Martin Gardner Mathematical Wonders and Mysteries. Martin gardner
Like many other subjects at the intersection of two disciplines, mathematical tricks are not used special attention neither mathematicians nor magicians. The first tend to regard them as empty fun, the second neglect them as too boring. Mathematical tricks, to put it bluntly, do not belong to the category of tricks that can keep an audience of non-mathematicians spellbound; such tricks usually take a lot of time, and they are not very effective; on the other hand, there is hardly a person who is going to draw deep mathematical truths from their contemplation.
And yet, mathematical tricks, like chess, have their own special charm. Chess combines the elegance of mathematical construction with the pleasure that the game can deliver. In mathematical tricks, the elegance of mathematical constructions is combined with amusement. It is not surprising, therefore, that they bring the greatest pleasure to those who are simultaneously familiar with both of these areas.
The present book, to the best of my knowledge, is the first attempt at a survey of the entire field of contemporary mathematical focus. Much of the material in this book is taken from specialized magic literature, not from entertaining mathematical literature. For this reason, students of recreational mathematical literature, but unfamiliar with the modern specialized literature on tricks, are likely to encounter in this book a new field of recreational knowledge - a rich new field, the existence of which they may have been completely unaware of.
Editor's preface to the Russian edition
From the author's preface
Chapter first
MATHEMATICAL TRICKS WITH CARDS
Five piles of cards (9).
Cards as units of account. Guessing the number of cards taken from the deck (10). Using the Numerical Values of the Cards
Focus with four cards (11). Amazing prediction (12). Focus with the intended card (13). Cyclic number (14). Missing card (15).
Tricks based on differences in colors and suits Trick with kings and queens (19). Use of the front and back sides of the cards. Comparison of the number of cards of black and red suit (20). Trick with flipping cards (20).
Tricks depending on the initial arrangement of cards in the deck
Trick with four aces (21). "Manhattan Wonders" (22). How many cards are transferred? (22). Focus with finding the card (23).
Chapter Two
FOCUSES WITH SMALL OBJECTS
Dice
Guessing the sum (25). Guessing the dropped number of points (27).
Domino Chain with a gap (27). A row of thirteen bones (28).
Calendars. Mysterious squares (29). Focus with marked dates (29). Prediction (30).
Watch. Guessing the planned number on the dial (31). Focus with clock and dice (32).
Matches. Three piles of matches (33). How many matches are clenched in a fist?
(34). Who took what? (34). Coins Mysterious Nine (36). Which hand is the coin in? (36). Coat of arms or "bar" (37). Chess board. Focus with three checkers (38) Small objects. Focus with three objects (39). Trick with guessing one of the four objects (40).
Chapter Three
TOPOLOGICAL PUZZLE
Paper rings (44).
Tricks with a handkerchief
Focus with finger cutting (48). Focus with interlocked handkerchiefs (50). The problem of tying knots (51).
Cords and twine
Tricks with cord or twine (52). Other cord tricks (56).
clothing
Mysterious loop (58). Turning the vest inside out (59). Removing the vest (60).
Rubber rings Jumping ring (60). Twisted ring (61).
Chapter Four
FOCUSES WITH SPECIAL EQUIPMENT
Cards with numbers (64). Cards with holes (65). Tricks with "touches"
Focus with six squares (66). Color map (67).
Think animal (69). Tricks with dice and dominoes 70. Trick with three-digit numbers (70). Domino trick box (70). Focus with chips (71).
Chapter Five
DISAPPEARANCE OF FIGURES. SECTION I
Paradox with lines (73). Facial disappearance (75). "Vanishing Warrior" (76). Lost rabbit (78).
Chapter six
DISAPPEARANCE OF FIGURES. SECTION II
The chessboard paradox (79). Paradox with area (81). Square variant (82). Fibonacci numbers (83).
Rectangle variant (85). Another variant of the paradox (87). Triangle variant (90). Four-piece squares (93). Three-piece squares (95). Two-piece squares (95). Curvilinear and 3D variants (96).
Chapter Seven
PUZZLE WITH NUMBERS
Quick cube root (98). Addition of Fibonacci numbers (100). Number prediction (101). Guessing the number (102). The Secret of the Nine (105). Digital roots (105). Digital root stability (107). Guessing age (108). Focus with addition (109). Focus with multiplication (109). Secret of the Seven (100). Sum prediction (112). "Psychological Moments" (114).
