## Mathematical Tricks With Cards

"Do you like card tricks?" "No, I hate card tricks," I answered. "Well, I'll just show you this one." fíe showed me three.

—Somerset Maugham, Mr. Know-Ail

Maugham's experience with card magicians is all too familiar. "I don't really like people who do card tricks," Elsa Maxwell once wrote (I quote from an autobiography of a lady magician, You Don't Have to Be Crazy, by Frances Ireland). "They never stop at one or two, but go on and on and on, and always make you take cards, or turn up cards, or cover cards, until you are worn out."

Mathematical card tricks, let it be admitted at once, are precisely the kind of tricks that are the most boring to most people. Nevertheless, they have a curious appeal to mathematicians and mathematically minded magicians.

Many excellent card deceptions are based on a parity principle, but the underlying even-odd structure is usually concealed so ingeniously that if you follow the directions with cards in hand you are likely to astonish yourself. Consider the following trick invented about 1946 by the Chicago card expert Ed Mario. Magicians classify it as an "oil and water" effect, for reasons that will be apparent in a moment. There are many ways of achieving the same effect by secret and difficult "moves," but this version is entirely self-working.

Remove 10 red and 10 black cards from the deck and arrange them in two face-up piles, side by side, with all red cards. on the left and all black cards on the right. First you tell your watchers that you will demonstrate what you intend to do by using only five cards of each color. With both hands simultaneously remove the top card from each pile and place them, still face up, on the table at the bottom of each pile. Do the same with the next two top cards, but this time cross your arms before you place the two cards on the two new piles you are starting. This puts a black card on the red one and a red card on the black one. The next transfer of a pair of cards is made with uncrossed arms, the next with crossed arms, and the fifth and last pair is dealt with arms uncrossed. In other words, five simultaneous deals are made, with arms crossed only on alternate deals. On each side you now have a pile of five face-up cards with their colors alternating. Put either pile on the other one. Spread the 10 cards to show that colors alternate throughout.

Square the cards and turn the packet face down. From its top deal the cards singly and face up to form two piles again, dealing alternately to the left and right. Call attention to the fact that this procedure naturally separates the colors. At the finish you will have five reds on the left and five blacks on the right.

State that you will repeat this simple series of operations with all 20 cards. Begin as before, with 10 face-up reds on the left and 10 face-up blacks on the right. Transfer the cards to form two new piles, just as you did before, crossing your arms on alternate deals so that the colors alternate in each pile. After all 20 cards are dealt put one pile on the other, square the cards, turn the packet over and hold it face down in your left hand.

Deal 10 cards face up to form two piles, dealing from left to right and observing aloud that this brings the reds together on the left and the blacks together on the right. After the 10 cards have been dealt face up do not pause but continue smoothly and deal the remaining 10 cards face down. It is best to put down the cards so that they overlap in two vertical rows [see Figure 124],

Pick up the five face-down cards on the left with your left hand and the five face-down cards on the right with your right hand. Cross your arms and put the cards down. You explain that you have transferred half of the cards of each pile to the pile of the opposite color but that like oil and water the colors mysteriously refuse to mix. Turn over the face-down cards. To everyone's surprise (you hope) the reds are back with the reds and the blacks are back with the blacks! Readers should have little difficulty discovering why it works with any set of cards

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containing an even number of cards of each color and why it did not work when you demonstrated it with 10 cards.

After you have finished the oil-and-water trick put the two piles together with either color on top. Turn the packet face down and spread it in a fan. You are ready to perform a red-black trick invented by Karl Fulves and published in his magic periodical, The Pallbearers Review, September, 1971.

Ask someone to pull slightly forward any 10 cards he pleases. The fan will resemble the one shown in Figure 125. With your right hand count the jogged (protruding) cards to make sure there are 10. Do this by removing the cards one at a time from right to left, putting them into a face-down pile as you count from one to 10. Close up the 10 cards remaining in your left hand and place them in a second face-down pile alongside the first.

