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Mathematics, Magic and Mystery

Mathematics, Magic and Mystery

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by Martin Gardner

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Why do card tricks work? How can magicians do astonishing feats of mathematics mentally? Why do stage "mind-reading" tricks work? As a rule, we simply accept these tricks and "magic" without recognizing that they are really demonstrations of strict laws based on probability, sets, number theory, topology, and other branches of mathematics.


Why do card tricks work? How can magicians do astonishing feats of mathematics mentally? Why do stage "mind-reading" tricks work? As a rule, we simply accept these tricks and "magic" without recognizing that they are really demonstrations of strict laws based on probability, sets, number theory, topology, and other branches of mathematics.
This is the first book-length study of this fascinating branch of recreational mathematics. Written by one of the foremost experts on mathematical magic, it employs considerable historical data to summarize all previous work in this field. It is also a creative examination of laws and their exemplification, with scores of new tricks, insights, and demonstrations. Dozens of topological tricks are explained, and dozens of manipulation tricks are aligned with mathematical law.
Nontechnical, detailed, and clear, this volume contains 115 sections discussing tricks with cards, dice, coins, etc.; topological tricks with handkerchiefs, cards, etc.; geometrical vanishing effects; demonstrations with pure numbers; and dozens of other topics. You will learn how a Moebius strip works and how a Curry square can "prove" that the whole is not equal to the sum of its parts.
No skill at sleight of hand is needed to perform the more than 500 tricks described because mathematics guarantees their success. Detailed examination of laws and their application permits you to create your own problems and effects.

Product Details

Dover Publications
Publication date:
Dover Recreational Math Series
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Product dimensions:
5.41(w) x 7.99(h) x 0.43(d)
1110L (what's this?)
Age Range:
12 Years

Read an Excerpt

Mathematics, Magic and Mystery

By Martin Gardner

Dover Publications, Inc.

Copyright © 1956 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-80117-9


Tricks with Cards—Part One

Playing cards possess five basic features that can be exploited in devising card tricks of a mathematical character.

(1) They can be used as counting units, without reference to their faces, as one might use pebbles, matches, or pieces of paper.

(2) The faces have numerical values from one to thirteen (considering the jack, queen, and king as 11, 12, and 13 respectively).

(3) They are divided into four suits, as well as into red and black cards.

(4) Each card has a front and back.

(5) Their compactness and uniform size make it easy to arrange them into various types of series and sets, and conversely, arrangements can be destroyed quickly by shuffling.

Because of this richness of appropriate properties, mathematical card tricks are undoubtedly as old as playing cards themselves. Although cards were used for gaming purposes in ancient Egypt, it was not until the fourteenth century that decks could be made from linen paper, and not until the early fifteenth century that card-playing became widespread in Europe. Tricks with cards were not recorded until the seventeenth century and books dealing entirely with card magic did not appear until the nineteenth. As far as I am aware, no book has yet been written dealing exclusively with card tricks based on mathematical principles.

The earliest discussion of card magic by a mathematician seems to be in PROBLÈMES PLAISANS ET DÉLECTABLES, by Claud Gaspard Bachet, a recreational work published in France in 1612. Since then, references to card tricks have appeared in many books dealing with mathematical recreations.

The Curiosities of Peirce

The first, perhaps the only, philosopher of eminence to consider a subject matter so trivial as card magic, was the American logician and father of pragmatism, Charles Peirce. In one of his papers (see THE COLLECTED PAPERS OF CHARLES SANDERS PEIRCE, 1931, Vol. 4, p. 473f) he confesses that in 1860 he "cooked up" a number of unusual card effects based on what he calls "cyclic arithmetic." Two of these tricks he describes in detail under the titles of "First Curiosity" and "Second Curiosity." To a modern magician, the tricks are curiosities in a sense unintended by Peirce.

The "First Curiosity," based on one of Fermat's theorems, requires thirteen pages for the mere description of how to perform it and fifty-two additional pages to explain why it works! Although Peirce writes that he performed this trick "with the uniform result of interesting and surprising all the company," the climax is so weak in proportion to the complexity of preparation that it is difficult to believe that Peirce's audiences were not half-asleep before the trick terminated.

