1、 t * r- I ; sti I t WHEELS,LIFE,- . . - i AND OTHER fh Tr 3 MATHEMATICAL - =. “ . - -. AMUSEMENTS - .- 2 L i - - * I , 1. - -. .*, elk - $ - - ;Q: a 0, 9 - 9- 0, +-. r7 9 54qp46 -2 * D- * 1 6 G 0 I , , MARTIN GARDNER - - WHEELS, LIFE AND OTHER MATHEMATICAL AMUSEMENTS W. H. Freeman and Company New Yo
2、rk Lilrar! of Corgr-e,s Cataloging in Public,ition Data Gardner, Xlartin, 191.1- Wheels, life, and othrr mathenlatical aniusemenrs. Inclutles bibliograpliirs .nd intlex. 1. Rlathernatical recl-eations. I. Title. (LA9.5.(;333 1983 793.74 83-1 1.592 ISBN 0-7167-1588-0 ISBN 0-5 167- 1.589-1) (pbk.) Lif
3、e configurations conrtes! of R. LVlliarn Goaper of Snit)olic. Jacket photograp11 b Jennter L17alsli Copright C 1983 h! 1V.H. Freeman and Cornpan) No part of ttiis book ma! be reprodrlced h- an! mechanical, plotographic. or electronic process, or it1 the form of a phonog-I-aplic recortling, nur nla i
4、t be stored in a retrieval s!stem, trarirriitted. or othervise copird for public or 1)l.i- vate use, vithout witten permissiol from the puhli5her. PRINTED IS THE LSITEI) SIAIES OF AAlt-KIC .a Ninth printing 1996, VB For Ronald L. Graham who juggles nunlbers and other mathematical objects as elegantl
5、y as he juggles balls and clubs, and trvirls himself on the trampoline CONTENTS Introduction vii Wheels 1 Diophantine Analysis and Fermats Last Theorem 10 The Knotted Molecule and Other Problems 20 Alephs and Supertasks 31 Nontransitive Dice and Other Probability Paradoxes 40 Geometrical Fallacies 5
6、1 The Combinatorics of Paper Folding 60 A Set of Quickies 74 Ticktacktoe Games 94 Plaiting Polyhedrons 106 The Game of Halma 115 Advertising Premiums 124 Salmon on Austins Dog 134 Nim and Hackenbush 142 Golombs Graceful Graphs 152 Charles Addams Skier and other Problems 166 Chess Tasks 183 Slither,
7、3X + 1, and Other Curious Questions 194 19. Mathematical Tricks With Cards 206 20. The Game of Life, Part l 214 21. The Game of Life, Part 11 226 22. The Game of Life, Part Ill 241 Name Index 259 INTRODUCTION “There remains one more game.“ “LVhat is it?“ “Ennui,“ I said. “The easiest of all. No rule
8、s, no boards, no equipment.“ “LVhat is Ennui?“ Amanda asked. “Ennui is the abqence of games.“ -Donald Barthelme, Guzltj Pltatulti Unfortunately, as recent studies of education in this country have made clear, one of the chief characteristics of mathemat-. ical classes, especially on the lover levels
9、 of public education, is ennui. Some teachers may be poorly trained in mathematics and others not trained at all. If mathematics bores them, can you blame their students for being bored? Like science, mathematics is a kind of game that we play vith the universe. The best mathematicians and the best
10、teachers of mathematics obviously are those who both understand the rules of the game, and who relish the excitement of playing it. Raymond Smullyan, who has enormous zest for the games of philosophy and mathematics, once taught an elementary course in geometry. In his delightful book 5000 U.C. and
11、0th Philosophical Fantasies (1983) he tells how- he once introduced his students to the Pythagorean theorem: I drew a right triangle on the board with squares on the hy- potenuse and legs and said, “Obviously, the square on the hy- potenuse has a larger area than either of the other two squares. Now
12、 suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. Which would you choose?“ Interestingly enough, about half the class opted for the one large square and half for the two small ones. A lively argu- ment began. Both groups
13、 were equally amazed when told that it would make no difference. INTRODUCTION It is this sense of surprise that all great mathematicians feel, and all great teachers of mathematics are able to communicate. I know of no better way to do this, especially fbr- beginning stu- dents, than by way of games
14、, puzzles, paradoxes, magic tricks, and all the other curious paraphernalia of “recreational mathematics.“ “Puzzles and ganies provide a rich source of example problems useful for illustrating and testing problem-solving methods,“ wrote Nils Nilsson in his widely used textbook Problem-Solving Met!od
15、,s in ArtlJicial Intelligence. He quotes Mar- vin Minsky: “It is not that the games and mathematical prob- lems are chosen because they are clear and simple; rather it is that they giveus, for the smallest initial structures, the greatest com- plexity, so that one can engage some really formidable s
16、ituations after a relatively rninirnal diversion into programming.“ Nilsson and Minsky had in mind the value of recreational mathematics in learning how to solve pr-ollerns by computers, but its value in learning how to solve problems by hand is just as great. In this book, the tenth collection of t
17、he Mathenlatical Games colurrins that I wrote for- Scic.ntzjic American, you will find an assortment of mathematical recreations of every vari- ety. Ihe last three chapters (the third was written especially for this volume) deal with John H. C;onways fantastic game of Life, the full wonders of which
18、 are still being explored. The two previously published articles on Life, in which I had the privilege of introducing this game for the first time, aroused more interest among computer buffs around the world than any other columns I have written. Now that Life software is beconlirlg available for ho
