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 kamp of the University of fields of applied math.” California at Berkeley. Arti- —William F. White, cles on recreational mathe- A Scrapbook of matics also appear with in- Elementary Mathematics creasing frequency in mathe- matical periodicals. The quarterly Journal of Recrea- tional Mathematics began y “Mathemati- publication in 1968. cal Games” col- The line between entertain- Mumn began in ing math and serious math is the December 1956 issue of a blurry one. Many profes- Scientific American with an sional mathematicians regard article on hexaflexagons. their work as a form of play, These curious structures, cre- in the same way professional ated by folding an ordinary golfers or basketball stars strip of paper into a hexagon might. In general, math is and then gluing the ends to- considered recreational if it gether, could be turned inside has a playful aspect that can out repeatedly, revealing one be understood and appreci- or more hidden faces. The Gamma Liaison ated by nonmathematicians. structures were invented in Recreational math includes 1939 by a group of Princeton elementary problems with University graduate students. DONNA BISE elegant, and at times surpris- Hexaflexagons are fun to MARTIN GARDNER continues to tackle mathematical puz- ing, solutions. It also encom- play with, but more impor- zles at his home in Hendersonville, N.C. The 83-year-old writ- passes mind-bending para- tant, they show the link be- er poses next to a Klein bottle, an object that has just one sur- doxes, ingenious games, be- face: the bottle’s inside and outside connect seamlessly. tween recreational puzzles wildering magic tricks and and “serious” mathematics: topological curiosities such one of their inventors was Richard Feyn- lation from the French of La Mathéma- as Möbius bands and Klein bottles. In man, who went on to become one of the tique des Jeux (Mathematical Recrea- fact, almost every branch of mathemat- most famous theoretical physicists of tions), by Belgian number theorist Mau- ics simpler than calculus has areas that the century. rice Kraitchik. But aside from a few other can be considered recreational. (Some At the time I started my column, only puzzle collections, that was about it. amusing examples are shown on the a few books on recreational mathemat- Since then, there has been a remark- opposite page.) ics were in print. The classic of the able explosion of books on the subject, genre—Mathematical Recreations and many written by distinguished mathe- Ticktacktoe in the Classroom Essays, written by the eminent English maticians. The authors include Ian Stew- mathematician W. W. Rouse Ball in art, who now writes Scientific Ameri- he monthly magazine published by 1892—was available in a version up- can’s “Mathematical Recreations” col- Tthe National Council of Teachers dated by another legendary figure, the umn; John H. Conway of Princeton of Mathematics, Mathematics Teacher, Canadian geometer H.S.M. Coxeter. University; Richard K. Guy of the Uni- often carries articles on recreational top- Dover Publications had put out a trans- versity of Calgary; and Elwyn R. Berle- ics. Most teachers, however, continue to 68 Scientific American August 1998 A Quarter-Century of Recreational Mathematics Copyright 1998 Scientific American, Inc. Four Puzzles from Martin Gardner (The answers are on page 75.) 1 2 ORPOLE TIONS BY IAN W A USTR ILL 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 47 48 49 50 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? A Quarter-Century of Recreational Mathematics Scientific American August 1998 69 Copyright 1998 Scientific American, Inc. a b e 2 1 √3 ORPOLE c d IAN W 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. ignore such material. For 40 years I “Interactive learning,” as it is called, is ly. I took no math courses in college. My have done my best to convince educa- substituted for lecturing. Although there columns grew increasingly sophisticat- tors that recreational math should be are some positive aspects of the new ed as I learned more, but the key to the incorporated into the standard curricu- new math, I was struck by the fact that column’s popularity was the fascinating lum. It should be regularly introduced the yearbook had nothing to say about material I was able to coax from some as a way to interest young students in the value of recreational mathematics, of the world’s best mathematicians. the wonders of mathematics. So far, which lends itself so well to cooperative Solomon W. Golomb of the Universi- though, movement in this direction has problem solving. ty of Southern California was one of been glacial. Let me propose to teachers the follow- the first to supply grist for the column. I have often told a story from my own ing experiment. Ask each group of stu- In the May 1957 issue I introduced his high school years that illustrates the di- dents to think of any three-digit num- studies of polyominoes, shapes formed lemma. One day during math study pe- ber—let’s call it ABC. Then ask the stu- by joining identical squares along their riod, after I’d finished my regular as- dents to enter the sequence of digits edges. The domino—created from two signment, I took out a fresh sheet of pa- twice into their calculators, forming the such squares—can take only one shape, per and tried to solve a problem that number ABCABC. For example, if the but the tromino, tetromino and pento- had intrigued me: whether the first play- students thought of the number 237, mino can assume a variety of forms: Ls, er in a game of ticktacktoe can always they’d punch in the number 237,237. Ts, squares and so forth. One of Gol- win, given the right strategy. When my Tell the students that you have the psy- omb’s early problems was to determine teacher saw me scribbling, she snatched chic power to predict that if they divide whether a specified set of polyominoes, the sheet away from me and said, “Mr. ABCABC by 13 there will be no remain- snugly fitted together, could cover a Gardner, when you’re in my class I ex- der.