Are you psychic?

The magic effect

You select a spectator who believes they have no psychic powers, and proceed to test their abilities. You have a pack containing the ace to five of Hearts, and the ace to five of Spades. The spectator mixes the cards and then you deal two piles of five. The challenge to the spectator is to choose pairs of unseen cards to eliminate, and they have to use whatever psychic powers they have to make sure the last two cards left match. For example, if they are really psychic they might be left with both threes: the red three of hearts and the black three of spades. After giving then completely free choices you show that, to their astonishment, they are psychic and the last two cards match. As their jaw drops you show them that their powers are even more potent then you expected: it turns out every pair of cards they eliminated matches in value too.

The mechanics

Before you start, set up the cards with the red 1-5 cards and black 1-5 cards in a single pile in that order. Spread the cards face down in your hand and have the spectator point to the back of a card of their choice. Cut the deck with a straight cut at that point. That just means split the pack at that point and place the top pile to the bottom of the pack underneath the others. Repeat doing this until the spectator is happy the cards are well mixed. Now you do the (oh so simple) secret move. Deal the top five cards, one at a time, into a pile on the table, thereby reversing their order. Place the remaining un-dealt cards in a second pile beside them. By putting this second pile straight down you have kept their order the same.

You explain to your spectator that as there are 5 cards in each pile you will give them 4 chances to use their psychic powers. They can have 4 swaps. A swap involves taking the top card on one of the piles and placing it on the bottom of the same pile. Explain that they can, for example, do all 4 swaps on one pile, 2 on each, or 3 on one and one on the other. It is their choice, remembering that their aim is to be left with two matching cards.

Once the 4 swaps are made remove the top card on each pile and place them aside. Point out that it does not matter what they are as they are being discarded. Now there are 4 cards in each pile. Offer the spectator 3 swaps in total, and once the swaps are done remove the top two cards from the piles. There are now three cards left in each pile, so give them two swaps this time, and again remove the top card from both piles. This leaves two cards in each pile. This is their final chance to get it right. One swap is left, and one card can make all the difference. They choose their swap, and the top two cards from each pile are discarded.

Now it's time to reveal the final two single cards left on the table. They match. The volunteer chose freely which cards to eliminate, so it was their secret psychic powers that came through to ensure a match at the end. Give their jaw time to drop, then dramatically reveal that all the pairs of cards they removed match in value too.

Prove it works!

Believe it or not, follow the instructions above and the trick works automatically. Let’s see why.

At the start we have the first pile of cards in order 1234512345. Cutting these cards with a single cut doesn’t disturb the cyclic order of the cards, so for example a cut between 2 and 3 may move the card order to 3451234512, but the 12345 and 12345 of the original is still there if we consider the stacked pack as being circular, with the cards being positions on a clock face. When we count round we will still get 12345 followed by 12345, they just start at different 'times' on the clock face.

For simplicity let's assume that the cards are in the original order. Our secret move, counting five cards and reversing their order gives us 12345 in one pile and 54321 in the other. This is called a palindromic stack. The values in one pile are exactly reversed in the other. To make it even simpler to explain, consider only 3 card values – so our piles are 123 and 321.

Now make 2 swaps. Remember the number of swaps is always one less than the number of cards in the piles. Swaps just rotate the cards. Think of that clock face, being on a dial you can rotate with the 12 o’clock position being the top card and the other two cards being at 4 o’clock and 8 o’clock (the bottom of the pack). See the diagram. We rotate the dial by the number of swaps. Rotating one less than the number of cards just means we rotate the bottom card up to the top: to 12 o’clock. If we do some of the swaps on one pile and some on the other, we are just rotating the numbers on the dials in opposite directions so that they meet. The mathematics behind this trick is called modular arithmetic...not surprisingly also known as clock arithmetic.

Let's look at all the possibilities to see how that happens.

If we do two swaps on the first pile, that takes the top card to bottom and takes us to 312. The other pile is unchanged with order 321. The top card of both piles is the 3 as we require. Remove it and the piles left are 12 and 21 – they still have the palindromic property so it will work again for the next round.

What happens if we do two swaps, one on each pile? The first pile goes from 123 to 231 and the second (reversed) pile from 321 to 213 and the top cards match again. We are left with piles of 31 and 13. The orders are still reversed so the next round will work again.

Finally, if we do two swaps on the second pile, that takes the top card to bottom and takes us from 321 to 132. The other pile is unchanged with order 123. The top card of both piles is 1 this time. The piles left are 32 and 23. Yet again they still have the palindromic property.

This mathematics works for any number of cards N that are placed in order in palindromic stacks. When we make N-1 swaps, as one card is moved down its equivalent rises in the other stack so that the ones meeting on top of the piles match.

The Showmanship

You can play up the psychic link by using Zener cards in place of normal cards. Rather than suits and numbers the cards each have one of five symbols: a circle, cross, wavy lines, square and star. These can be bought from magic shops or you could make your own (just google 'Zener' to get the images). They were actually invented by psychologist Karl Zener as a way to do rigorous experiments on extra-sensory perception. In a typical experiment the experimenter would turn over cards, writing the symbol down, and the subject would try and work out, using clairvoyance, what the card was. Sometimes to remove the possibility of bias or subtle forms of communication, the subject would be in a different room to the experimenter. Explain this and that you are going to do a variation to see if the subject can match two cards. The extra detail adds authenticity to your performance, casting you in the role of experimenter rather than of magician.

As an alternative, you can cast the trick as about 'psychic resonance and the power of crystals' demonstration. Give the volunteer a large 'crystal' (eg a piece of quartz – aquarium stockists sell it) to hold to 'magnify their psychic resonance'.