Editor's Notes
Gardner Martin
"MATHEMATICAL WONDERS AND MYSTERIES"
Editor's preface to the Russian edition
Before you is a regular square chess grid of 64 cells. Before your eyes, several cuts are made and a rectangle is made from the resulting parts, in which, however, there are only 63 cells!
You thought of a number - one of those that are written on the cards scattered on the table. Your partner touches the cards one by one with a pointer, and at the same time you spell the intended number to yourself, and when you get to the last letter, the pointer stops right at your number!
Focuses? Yes, if you like; or rather, experiments based on mathematics, on the properties of figures and numbers, and only clothed in a somewhat extravagant form. And to understand the essence of this or that experiment means to understand even a small but precise mathematical regularity.
It is with this hidden mathematics that Martin Gardner's book is interesting. Hidden - because for the most part the author himself does not formulate in the language of mathematics the patterns underlying his experiments, limiting himself to describing the actions of the showing, explicit and secret; but the reader, familiar with the elements of school algebra and geometry, will undoubtedly take pleasure in restoring the corresponding algebraic or geometric idea from the author's explanations. However, in separate, more interesting cases (marked by numbers with parentheses), we took the liberty of accompanying the author's presentation with small notes that reveal the mathematical essence of his constructions; these notes are placed at the end of the book.
Mathematical tricks are a very peculiar form of demonstrating mathematical patterns.
If in educational presentation they strive for the greatest possible disclosure of ideas, then here, in order to achieve efficiency and entertainment, on the contrary, they mask the essence of the matter as cunningly as possible. That is why, instead of abstract numbers, various objects or sets of objects associated with numbers are so often used: dominoes, matches, clocks, a calendar, coins, and even cards (of course, this use of cards has nothing to do with the senseless pastime of gamblers; as the author points out, here the cards are considered simply as identical objects that are convenient to count; the images on them do not play any role in this).
We hope that Gardner's book will be of interest to many readers: young participants and solo mathematical circles, adult "disorganized" lovers of mathematics, and perhaps one or another of the experiments described here will awaken a smile even in a serious scientist in a short moment of rest from big work.
G. E. Shilov
Like many other subjects that are at the intersection of two disciplines, mathematical tricks do not receive special attention from either mathematicians or magicians. The first tend to regard them as empty fun, the second neglect them as too boring. Mathematical tricks, to put it bluntly, do not belong to the category of tricks that can keep an audience of non-mathematicians spellbound; such tricks usually take a lot of time, and they are not very effective; on the other hand, there is hardly a person who is going to draw deep mathematical truths from their contemplation.
And yet, mathematical tricks, like chess, have their own special charm. Chess combines the elegance of mathematical construction with the pleasure that the game can deliver. In mathematical tricks, the elegance of mathematical constructions is combined with amusement. It is not surprising, therefore, that they bring the greatest pleasure to those who are simultaneously familiar with both of these areas.
The present book, to the best of my knowledge, is the first attempt at a survey of the entire field of contemporary mathematical focus. Much of the material in this book is taken from specialized magic literature, not from entertaining mathematical literature. For this reason, students of recreational mathematical literature, but unfamiliar with the modern specialized literature on tricks, are likely to encounter in this book a new field of recreational knowledge - a rich new field, the existence of which they may have been completely unaware of.
New York, 1955
Martin Gardner
Chapter first. MATHEMATICAL TRICKS WITH CARDS
Playing cards have some specific properties that can be used in the preparation of magic tricks of a mathematical nature. We indicate five such properties.
1. Cards can be considered simply as identical objects that are convenient to count; the images on them do not play any role.
With the same success one could use pebbles, matches or pieces of paper.
2. Cards can be assigned numerical values from 1 to 13, depending on what is shown on their front side (in this case, jack, queen and king are taken as 11, 12 and 13, respectively)).
3. They can be divided into four suits or into black and red cards.
4. Each card has a front and reverse side.
5. The cards are compact and uniform in size. This allows you to lay them out in various ways, grouping them in rows or making piles that can be easily upset right there by simply mixing the cards.
With so many possibilities, card tricks must have been around a long time ago, and one can say that mathematical card tricks are certainly as old as the game of cards itself.
Apparently, the earliest discussion of card tricks by a mathematician is found in Claude's entertaining book, Gaspard Basche ( Claud Gaspard Bachet"Problemes plaisants et delectables"), published in France in 1612. Subsequently, references to card tricks appeared in many books devoted to mathematical entertainment.