Figure 125

Figure 125

Tell your audience that an amazing thing has happened. Although 10 cards were selected randomly, the colors in the two piles are so ordered that every nth card in one pile has a color opposite to the color of the nth card in the other pile. To prove this, turn over the top cards of each pile simultaneously. One will be red and the other black. Place the black under the red, turn the pair over and put it aside to form a new face-down pile. Repeat the procedure with the cards now on top of the two original piles. They will be red-black too. Indeed, every pair you turn will be red-black!

As you show the pairs always put the black card under the red before you turn them over and place them on the third pile. When you finish, the cards in this face-down pile will have alternating colors.

Now you are ready to perform a truly mystifying trick in which parity is conserved in spite of repeated shuffling. Known as Color Scheme, it was invented by Oscar Weigle, an amateur magician who is now an editor at Grosset & Dunlap. It sold as a manuscript in magic stores in 1949.

Give the packet of 20 cards to someone and ask him to hold it under the table where neither he nor anyone else can see the cards. Tell him to mix the cards by the following procedure. (It is known as the Hummer shuffle, after Bob Hummer, the magician who first used it in tricks.) Turn over the top two cards (not one at a time but both together as if they were one card), place them on top and cut the packet. Your assistant is to keep repeating this procedure of turn two, cut, turn two, cut for as long as he wishes. The procedure will, of course, result in a packet containing an unknown number of randomly distributed reversed cards.

With the cards still held under the table, tell your assistant to do the following. Shift the top card to the bottom. Then turn over the next card, produce it from under the table and place it on the table. This procedure is repeated—card to bottom, reverse next card and deal—until 10 cards have been dealt to the table. It will be apparent that the cards have become mysteriously ordered. All the face-up cards are the same color and all the face-down ones are of the opposite color.

The second and climactic half of the trick, which Weigle confesses is a "bare-faced swindle," now unfolds. Your assistant is still holding 10 cards under the table. Ask him to shuffle them by separating them into two packets; then, keeping all the cards flat (no card must be allowed to turn over), weave the two packets into each other in a completely random way. You can demonstrate how to do this by using the 10 cards already dealt. After your assistant has executed the shuffle a few times, ask him to turn over the packet and shuffle the same way a few more times. If he likes, he can give the packet a final cut.

Now he continues with the dealing procedure he used before: card to bottom, next card reversed and dealt. (The final card is reversed and dealt.) In spite of the thorough mixing the result is exactly the same as before. All the face-up cards match the former face-up cards in color, and the same is true of all face-down cards.

One of the oldest themes in card magic is to produce in some startling fashion a card that has been randomly selected and replaced. Here is a simple method that exploits a binary sorting technique. Fulves published it in his periodical in November, 1970.

Take 16 cards from a shuffled deck and spread them face down on the table without mentioning how many cards you are using. A viewer selects a card, looks at it and places it on top of the deck. The remaining cards in the spread are squared and put on top of the deck above the chosen card. Ask him to cut off about half of the deck, give or take half a dozen cards. Actually he can take between 16 and 32 cards. He hands this packet to you.

Hold the packet in both hands. As your left thumb slides the cards one at a time to the right, move your right hand forward and back so that every other card, starting with the first one, is jogged forward. The resulting fan of cards will resemble the one in Figure 125 except that the jogged cards are not randomly distributed. Strip all the projecting cards from the fan and discard them. Square the remaining cards and repeat the procedure, jogging forward all the cards at odd positions, starting with the first card. Strip them out and discard. Continue in this way until one card is left. Before turning it over ask for the chosen card's name. It will be the card you hold.

A completely different method of locating a selected card can be found in several books on card magic. Turn your back and instruct someone to cut a shuffled deck into three approximately equal piles. He turns over any pile and then reassembles the deck by sandwiching the face-up pile between the other two, which remain face down. He is told to remember the top card of the face-up pile. With your back still turned, ask him to cut the deck several times, then give it one thorough riffle-shuffle. The shuffle will of course distribute the face-up cards randomly throughout the deck.

Turn around, reverse the pack and spread it in a row. Look for a long run of face-up cards, remembering that a cut may have split the run so that part of it is at each end of the spread. The first face-down card above the run is the chosen one. Slide it from the spread, have the card named and then turn it over.