About the turn of the present century card magic experienced an unprecedented growth. Most of it was concerned with the invention of "sleights" (ways of manipulating cards secretly), but the development also saw the appearance of hundreds of new tricks that depended wholly or in part on mathematical principles for their operation. Since 1900 card magic has steadily improved, and at present there are innumerable mathematical tricks that are not only ingenious but also highly entertaining.

One illustration will indicate how the principle of an old trick has been transformed in such a way as to increase enormously its entertainment value. W. W. Rouse Ball, in his MATHEMATICAL RECREATIONS, 1892, describes the following effect:

Sixteen cards are placed face up on the table, in the form of a square with four cards on each side. Someone is asked to select any card in his mind and tell the performer which of the four vertical rows his card is in. The cards are now gathered by scooping up each vertical row and placing the cards in the left hand. Once more the cards are dealt to the table to form a square. This dealing is by horizontal rows, so that after the square is completed, the rows which were vertical before are now horizontal. The performer must remember which of these rows contains the chosen card.

Once more the spectator is asked to state in which vertical row he sees his card. The intersection of this row with the horizontal row known to contain the card will naturally enable the magician to point to the card instantly. The success of the trick depends, of course, on the spectator's inability to follow the procedure well enough to guess the operating principle. Unfortunately, few spectators are that dense.

The Five Poker Hands

Here is how the same principle is used in a modern card effect:

The magician is seated at a table with four spectators. He deals five hands of five cards each. Each person is asked to pick up his hand and mentally select one card among the five. The hands are gathered up and once more dealt around the table to form five piles of cards. The magician picks up any designated pile and fans it so that the faces are toward the spectators. He asks if anyone sees his selected card. If so, the magician (without looking at the cards) immediately pulls the chosen card from the fan. This is repeated with each hand until all the selected cards are discovered. In some hands there may be no chosen cards at all. Other hands may have two or more. In all cases, however, the performer finds the cards instantly.

The working is simple. The hands are gathered face down, beginning with the first spectator on the left and going around the table, the magician's own hand going on top of the other four. The cards are then redealt. Any hand may now be picked up and fanned. If spectator number two sees his selected card, then that card will be in the second position from the top of the fan. If the fourth spectator sees his card, it will be the fourth in the hand. In other words, the position of the chosen card will correspond to the number of the spectator, counting from left to right around the table. The same rule applies to each of the five hands.

A little reflection and you will see that the principle of intersecting sets is involved in this version exactly as in the older form. But the newer handling serves better to conceal the method and also adds considerably to the dramatic effect. The working is so simple that the trick can be performed even while blindfolded, a method of presentation that elevates the trick to the rank of first-class parlor magic.

In the pages to follow we shall consider representative samples of modern mathematical card tricks. The field is much too vast to permit an exhaustive survey, so I have chosen only the more unusual and entertaining effects, with a view to illustrating the wide variety of mathematical principles that are employed. Although most of these tricks are known to card magicians, very few of them have found their way into the literature of mathematical recreations.


Under this heading we shall consider only the type of trick in which cards are used as units, without combination with other properties of the deck. Any collection of small objects, such as coins, pebbles, or matches could be similarly employed, but the compactness of cards makes them easier than most objects to handle and count.

The Piano Trick

The magician asks someone to place his hands palm down on the table. A pair of cards is inserted between each two adjacent fingers (including the thumbs), with the exception of the fourth and fifth fingers of the left hand. Between these fingers the magician places a single card.

The first pair on the magician's left is removed, the cards separated, and placed side by side on the table. The next pair is treated likewise, the cards going on top of the first two. This is continued with all the pairs, forming two piles of cards on the table.

The magician picks up the remaining single card and asks, "To which pile shall I add this odd card?" We will assume that the pile on the left is designated. The card is dropped on this pile.

The performer announces that he will cause this single card to travel magically from the left pile to the pile on the right. The left pile is picked up and the cards dealt off by pairs. The pairs come out even, with no card left over. The right hand pile is picked up and the cards taken off by pairs as before. After all the pairs are dealt, a single card remains!

Method: The working is due to the fact that there are seven pairs of cards. When these pairs are separated, each pile will contain seven cards—an odd number. Adding the extra card, therefore, makes this an even pile. If the cards are dealt by pairs, without counting them aloud, no one will notice that one pile contains one more pair than the other.

