M Gardner mathematical wonders and secrets. Mathematical Miracles and Secrets

Gardner Martin

"MATHEMATICAL MIRACLES AND MYSTERIES"

Editor's note to the Russian edition

Here is a regular square chess grid of 64 squares. Before your eyes, several cuts are made and a rectangle is made of the resulting parts, in which, however, there are only 63 cells!

You conceived a number - one of those written on cards scattered across the table. Your partner alternately touches the cards with a pointer, and at that time you spell the conceived number to yourself, and when you reach the last letter, the pointer stops just on your number!

Tricks Yes, if you want; but 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 an experiment is to understand even a small but exact mathematical regularity.

This hidden mathematics is an interesting book by Martin Gardner. Hidden - because for the most part the author himself does not formulate in the language of mathematics the laws underlying his experiments, limiting himself to describing the actions of the revealing, explicit and secret; but a reader familiar with the elements of school algebra and geometry will undoubtedly take pleasure in reconstructing the corresponding algebraic or geometric idea from the author’s explanations. However, in separate, more interesting cases (marked with numbers with a parenthesis) we allowed ourselves to accompany the author’s presentation with small notes revealing the mathematical essence of his constructions; these notes are placed at the end of the book.

Mathematical tricks are a very peculiar form of demonstration of mathematical laws.

If in the educational presentation they strive for the greatest possible disclosure of the idea, then here, in order to achieve efficiency and amusement, 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, calendar, coins and even cards (of course, such use of cards has nothing to do with the meaningless pastime of gambling players; as the author indicates, here, the cards are viewed simply as identical objects that are convenient to read; the images on them do not play any role at the same time ”).

We hope that Gardner’s book will be of interest to many readers: young participants in the math classes, adult “unorganized” math lovers, or maybe one or another of the experiments described here will awaken the smile of a serious scientist in a short moment of rest from a lot of work.

Nevertheless, mathematical tricks, like chess, have their own special charm. In chess, the grace of mathematical constructions is combined with the pleasure that a 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 one who is simultaneously familiar with both of these areas.

This book, as far as I know, represents the first attempt to review the entire field of modern mathematical focus. Most of the material in the book is taken from special literature on tricks, and not from entertaining mathematical literature. For this reason, people who study entertainment mathematical literature, but are unfamiliar with modern special literature on tricks, are likely to find in this book a new field of entertaining knowledge - a new rich field, the existence of which they might not have suspected at all.

New York, 1955

Martin Gardner

Chapter first. MATHEMATICAL FOCUSES WITH CARDS

Playing cards have some specific properties that can be used in drawing up tricks of a mathematical nature. We indicate five such properties.

1. Cards can be considered simply as identical objects that are convenient to read; the images on them do not play any role.

With the same success it would be possible to use pebbles, matches or pieces of paper.

2. Cards can be assigned numerical values \u200b\u200bfrom 1 to 13 depending on what is shown on their front side (in this case the jack, queen and king are taken as 11, 12 and 13, respectively)).

3. They can be divided into four suits or 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 different ways, grouping in rows or making up piles that can be easily upset right away just by mixing cards.

Due to such an abundance of possibilities, card tricks should have appeared a very long time ago, and it can be considered that mathematical tricks with cards are certainly as old as the game of cards.

Apparently, the earliest discussion of card tricks by a mathematician is found in Claud Gaspard Bachet’s entertaining book, 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 to come down to the consideration of card tricks was the American Charles Peirce. In one of his articles, he admits that in 1860 he “concocted” several unusual card tricks, based, using his terminology, on “cyclic arithmetic”. He describes two such tricks in detail under the name “first curiosity” and “second curiosity”.

The First Curious is based on Fermat's theorem. It took 13 pages to describe only how to demonstrate it, and an additional 52 pages were devoted to explaining its essence. And although Pairs reports about “the constant interest and amazement of the public” caused by his focus, the culmination effect of this focus seems so inconsistent with the complexity of the preparations that it is hard to believe that the audience did not go into sleep long before it ended

Lovers of mathematical puzzles will find in this book many fascinating tasks, entertaining episodes from the history of science and mathematical oddities from the outstanding popularizer Martin Gardner.