Our last trick, based on a curious shuffling principle discovered by Fulves, is presented as a gambling proposition. All cards of one suit (the suit can be chosen by the victim) are removed from the deck. Assume that the discarded suit is diamonds. The remaining cards are arranged so that each triplet has three different suits in the same order. (Card values are ignored.) Again the victim may specify the ordering. Suppose he chooses spades, hearts and clubs. The 39-card deck is arranged from the top down so that the suits follow the sequence spades, hearts, clubs, spades, hearts, clubs and so on.

Place the deck face up in front of the victim. Ask him to cut it in two packets and riffle-shuffle them together. As he makes the cut, note the suit exposed on top of the lower half. We shall call this suit k. After the single shuffle the deck is turned face down. The cards are now taken from the top three cards at a time, and each triplet is checked to see if it contains two cards of the same suit.

It is hard to believe, but:

(1) If K is spades, no triplet will contain two spades.

(2) If k is hearts, no triplet will contain two clubs.

(3) If k is clubs, no triplet will contain two hearts.

This assumes, of course, a spades-hearts-clubs ordering. If the ordering is otherwise, the three rules must be modified accordingly; that is, spades must be changed to whatever suit is at the top of each triplet, and so on. Let m stand for the suit that you know cannot show twice in any triplet, and a and b for the suits that can.

Before dealing through the deck to inspect the triplets, make the following betting proposition. For every triplet containing a pair of m's you will pay the victim $10. In return he must agree to pay you 10 cents for every pair of a's or b's. It seems like a good bet for the victim, but it is impossible for you to lose, and the swindle can be repeated as often as you please. Just arrange the cards again and allow the victim to make the single riffle-shuffle. Naturally you always promise to pay him for doublets of the suit that you know cannot show. The fact that this suit may vary from deal to deal makes the bet particularly mystifying.

As Fulves has observed, the triplets have other unexpected properties. Of the triplets containing pairs the a's and b's will alternate; after a pair of as the next pair will be b's and vice versa. Pairs of one suit always include a top card of the triplet. Pairs of the other suit always include a bottom card.

No explanation of these tricks will be given. Readers will find it stimulating, however, to analyze each trick to see if they can comprehend exactly why it operates with such uncanny precision.

### ADDENDUM

Peter T. Sarjeant extended Fulves' shuffling trick to the four suits of a full deck. Arrange the cards so that from top down the sequence is a repetition of clubs, diamonds, hearts, spades. As before, the deck is placed face up and cut about in half. Note the suit on the top of the bottom half. Call it k. The halves are then interlaced with a single riffle-shuffle.

When cards are taken four at a time from the top you will find the following true of each quadruplet:

(1) If k is clubs, there will be no pair of hearts and no pair of clubs.

(2) If k is diamonds, any suit may be paired.

(3) If k is hearts, there will be no pair of diamonds and no pair of spades.

Knowledge of these facts can, of course, be the basis of a variety of betting swindles.

Edward M. Cohen proposed the following variation of Fulves' trick involving a selected card that goes sixteenth from the top of the deck. He likes to begin by forming a square array of 16 cards, face down on the table. A spectator picks a row. Another person picks a column. The card at the intersection is turned face up and remembered. This card goes to the bottom of the deck. The remaining 15 cards are swept into a pile and the deck placed on top of them. The chosen card is now sixteenth from the bottom.

Anyone may now cut the deck about in half (it is only necessary that the lower portion contain more than 16 and less than 32 cards). The top half is discarded. Hand the lower half to someone with the request that he deal it into two piles, alternating piles as he deals. The pile that gets the last card is discarded. This procedure is repeated until only one card remains. It will be the chosen card.

Hundreds of more elaborate card tricks have been based on the binary principles that underlie this trick, but the one just described is as simple, effective, and as easy to perform as any.

### BIBLIOGRAPHY

Scarne on Card Tricks. John Scarne. Crown, 1950 Mathematics, Magic and Mystery. Martin Gardner. Dover, 1956. Mathematical Magic. William Simon. Scribner's, 1964. Self-Working Card Tricks. Karl Fulves. Dover, 1976.

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