COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. ScientificAmerican.com MATHEMATICAL AMERICAN exclusive online issue no. 10 “Mathematics, rightly viewed, possesses not only truth, but supreme beauty." So wrote British philosopher and logician Bertrand Russell nearly 100 years ago. He was not alone in this sentiment. French mathematician Henri Poincaré declared that "the mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." Likewise, Einstein described pure mathematics as "the poetry of logical ideas." Indeed, many a scholar has remarked on the elegance of the science. It is in this spirit that we have put together a collection of Scientific American articles about math. In this exclusive online issue, Martin Gardner, longtime editor of the magazine’s Mathematical Games column, reflects on 25 years of fun puzzles and serious discoveries; other scholars explore the concept of infinity, the fate of mathematical proofs in the age of computers, and the thriving of native mathematics during Japan’s period of national seclusion. The anthology also includes articles that trace the long, hard roads to resolving Fermat’s Last Theorem and Zeno’s paradoxes, two problems that for centuries captivated--and tormented--some of the discipline’s most beautiful minds. —The Editors TABLE OF CONTENTS 2 A Quarter-Century of Recreational Mathematics BY MARTIN GARDNER; SCIENTIFIC AMERICAN, AUGUST 1998 The author of Scientific American's column "Mathematical Games" from 1956 to 1981 recounts 25 years of amusing puzzles and serious discoveries 10 The Death of Proof BY JOHN HORGAN; SCIENTIFIC AMERICAN, OCTOBER 1993 Computers are transforming the way mathematicians discover, prove and communicate ideas, but is there a place for absolute certainty in this brave new world? 16 Resolving Zeno's Paradoxes BY WILLIAM I. MCLAUGHLIN; SCIENTIFIC AMERICAN, NOVEMBER 1994 For millennia, mathematicians and philosophers have tried to refute Zeno's paradoxes, a set of riddles suggesting that motion is inherently impossible. At last, a solution has been found 21 A Brief History of Infinity BY A. W. MOORE; SCIENTIFIC AMERICAN, APRIL 1995 The infinite has always been a slippery concept. Even the commonly accepted mathematical view, developed by Georg Cantor, may not have truly placed infinity on a rigorous foundation 25 Fermat's Last Stand BY SIMON SINGH AND KENNETH A. RIBET; SCIENTIFIC AMERICAN, NOVEMBER 1997 His most notorious theorem baffled the greatest minds for more than three centuries. But after 10 years of work, one mathematician cracked it 29 Japanese Temple Geometry BY TONY ROTHMAN; SCIENTIFIC AMERICAN, MAY 1998 During Japan's period of national seclusion (1639-1854), native mathematics thrived, as evidenced in "sangaku"— wooden tablets engraved with geometry problems hung under the roofs of shrines and temples 1 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Originally published in August 1998 A Quarter-Century of Recreational Mathematics The author of Scientific American’s column “Mathematical Games” from 1956 to 1981 recounts 25 years of amusing puzzles and serious discoveries by Martin Gardner “Amusement is one of the fields of ics were in print. The classic of the and serious math is a blurry one. Many applied math.” genre—Mathematical Recreations and professional mathematicians regard —William F. White, Essays, written by the eminent English their work as a form of play, in the A Scrapbook of mathematician W. W. Rouse Ball in same way professional golfers or bas- Elementary Mathematics 1892—was available in a version up- ketball stars might. In general, math is dated by another legendary figure, the considered recreational if it has a play- Canadian geometer H.S.M. Coxeter. ful aspect that can be understood and Dover Publications had put out a trans- appreciated by nonmathematicians. y “Mathematical Games” lation from the French of La Mathéma- Recreational math includes elementary column began in the De- tique des Jeux (Mathematical Recrea- problems with elegant, and at times Mcember 1956 issue of Sci- tions), by Belgian number theorist Mau- surprising, solutions. It also encompass- entific American with an article on rice Kraitchik. But aside from a few other es mind-bending paradoxes, ingenious hexaflexagons. These curious struc- puzzle collections, that was about it. games, bewildering magic tricks and tures, created by folding an ordinary Since then, there has been a remark- topological curiosities such as Möbius strip of paper into a hexagon and then able explosion of books on the subject, bands and Klein bottles. In fact, almost