This trick is at least fifty years old. It is known as "The Piano Trick" because the position of the spectator's hands suggests playing a piano.

The Estimated Cut

The performer asks someone to cut a small packet of cards from the deck. He then cuts a larger packet for himself. The magician counts his cards. We will assume he has twenty. He now announces, "I have as many cards as you have, plus four cards, and enough left to make 16." The spectator counts his cards. Let us say he has eleven. The magician deals his cards to the table, counting up to eleven. He then places four cards aside, according to his statement, and continues dealing, counting 12, 13, 14, 15, 16. The sixteenth card is the last one, as he predicted.

The trick is repeated over and over, though the prediction varies each time in the number of cards to be placed aside—sometimes three, sometimes five, and so on. It seems impossible for the magician to make his prediction without knowing the number of cards taken by the spectator.

Method: It is not necessary for the performer to know the number held by the spectator. He simply makes sure that he takes more cards than the other person. He counts his cards. In the example given, he has twenty. He then arbitrarily picks a small number such as 4, subtracting it from 20 to get 16. The statement is worded, "I have as many cards as you have, plus four extra cards, and enough left to make sixteen." The cards are counted, as previously explained, and the statement proves to be correct.

The method of counting seems to involve the spectator's number, though actually the magician is simply counting his own cards, with the exception of the four which are placed aside. Varying the number to be set aside each time serves to impress the spectator with the fact that somehow the formula is dependent on the number of cards he is holding.


Findley's Four-Card Trick

A deck of cards is shuffled by the audience. The magician places it in his pocket and asks someone to call out any card that comes to mind. For example, the Queen of Spades is named. He reaches into his pocket and removes a spade. This, he explains, indicates the suit of the chosen card. He then removes a four and an eight which together total 12, the numerical value of the queen.

Method: Previous to showing the trick, the magician removes from the deck the Ace of Clubs, Two of Hearts, Four of Spades, and Eight of Diamonds. He places these four cards in his pocket, remembering their order. The shuffled pack is later placed in the pocket beneath these four cards so that they become the top cards of the pack. The audience, of course, is not aware of the fact that four cards are in the magician's pocket while the deck is being shuffled.

Because the four cards are in a doubling series, each value twice the previous one, it is possible to combine them in various ways to produce any desired total from 1 to 15. Moreover, each suit is represented by a card.

The card with the appropriate suit is removed from the pocket first. If this card is also involved in the combination of cards used to give the desired total, then the additional card or cards are removed and the values of all the cards are added. Otherwise, the first card is tossed aside and the card or cards totaling the desired number are then taken from the pocket. As we shall see in later chapters, the doubling principle involved in this trick is one that is used in many other mathematical magic effects.

Occasionally one of the four cards will be named. In this case the magician takes the card itself from his pocket—a seeming miracle! The trick is the invention of Arthur Findley, New York City.

A Baffling Prediction

A spectator shuffles the deck and places it on the table. The magician writes the name of a card on a piece of paper and places it face down without letting anyone see what has been written.

Twelve cards are now dealt to the table, face down. The spectator is asked to touch any four. The touched cards are turned face up. The remaining cards are gathered and returned to the bottom of the pack.

We will assume the four face-up cards to be a three, six, ten, and king. The magician states that he will deal cards on top of each of the four, dealing enough cards to bring the total of each pile up to ten. For example, he deals seven cards on the three, counting "4, 5, 6, 7, 8, 9, 10." Four cards are dealt on the six. No cards are dealt on the ten. Each court card counts as ten, so no cards are placed on the king.

The values of the four cards are now added: 3, 6, 10, and 10 equals 29. The spectator is handed the pack and asked to count to the 29th card. This card is turned over. The magician's prediction is now read. It is, of course, the name of the chosen card.

Method: After the deck is shuffled the magician casually notes the bottom card of the pack. It is the name of this card that he writes as his prediction. The rest works automatically. Gathering the eight cards and placing them on the bottom of the pack places the glimpsed card at the 40th position. After the cards are properly dealt, and the four face-up cards totaled, the count will invariably fall on this card. The fact that the deck is shuffled at the outset makes the trick particularly baffling.