The first and perhaps the only philosopher who condescended to consider card tricks was the American Charles Peirce. In one of his articles, he admits that in 1860 he "concocted" several extraordinary card tricks based, using his terminology, on "cyclic arithmetic." He describes two such tricks in detail under the title "first curiosity" and "second curiosity."
The "First Curiosity" is based on Fermat's Theorem. It took 13 pages just to describe the way it was demonstrated, and an additional 52 pages were taken up with an explanation of its essence. Although Peirce reports the “continuous interest and amazement of the public” at his trick, the climactic effect of this trick seems so out of proportion to the complexity of the preparations that it is hard to believe that the audience did not fall asleep long before the end of his performance.
Here is an example of how, as a result of a modification in the way of demonstrating an old trick, its entertainment increased unusually.
Sixteen cards are laid out on the table face up in the form of a square with four cards in a row. Someone is invited to think of one card and tell the demonstrator in which vertical row it lies. Then the cards are collected with the right hand in vertical rows and sequentially stacked in the left hand. After that, the cards are again laid out in the form of a square sequentially along the horizontals; thus, cards that were originally laid out in the same vertical row now appear in the same horizontal row. The demonstrator needs to remember which of them contains the now conceived card. Next, the viewer is asked once again to indicate in which vertical row he sees his card. It is clear that after that the demonstrator can immediately indicate the intended card, which will lie at the intersection of the just named vertical row and the horizontal row in which, as is known, it should be located. The success of this trick, of course, depends on whether the spectator follows the procedure closely enough to recognize the essence of the matter.
Five piles of cards
And now we will tell how the same principle is used in another case.
The demonstrator sits down at the table along with four spectators. He deals five cards to everyone (including himself), invites everyone to look at them and think of one. Then he collects the cards, lays them out on the table in five piles and asks someone to point him to one of them. Then he takes this pile in his hands, opens the cards in a fan, facing the audience, and asks if any of them see the intended card. If so, then the one showing (without looking even once at the cards) immediately pulls it out. This procedure is repeated with each of the heaps until all the intended cards are found. In some heaps of conceived cards, there may not be any at all, while in others there may be two or more, but in any case, the cards are guessed by the showing unmistakably.
Gardner Martin
"MATHEMATICAL WONDERS AND MYSTERIES"
Editor's preface to the Russian edition
Before you is a regular square chess grid of 64 cells. Before your eyes, several cuts are made and a rectangle is made from the resulting parts, in which, however, there are only 63 cells!
You thought of a number - one of those that are written on the cards scattered on the table. Your partner touches the cards one by one with a pointer, and at the same time you spell the intended number to yourself, and when you get to the last letter, the pointer stops right at your number!
Focuses? Yes, if you like; or rather, experiments based on mathematics, on the properties of figures and numbers, and only clothed in a somewhat extravagant form. And to understand the essence of this or that experiment means to understand even a small but precise mathematical regularity.
It is with this hidden mathematics that Martin Gardner's book is interesting. Hidden - because for the most part the author himself does not formulate in the language of mathematics the patterns underlying his experiments, limiting himself to describing the actions of the showing, explicit and secret; but the reader, familiar with the elements of school algebra and geometry, will undoubtedly take pleasure in restoring the corresponding algebraic or geometric idea from the author's explanations. However, in separate, more interesting cases (marked by numbers with parentheses), we took the liberty of accompanying the author's presentation with small notes that reveal the mathematical essence of his constructions; these notes are placed at the end of the book.
Mathematical tricks are a very peculiar form of demonstrating mathematical patterns.
If in educational presentation they strive for the greatest possible disclosure of ideas, then here, in order to achieve efficiency and entertainment, on the contrary, they mask the essence of the matter as cunningly as possible. That is why, instead of abstract numbers, various objects or sets of objects associated with numbers are so often used: dominoes, matches, clocks, a calendar, coins, and even cards (of course, this use of cards has nothing to do with the senseless pastime of gamblers; as the author points out, here the cards are considered simply as identical objects that are convenient to count; the images on them do not play any role in this).
We hope that Gardner's book will be of interest to many readers: young participants and solo mathematical circles, adult "disorganized" lovers of mathematics, and perhaps one or another of the experiments described here will awaken a smile even in a serious scientist in a short moment of rest from big work.