19、me-computer screens, there has been a renewed interest in this remarkable recrea- tion. Although Life rules are incredibly simple, the complexity of its structure is so awesome that no one can experiment with its “life forms“ without being overwhelmed by a sense of the infinite range and depth and m
20、ystery of mathematical struc- ture. Few have expressed this emotion Inore colorfully than the British-American mathematician .James J. Sylvester: Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs orily patience . . to ransack; it is not a rnin
21、e, whose treasures rnay take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose terrility can be ex- hausted by the yield of successive harvests; it is not a conti- nent or an ocean, whose area can be mapped out and its ron- tour defined:
22、 it is linlitless as that space which it finds too INTRODUCTION narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomers gaze; it is as incapable of being restricted within assigned boundaries or being reduced to d
23、efinitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence. Martin Gardner WHEELS The miraculous paradox of smooth
24、 round objects conquering space by simply tumbling over and over, instead of laboriously lifting heavy limbs in order to progress, must have given young mankind a most salutary shock. Things would be very different without the wheel. Transpor- tation aside, if we consider wheels as simple machines-p
25、ulleys, gears, gyroscopes and so on-it is hard to imagine an) ad- vanced society without them. H. G. Wells, in The War of the Worlds, describes a Martian civilization far ahead of ours but using no wheels in its intricate machinery. Wells may have in- tended this to be a put-on; one can easily under
26、stand how the American Indian could have missed discovering the wheel, but a society capable of sending spaceships from Mars to the earth? Until recently the wheel was believed to have originated in Mesopotamia. Pictures of wheeled Mesopotamian carts date back to 3000 B.C. and actual remains of mass
27、ive disk wheels have been unearthed that date back to 2700 B.C. Since FYorld War 11, however, Russian archaeologists have found potter models of wheeled carts in the Caucasus that suggest the wheel may have originated in southern Russia even earlier than it did in Mesopotamia. There could have been
28、two or more inde- pendent inventions. Or it may have spread by cultural diffu- sion as John Updike describes it in a stanza of his poem, Whel: CHAPTER 1 The Eskimos had never heard Of centripetal force when Byrd Bicycled up onto a floe And told them, “This how white man go.“ It seems surprising that
29、 evolution never hit on the wheel as a means for making animals go, but on second thought one re- alizes how difficult it would be for biological mechanisms to make wheeled feet rotate. Perhaps the tumbleweed is the clos- est nature ever came to wheeled transport. (On the other hand, the Dutch artis
30、t Maurits C. Escher designed a creature capable of curling itself into a wheel and rolling along at high speeds. Who can be sure such creatures have not evolved on other planets?) There may also be submicroscopic swivel devices in- side the cells of living bodies on the earth, designed to unwind and
31、 rewind double-helix strands of DNA, but their existence is still conjectural. A rolling wheel has many paradoxical properties. It is easy to see that points near its top have a much faster ground speed than points near its bottom. Maximum speed is reached by a point on the rim when it is exactly at
32、 the top, minimum speed (zero) when the point touches the ground. On flanged train wheels whose rims extend slightly below a track, there is even a short segment in which a point on the rim moves backward. G. K. Chesterton, in an essay on wheels in his book Alarms and Discursions, likens the wheel t
33、o a healthy society in having “a part that perpetually leaps helplessly at the sky; and a part that perpetually bows down its head into the dust.“ He reminds his readers, in a characteristically Chestertonian remark, that “one cannot have a Revolution without revolving.“ The most subtle of all wheel
34、 paradoxes is surprisingly little known, considering that it was first mentioned in the Mechan- ica, a Greek work attributed to Aristotle but more likely written by a later disciple. “Aristotles wheel,“ as the paradox is called, is the subject of a large literature to which such eminent math- ematic
35、ians as Galileo, Descartes, Fermat and many others con- tributed. As the large wheel in Figure 1 rolls from A to B, the rim of the small wheel rolls along a parallel line from C to D. (If the two lines are actual tracks, the double wheel obviously cannot roll smoothly along both. It either rolls on
36、the upper track while the large wheel continuously slides backward on the lower track, or it rolls on the lower track while the small wheel slips forward on the upper track. This is not, however, the heart of the paradox.) Assume that the bottom wheel rolls without slipping from A to B. At every ins
37、tant that a unique WHEELS Figure 1 Aristotles wheel paradox point on the rim of the large wheel touches line AB, a unique point on the small wheel is in contact with line CD. In other words, all points on the small circle can be put into one-to-one correspondence with all points on the large circle.
