The third card (up) turns over (down) and is placed on top of the second card --- also down. And the fourth card (down)turns over (up), and is placed on top of the first card --- also up.

Now you have a rectangle, made from cards or small piles of cards, and it has the same chessboard pattern of ups and downs. So the same thing happens for the next fold, and the next... By the time you reach a single pile, all of the cards in the pile face the same way.

However, when you started, the cards on the diagonal were the wrong way up compared to the chessboard pattern. After folding, they will again be the wrong way up. So instead of a pile of 16 cards all facing the same way, you will have a pile with 12 cards facing one way, and the four aces facing the other.

Mathematically, the chessboard pattern has ‘colour symmetry’. The fold lines act like mirrors, and the mirror-image of each card sits on top of a card that faces the other way. This idea is used to study how the atoms in crystals are arranged. The cunning bit is to turn the maths into an effective card trick.

Taken from The telegraph