gluing the ends together, could be many written by distinguished mathe- every branch of mathematics simpler turned inside out repeatedly, revealing maticians. The authors include Ian Stew- than calculus has areas that can be con- one or more hidden faces. The struc- art, who now writes Scientific Ameri- sidered recreational. (Some amusing ex- tures were invented in 1939 by a group can’s “Mathematical Recreations” col- amples are shown on the following of Princeton University graduate stu- umn; John H. Conway of Princeton page.) dents. Hexaflexagons are fun to play University; Richard K. Guy of the Uni- with, but more important, they show versity of Calgary; and Elwyn R. Berle- Ticktacktoe in the Classroom the link between recreational puzzles kamp of the University of California at and “serious” mathematics: one of their Berkeley. Articles on recreational math- he monthly magazine published by inventors was Richard Feynman, who ematics also appear with increasing fre- Tthe National Council of Teachers went on to become one of the most fa- quency in mathematical periodicals. of Mathematics, Mathematics Teacher, mous theoretical physicists of the cen- The quarterly Journal of Recreational often carries articles on recreational top- tury. Mathematics began publication in ics. Most teachers, however, continue to At the time I started my column, only 1968. ignore such material. For 40 years I a few books on recreational mathemat- The line between entertaining math have done my best to convince educa- 2 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Four Puzzles from Martin Gardner (The answers are on page 75.) 1 2 ILLUSTRATIONS BY IAN WORPOLE ILLUSTRATIONS 1 1 1 3 3 3 ? ? r. Jones, a cardsharp, puts three cards face down on a table. he matrix of numbers above is a curious type of magic square. MOne of the cards is an ace; the other two are face cards. You TCircle any number in the matrix, then cross out all the numbers place a finger on one of the cards, betting that this card is the ace. in the same column and row. Next, circle any number that has not 1 The probability that you’ve picked the ace is clearly /3. Jones now se- been crossed out and again cross out the row and column containing cretly peeks at each card. Because there is only one ace among the that number. Continue in this way until you have circled six numbers. three cards, at least one of the cards you didn’t choose must be a face Clearly, each number has been randomly selected. But no matter card. Jones turns over this card and shows it to you. What is the prob- which numbers you pick, they always add up to the same sum. What ability that your finger is now on the ace? is this sum? And, more important, why does this trick always work? 3 4 12 3 456 78910 CUT THE DECK 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 RIFFLE- 33 34 35 36 37 38 39 SHUFFLE 40 41 42 43 44 45 46 4847 5049 rinted above are the first three verses of Genesis in the King magician arranges a deck of cards so that the black and PJames Bible. Select any of the 10 words in the first verse: “In the A red cards alternate. She cuts the deck about in half, making sure beginning God created the heaven and the earth.” Count the num- that the bottom cards of each half are not the same color. Then she ber of letters in the chosen word and call this number x. Then go to allows you to riffle-shuffle the two halves together, as thoroughly or the word that is x words ahead. (For example, if you picked “in,” go to carelessly as you please. When you’re done, she picks the first two “beginning.”) Now count the number of letters in this word—call it cards from the top of the deck. They are a black card and a red card n—then jump ahead another n words. Continue in this manner until (not necessarily in that order). The next two are also a black card and your chain of words enters the third verse of Genesis. a red card. In fact, every succeeding pair of cards will include one of On what word does your count end? Is the answer happenstance each color. How does she do it? Why doesn’t shuffling the deck pro- or part of a divine plan? duce a random sequence? 3 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. ab e 2 1 √3 cd IAN WORPOLE LOW-ORDER REP-TILES fit together to make larger replicas of themselves. The isosceles right triangle (a) is a rep-2 figure: two such triangles form a larger triangle with the same shape. A rep-3 triangle (b) has angles of 30, 60 and 90 degrees. Other rep- tiles include a rep-4 quadrilateral (c) and a rep-4 hexagon (d). The sphinx (e) is the only known rep-4 pentagon.