Mathematical tricks are a very peculiar form of demonstration of mathematical laws.
If in the educational presentation they strive for the greatest possible disclosure of the idea, then here, in order to achieve efficiency and entertaining, 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, calendar, coins and even cards (of course, such use of cards has nothing to do with the meaningless pastime of gambling players; as the author indicates, here, the cards are considered simply as identical objects that are convenient to read; the images on them do not play any role at the same time ”).

Download and Read Mathematical Miracles and Secrets, Gardner M.

New puzzles, games, paradoxes and other mathematical entertainments from the Science American magazine with a preface by Donald Knuth, an afterword by the author and 105 drawings and diagrams.

Welcome to the greatest mathematical representation on earth! Martin Gardner again acts as an experienced entertainer, presenting both simple problems about matches and dollar bills, and the fundamental problems of physics, mathematics, astronomy and philosophy. Like all books by M. Gardner, this publication is simultaneously accessible to a wide range of readers and interesting to professional mathematicians.

Download and read The best math games and puzzles, or the real math circus, Gardner M., 2009

Title: Classic puzzles.

All the riddles in this book belong to the type that we call "riddles for comprehensive thinking" or "riddles-situations."

Like many other subjects at the junction of the two disciplines, mathematical tricks do not receive special attention from either mathematicians or magicians. The former tend to regard them as empty fun, the latter neglect them as too boring. Mathematical tricks, frankly, do not belong to the category of tricks that can keep an audience from inexperienced in mathematics spectators enchanted; such tricks usually take a lot of time, and they are not too spectacular; on the other hand, there is hardly a person who is about to draw deep mathematical truths from their contemplation.
Nevertheless, mathematical tricks, like chess, have their own special charm. In chess, the grace of mathematical constructions is combined with the pleasure that a 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 one who is simultaneously familiar with both of these areas.
This book, as far as I know, represents the first attempt to review the entire field of modern mathematical focus. Most of the material in the book is taken from special literature on tricks, and not from entertaining mathematical literature. For this reason, people who study entertainment mathematical literature, but are unfamiliar with modern special literature on tricks, are likely to find in this book a new field of entertaining knowledge - a new rich field, the existence of which they might not have suspected at all.

Editor's note to the Russian edition
   From the author's foreword
   Chapter first
MATHEMATICAL FOCUSES WITH CARDS
   Five piles of cards (9).
   Cards as counting units. Guessing the number of cards removed from the deck (10). Using numeric card values
   Focus with four cards (11). Amazing prediction (12). Focus with a conceived map (13). The cyclic number (14). Missing card (15).
   Tricks based on the difference in colors and suits. Focus with kings and queens (19). Using the front and back sides of cards. Comparison of the number of cards of black and red suits (20). Card overturn focus (20).
   Tricks depending on the initial location of cards in the deck
   Focus with four aces (21). "Manhattan Miracles" (22). How many cards are there? (22). Focus with finding a map (23).
   Chapter two
FOCUSES WITH SMALL ITEMS
   Dice
   Guessing the sum (25). Guessing the number of points (27).
   Domino Chain with a break (27). A row of thirteen seeds (28).
   Calendars Mysterious Squares (29). Focus with marked dates (29). Prediction (30).
   Clock. Guessing the conceived 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 lattice (37). Chess board. Focus with three checkers (38) Small objects. Focus with three subjects (39). Focus with guessing one of four objects (40).
   Chapter three
   TOPOLOGICAL PUZZLE
   Paper rings (44).
   Handkerchief Tricks
   Finger cutting focus (48). Focus with clutching handkerchiefs (50). The problem of knotting (51).
   Cords and Strings
   Tricks with a cord or twine (52). Other tricks with a cord (56).
   clothing
   Mysterious Loop (58). Turning the vest inside out (59). Removing the vest (60).
   Rubber rings Galloping ring (60). Twisted ring (61).
   Chapter four
FOCUSES WITH SPECIAL EQUIPMENT
   Cards with numbers (64). Cards with holes (65). Touch Tricks
   Focus with six squares (66). Flower Card (67).
   Think of an animal (69). Tricks with dice and dominoes 70. Focus with three-digit numbers (70). Box for focus with dominoes (70). Focus with chips (71).
   Chapter five
DISAPPEARANCE OF FIGURES. SECTION I
   The paradox with lines (73). The disappearance of the face (75). The Vanishing Warrior (76). Missing Rabbit (78).
   Chapter six
DISAPPEARANCE OF FIGURES. SECTION II
   Checkerboard paradox (79). A paradox with an area of \u200b\u200b(81). Variant with a square (82). Fibonacci numbers (83).
   Variant with a rectangle (85). Another version of the paradox (87). Variant with a triangle (90). Four-piece squares (93). Three-piece squares (95). Two-piece squares (95). Curvilinear and three-dimensional variants (96).
   Chapter seven
PUZZLE PUZZLE
   Quick cube root extraction (98). Addition of Fibonacci numbers (100). Prediction of number (101). Guessing the number (102). 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). Prediction of the sum (112). “Psychological moments” (114).
   Editor's Notes