It is interesting to note that in this trick, as well as in others based on the same principle, you may permit the spectator to assign any value, from 1 to 10, to the jacks, kings, and queens. For example, he may decide to call each jack a 3, each queen a 7, and each king a 4. This has no effect whatever on the working of the trick, but it serves to make it more mysterious. Actually, the trick requires only that the deck consist of 52 cards—it matters not in the least what these cards are. If they were all deuces the trick would work just as well. This means that a spectator can arbitrarily assign a new value to any card he wishes without affecting the success of the trick!

Further mystification may be added by stealing two cards from the pack before showing the trick. In this case ten cards are dealt on the table instead of twelve. After the trick is over, the two cards are secretly returned to the pack. Now if a spectator tries to repeat the trick exactly as he saw it, it will not work.

Henry Christ's Improvement

A few years ago Henry Christ, New York City amateur magician, made a sensational improvement on this effect. As in the original version, the count ends on the card that is ninth from the bottom of the deck. Instead of predicting this card, however, the spectator is allowed to select a card which is then brought to the desired position in the following manner. After the pack has been shuffled, the magician deals nine cards in a face-down heap on the table. A spectator selects one of these cards, notes its face, then returns it to the top of the heap. The deck is replaced on the pile, thus bringing the chosen card to the position of ninth from the bottom.

The spectator now takes the pack and starts dealing the cards one at a time into a face-up pile, at the same time counting aloud and backward from 10 to 1. If he by chance deals a card that corresponds with the number he is calling (for example, a four when he counts 4), then he stops dealing on that pile and starts a new pile next to it. If there is no coincidence of card and number by the time he reaches the count of 1, the pile is "killed" by covering it with a face-down card taken from the top of the deck.

Four piles are formed in this manner. The exposed top cards of the piles which have not been "killed" are now added. When the spectator counts to that number in the deck, he will end the count on his selected card. This is a much more effective handling than the older version because the selection of the cards to be added seems to be completely random and the compensation principle involved is more deeply hidden. The trick was first described in print by John Scarne as trick No. 30 in his book SCARNE ON CARD TRICKS, 1950. (See also Trick No. 63 for a slightly different handling by the Chicago magician Bert Allerton.)


Excerpted from Mathematics, Magic and Mystery by Martin Gardner. Copyright © 1956 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Meet the Author

Martin Gardner was a renowned author who published over 70 books on subjects from science and math to poetry and religion. He also had a lifelong passion for magic tricks and puzzles. Well known for his mathematical games column in Scientific American and his "Trick of the Month" in Physics Teacher magazine, Gardner attracted a loyal following with his intelligence, wit, and imagination.

Martin Gardner: A Remembrance
The worldwide mathematical community was saddened by the death of Martin Gardner on May 22, 2010. Martin was 95 years old when he died, and had written 70 or 80 books during his long lifetime as an author. Martin's first Dover books were published in 1956 and 1957: Mathematics, Magic and Mystery, one of the first popular books on the intellectual excitement of mathematics to reach a wide audience, and Fads and Fallacies in the Name of Science, certainly one of the first popular books to cast a devastatingly skeptical eye on the claims of pseudoscience and the many guises in which the modern world has given rise to it. Both of these pioneering books are still in print with Dover today along with more than a dozen other titles of Martin's books. They run the gamut from his elementary Codes, Ciphers and Secret Writing, which has been enjoyed by generations of younger readers since the 1980s, to the more demanding The New Ambidextrous Universe: Symmetry and Asymmetry from Mirror Reflections to Superstrings, which Dover published in its final revised form in 2005.

To those of us who have been associated with Dover for a long time, however, Martin was more than an author, albeit a remarkably popular and successful one. As a member of the small group of long-time advisors and consultants, which included NYU's Morris Kline in mathematics, Harvard's I. Bernard Cohen in the history of science, and MIT's J. P. Den Hartog in engineering, Martin's advice and editorial suggestions in the formative 1950s helped to define the Dover publishing program and give it the point of view which — despite many changes, new directions, and the consequences of evolution — continues to be operative today.

In the Author's Own Words:
"Politicians, real-estate agents, used-car salesmen, and advertising copy-writers are expected to stretch facts in self-serving directions, but scientists who falsify their results are regarded by their peers as committing an inexcusable crime. Yet the sad fact is that the history of science swarms with cases of outright fakery and instances of scientists who unconsciously distorted their work by seeing it through lenses of passionately held beliefs."

"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?" — Martin Gardner

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