Like many other subjects that are at the intersection of two disciplines, mathematical tricks do not receive special attention from either mathematicians or magicians. The first tend to regard them as empty fun, the second neglect them as too boring. Mathematical tricks, to put it bluntly, do not belong to the category of tricks that can keep an audience of non-mathematicians spellbound; such tricks usually take a lot of time, and they are not very effective; on the other hand, there is hardly a person who is going to draw deep mathematical truths from their contemplation.
And yet, mathematical tricks, like chess, have their own special charm. Chess combines the elegance of mathematical construction with the pleasure that the game can deliver. In mathematical tricks, the elegance of mathematical constructions is combined with amusement. It is not surprising, therefore, that they bring the greatest pleasure to those who are simultaneously familiar with both of these areas.
The present book, to the best of my knowledge, is the first attempt at a survey of the entire field of contemporary mathematical focus. Much of the material in this book is taken from specialized magic literature, not from entertaining mathematical literature. For this reason, students of recreational mathematical literature, but unfamiliar with the modern specialized literature on tricks, are likely to encounter in this book a new field of recreational knowledge - a rich new field, the existence of which they may have been completely unaware of.
New York, 1955
Martin Gardner
Chapter first. MATHEMATICAL TRICKS WITH CARDS
Playing cards have some specific properties that can be used in the preparation of magic tricks of a mathematical nature. We indicate five such properties.
1. Cards can be considered simply as identical objects that are convenient to count; the images on them do not play any role.
With the same success one could use pebbles, matches or pieces of paper.
2. Cards can be assigned numerical values from 1 to 13, depending on what is shown on their front side (in this case, jack, queen and king are taken as 11, 12 and 13, respectively)).
3. They can be divided into four suits or into black and red cards.
4. Each card has a front and back side.
5. The cards are compact and uniform in size. This allows you to lay them out in various ways, grouping them in rows or making piles that can be easily upset right there by simply mixing the cards.
With so many possibilities, card tricks must have been around a long time ago, and one can say that mathematical card tricks are certainly as old as the game of cards itself.
Apparently, the earliest discussion of card tricks by a mathematician is found in an entertaining book by Claud Gaspard Bachet (Problemes plaisants et delectables), published in France in 1612. Subsequently, references to card tricks appeared in many books devoted to mathematical entertainment.
The first and perhaps the only philosopher who condescended to consider card tricks was the American Charles Peirce. In one of his articles, he admits that in 1860 he "concocted" several extraordinary card tricks based, using his terminology, on "cyclic arithmetic." He describes two such tricks in detail under the title "first curiosity" and "second curiosity."
The "First Curiosity" is based on Fermat's Theorem. It took 13 pages just to describe the way it was demonstrated, and an additional 52 pages were taken up with an explanation of its essence. And although Peirce reports the “continuous interest and amazement of the public” caused by his trick, the climactic effect of this trick seems so out of proportion to the complexity of the preparations that it is hard to believe that the audience did not fall asleep long before it ended.
Fans of mathematical puzzles will find in this book many fascinating problems, entertaining episodes from the history of science and mathematical curiosities from the outstanding popularizer Martin Gardner.
Mathematical tricks are a very peculiar form of demonstrating mathematical patterns.
If in educational presentation they strive for the greatest possible disclosure of ideas, then here, in order to achieve efficiency and entertainment, on the contrary, they mask the essence of the matter as cunningly as possible. That is why, instead of abstract numbers, various objects or sets of objects associated with numbers are so often used: dominoes, matches, clocks, a calendar, coins, and even cards (of course, this use of cards has nothing to do with the senseless pastime of gamblers; as the author points out, here the cards are considered simply as identical objects that are convenient to count; the images on them do not play any role in this).
Download and read Mathematical Wonders and Mysteries, Gardner M.
New puzzles, games, paradoxes and other mathematical entertainment from the Scientific American magazine with an introduction by Donald Knuth, an afterword by the author and 105 figures and diagrams.
Welcome to the greatest math show on earth! Martin Gardner is once again the seasoned entertainer, presenting both simple match and dollar bill problems as well as fundamental problems in physics, mathematics, astronomy and philosophy. Like all books by M. Gardner, this edition is both accessible to the widest circle of readers and interesting to professional mathematicians.
Title: Classic puzzles.
All the riddles in this book are of the type we call "out-of-the-box" or "situational" riddles.