Gardner Martin

"MATHEMATICAL MIRACLES AND MYSTERIES"

Editor's note to the Russian edition

Here is a regular square chess grid of 64 squares. Before your eyes, several cuts are made and a rectangle is made of the resulting parts, in which, however, there are only 63 cells!

Mathematical tricks are a very peculiar form of demonstration of mathematical laws.

G. E. Shilov

New York, 1955

Martin Gardner

Chapter first. MATHEMATICAL FOCUSES WITH CARDS

Playing cards have some specific properties that can be used in drawing up tricks of a mathematical nature. We indicate five such properties.

1. Cards can be considered simply as identical objects that are convenient to read; the images on them do not play any role.

With the same success it would be possible to use pebbles, matches or pieces of paper.

2. Cards can be assigned numerical values \u200b\u200bfrom 1 to 13 depending on what is shown on their front side (in this case the jack, queen and king are taken as 11, 12 and 13, respectively)).

3. They can be divided into four suits or 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 different ways, grouping in rows or making up piles that can be easily upset right away just by mixing cards.

Apparently, the earliest discussion of card tricks by a mathematician is found in the entertainment book of Claude, Gaspard Bachet ( 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 Curious is based on Fermat's theorem. It took 13 pages to describe only how to demonstrate it, and an additional 52 pages were devoted to explaining its essence. And although Pairs reports about the “constant interest and amazement of the public” caused by his focus, the climax of this focus seems so inconsistent with the complexity of the preparations that it is hard to believe that the audience did not go into sleep long before the end of its demonstration.

Here is an example of how, as a result of a modification of the way of demonstrating one old focus, its amusement 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 conceive one card and tell the one showing in which vertical row it lies. Then the cards are collected with the right hand in vertical rows and successively folded into the left hand. After this, the cards are again laid out in the form of a square, horizontally sequentially; thus, the cards lying in the same vertical row during the initial layout now appear in the same horizontal row. The indicating person needs to remember in which one the now conceived map lies. Further, the viewer is asked once again to indicate in which vertical row he sees his card. It is clear that after this the showing can immediately indicate the planned map that will lie at the intersection of the just named vertical row and the horizontal row in which, as you know, it should be. The success of this focus, of course, depends on whether the viewer follows the procedure so closely as 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 showman sits at a table with four spectators. He hands everyone (including himself) five cards, invites everyone to see them and think of one. Then he collects the cards, puts them on the table in five piles and asks someone to point him to one of them. Then he takes this handful in his hands, reveals the cards with a fan, facing the audience, and asks if any of them see the conceived card. If so, then the showing one (without ever glancing at the cards) immediately pulls it out. This procedure is repeated with each of the heaps until all conceived cards are discovered. In some piles of conceived cards it may not appear at all, in others there may be two or more, but in any case, the cards are guessed correctly.