Magic Trick with Cards and Algebra

Summary

Use an ordinary deck of playing cards to complete this trick.  While knowledge of algebra is not required, the explanation of how the trick works uses basic algebra.

Instructions

Use twelve cards from a deck of cards to complete this trick. While any twelve cards may be used, it is suggested to use the twelve face cards – the four kings, queens and jacks.

Progression of the “trick”

• Place the twelve cards face-down on a table.

• Ask one of your audience members to turn five of the cards face-up.

• Turn around and face away from the table.

• Now ask your audience member to mix the twelve cards but not to change their orientation (there should still be five face-up cards). 

• Next, ask to have cards divided into two piles; one pile with five cards and the other with seven. It is their choice how to distribute the face-up cards between the two piles.

• You remain facing away from the table. Reach behind you to grab the two piles of cards – one pile in each hand.

• Ask your audience to confirm which hand is holding the short pile of five cards. You might be able to tell by the feel of the cards, but it is good to confirm the one with the short pile.

• Announce that for your trick that without looking you will arrange the cards so each pile will have the same number of cards face up. However, since we started with an odd number of cards, you say that you will need to move cards from one pile to the other or to flip one or more cards over (of course, without looking at the cards).

• Turn around so you are facing the table, but the cards remain behind your back in your two hands.

• Make the adjustments to the pile behind your back (exactly what you do will be described in the next section).

• Place the two piles back on the table and confirm that the two piles have the same number of face up cards.

Your magician’s instructions

• No sleight of hand or deception is required for this trick.

• At the point where you are holding the two piles and you have turned around to face the table, simply flip over the short pile before placing the two piles back on the table. The tall pile should be placed on the table unchanged in its original orientation.

Example 1

Your audience put two face-up cards in the pile of seven and three in the pile of five.

When you reverse, or turn over, the short pile you now have two face-up cards in that pile matching the two face-up cards in the taller pile.

Example 2

Your audience put four face-up cards in the pile of seven and one in the pile of five.

When you reverse, or turn over, the short pile you now have four face-up cards in that pile matching the four face-up cards in the taller pile. Now, you can probably see the pattern of what is happening.

Additional Examples

Try it for yourself – place any number of the face-up cards in the tall pile and flip over the short pile to confirm that this works for any number of the face-up cards starting in the tall pile. It should work even if you place all five or none in the tall pile.

For advanced thinkers - How does it work?

Understanding this trick requires some simple algebra. Let X=number of face-up cards in the tall pile.

When your audience splits up the cards into two piles of seven and five cards, you know that the tall pile has X face-up cards.

In total you started with five face-up cards, so if the tall pile has x face-up cards then the remainder (5-X) are in the short pile. Since the short pile has 5 cards in total and 5-X are face-up cards, then it has x   number of face-down cards. Regardless of how many of the original face-up cards are placed in the tall pile, there will be a matching number of face-down cards in the short pile. Simply turning over the short pile converts all its face-down cards to be face-up so now the number of face-up cards in both piles is the same in both piles.

The table below also shows this:

	Tall Pile	Short Pile	Both Piles Combined
Total number of cards	7	5	12
Face-up cards	X	5-X	5
Face-down cards	7-X	5-(5-X)=X	7
Face-up cards after flipping over Short Pile	X	X	2X

Note: the variable X represents the number of face-up cards in the tall pile.

Additional ways to perform the trick

Variation 1: Turn your back to the table. Ask the audience to split the cards into the two piles of seven and five cards. Then have them cover the piles with a large napkin. When you turn around, place your hands under the napkin and announce that you will rearrange the cards so both will have the same number of face-up cards. You complete the trick simply by turning over the short pile.

Variation 2: Turn your back to the table. Ask the audience to split the cards into the two piles of seven and five cards. Turn around and hold your hands directly over the two piles. Of course, you can see the top card of each pile but not the others. Announce you “sense” the orientation of the cards underneath the top cards. After a few seconds, turn your back to the table again. Ask your audience to exchange the bottom two cards of each pile without changing their orientation (actually, you can ask them to exchange any number of ways between the two piles – for example, exchange the bottom three cards of the short pile with the top three cards of the tall pile). Next ask them to flip the short pile over. Turn around and confirm with your audience that the two piles now have the same number of face-up cards.

Variation 3: Start with a different number of cards in total. To make the “trick” work, the number of starting face-up cards must match the size of the short pile. Use a table like the one above to make sure the trick will work for the number of cards you select.

Variation 4: Use coins instead of cards. Instead of face-up and face-down, use the heads and tails side of the coins for the trick. This works a little better using the napkin cover; however, some audience members may suspect that you are feeling each coin to determine their orientation.