A blog for the mathematically curious

Tag Archives: permutations

Who doesn’t love a good card trick? Many card tricks rely on math and can be a way to demonstrate mathematical concepts, or they can just be a fun way to amaze your friends. Here’s one that was invented by Bob Hummer.

This trick illustrates the concept of permutations. A permutation is basically the order of arrangement in a set. For the set {1,2,3}, there are 6 possible permutations: {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, and {3,2,1}. In the first step of the trick, the magician asks the volunteer to make an unknown transposition, which is when you swap one element with another. For example, transposing 1 with 3 in {1,2,3} gives {3,2,1}.

Procedure:

Place an ace, 2, and 3 on the table as shown above. (You can use any three cards, but it is easiest to remember 1,2,3.)

Turn your back and ask your friend to choose one of the cards without telling you, remember what it is, and flip it over. Then ask your friend switch the positions of the 2 remaining cards and flip those over.

Now, pick up the cards in this order: left card on the bottom, middle card in the middle, right card on the top.

“Shuffle” the cards, by moving top cards to the bottom 1 or 2 cards at a time. Stop when you have moved 4, 7, or 10 cards from the top to the bottom.

Place the cards facedown on the table in this order: top card in the middle, next card on the right, and last card on the left.

Imagine the cards in the reverse order as the original positions (3, 2, 1).

Have the friend guess which card is theirs and flip it over, but not tell you whether they are right or not.

If the card is in the expected position (i.e., they flip over the first card and it is the 3), you know they picked the correct card and you can say “Congratulations!”

If the card your friend flips is not in the expected position, it must have been one of the switched cards. This card, and the card in the place you would have expected the flipped card to be in, are not the correct cards. For example, if they flip over the first card and it is the ace, this card and the 3rd card (the expected position of the ace) are not the correct cards. (Remember, the cards should be in reverse order at this point.) So, flip the 2nd Say, “I think you were looking for this one.”

This trick works because you are picking up the cards in a way that reverses their order, which is equivalent to the transposing 1 and 3. By revealing one the cards allows, you can determine which two cards were switched by your friend.