38、 No points on either circle are left out. This seems to prove that the two circumferences have equal lengths. Aristotles wheel is closely related to Zenos well-known par- adoxes of motion, and it is no less deep. Modern mathemati- cians are not puzzled by it because they know that the number of poin
39、ts on any segment of a curve is what Georg Cantor called c, the transfinite number that represents the “power of the continuum.“ All points on a one-inch segment can be put in one-to-one correspondence with all points on a line a million miles long as well as on a line of infinite length. Moreover,
40、it is not hard to prove that there are aleph-one points within a square or cube of any size, or within an infinite Euclidian space having any finite number of dimensions. Of course, mathema- ticians before Cantor were not familiar with the peculiar prop- erties of transfinite numbers, and it is amus
41、ing to read their fumbling attempts to resolve the wheel paradox. Galileos approach was to consider what happens when the two wheels are replaced by regular polygons such as squares see Figure 21. After the large square has made a complete turn along AB, the sides of the small square have coincided
42、with CD Figure 2 Galileos approach to the wheel paradox CHAPTER 1 in four segments separated by three jumped spaces. If the wheels are pentagons, the small pentagon will jump four spaces on each rotation, and so on for higher-order polygons. As the number of sides increases, the gaps also increase i
43、n number but decrease in length. When the limit is reached-the circle with an infinite number of sides-the gaps will be infinite in number but each will be infinitely short. These Galilean gaps are none other than the mystifying “infinitesimals“ that later so muddied the early development of calculu
44、s. And now we are in a quagmire. If the gaps made by the small wheel are infinitely short, why should their sum cause the wheel to slide a finite distance as the large wheel rolls smoothly along its track? Readers interested in how later mathematicians replied to Galileo, and argued with one another
45、, will find the details in the articles listed in this chapters bibliography. As a wheel travels a straight line, any point on its circumfer- ence generates the familiar cycloid curve. When a wheel rolls on the inside of a circle, points on its circumference generate curves called hypocycloids. When
46、 it rolls on the outside of a circle, points on the circumference generate epicycloids. Let Rlr be the ratio of the radii, R for the large circle, r for the small. If Rlr is irrational, a point a on the rolling circle, once in contact with point b on the fixed circle, will never touch b again even t
47、hough the wheel rolls forever. The curve generated by a will have an aleph-null infinity of cusps. If Rlr is rational, a and b will touch again after a finite number of revolutions. If Rlr is integral, a returns to b after exactly one revolution. Consider hypocycloids traced by a circle of radius r
48、as it rolls inside a larger circle of radius R. When Rlr is 2, 3, 4, . . . , points a and b touch again after one revolution and the curve will have Rlr cusps. For example, a three-cusped deltoid results when Rlr equals 3 see Figure 3, left. The same deltoid is pro- duced when Rlr is 312; that is, w
49、hen the rolling circles radius Figure 3 The deltoid The astroid “Two-cusped“ hypocycloid WHEELS is two-thirds that of the fixed circle. All line segments tangent to the deltoid, with ends on the curve, have the same length. A four-cusped astroid is generated when Rlr equals 4 or 413 see Fzgure 3, rnzcldle. The two ratios appl) to all higher-order hy- pocycloids of this type: when Rlr is either n or ni(n- 1), the rolling circle produces an n-cusped curve. There is a surprising result when Rlr equals 2 see Fzgure 3, rzght. The hypo
