Martin Book of ' Gardner's Mathematical A Diversions from . - Scientzfic American For my brother-in-law, James B. Weaver Contents Introduction ix 1. The Helix 1 2. Klein Bottles and Other Surfaces 9 3. Combinatorial Theory 19 4. Bouncing Balls in Polygons and Polyhedrons 29 5. Four Unusual Board Games 39 6. The Rigid Square and Eight Other Problems 48 7. Sliding-Block Puzzles 64 8. Parity Checks 71 9. Patterns and Primes 79 10. Graph Theory 91 11. The Ternary System 104 12. The Trip around the Moon and Seven Other Problems 113 13. The Cycloid: Helen of Geometry 127 14. Mathematical Magic Tricks 135 15. Word Play 143 16. The Pythagorean Theorem 152 17. Limits of Infinite.Series 163 18. Polyiamonds 173 19. Tetrahedrons 183 20. Coleridge's Apples and Eight Other Problems 195 21. The Lattice of Integers 208 22. Infinite Regress 220 23. O'Gara, the Mathematical Mailman 230 24. OpArt 239 25. Extraterrestrial Communication 253 Introduction Ten years ago the writer of a mathematics educators that students learn best who are textbook would have been considered motivated best. Mathematics has never frivolous by his colleagues if his book in- been a dreary topic, although too often it cluded puzzles and other entertaining has been taught in the dreariest possible topics. This is no longer true. Exercises in way. There is no better way to relieve the the first two volumes of Donald E. Knuth's tedium than by injecting recreational top- monumental work in progress, The Art ics into a course, topics strongly tinged of Computer Programming (Reading: with elements of play, humor, beauty, and Addison-Wesley, 1968, 1969), are filled surprise. The greatest mathematicians al- with recreational material. There are even ways looked upon their subject as a source textbooks in which a recreational emphasis of intense intellectual delight and seldom is primary. A delightful instance is Harold hesitated to pursue problems of a recre- R. Jacobs's Mathematics: A Human En- ational nature. If you flip the leaves of deavor, subtitled A Textbook for Those W. W. Rouse Ball's classic British work, Who Think They Don't Like the Subject Mathematical Recreations and Essays (San Francisco: W. H. Freeman and Co., (first published by Macmillan in 1892 and 1970). Richard Bellman, Kenneth L. Cooke, soon to be issued in a twelfth revised edi- and Jo Ann Lockett, authors of Algorithnzs, tion), you will find the names of celebrated Graphs, and Computers (New York: Aca- mathematicians on almost every page. demic Press, 1970), write in their preface, Euclid himself, among the earliest of "The principal medium we have chosen to the mathematical giants, wrote an entire achieve our goals is the mathematical book (unfortunately it did not survive) on puzzle." geometrical fallacies. This is a topic cov- The trend is not hard to understand. It ered in standard works on recreational is part of the painfully slow recognition by mathematics but curiously avoided in most Introduction geometry textbooks. One of these days high munication systems." (Reprinted in Annuls school teachers of geometry will discover of the Computation Laboratory of Hurcurd that an excellent way to impress their stu- Unicersity, Vol. 30, 1959; pages 285-292.) dents with the need for rigor in deduction Need I remind readers that the maze is a is to "prove" on the blackboard that, say, topological puzzle older than Euclid's a right angle equals an obtuse angle, then geometry, and that topology itself had its challenge the class to explain where the origin in Leonhard Euler's famous analysis reasoning went wrong. of a route-tracing puzzle iilvolving the The value of recreational mathematics is seven bridges of Konigsberg? not limited to pedagogy. There are endless This is the sixth anthology of my arti- historical examples of puzzles, believed cles for the Scientific American department to be utterly trivial, the solving of which called Slathematical Games. As in previous led to significailt new theorems, often with collections, the articles have been ex- useful applications. I cite only one recent panded, errors corrected, bibliographies instance. Edward F. Sloore writes, in an added. I am grateful to the magazine for important paper on "The Shortest Path the great privilege of contributing regu- through a hlaze": "The origin of the present larly to its pages, to my wife for unfailing methods provides an interesting illustra- help in proofing, and as always to the hun- tion of the value of basic research on puz- dreds of Scientific Americccn readers whose zles and games. Although such research is suggestions have added so much to the often frowned upon as being frivolous, it value of the original articles. seems plausible that these algorithms might eventually lead to savings of very large MARTING ARDNER sums of money by permitting more efficient February, 1971 use of congested trallsportatio~l or com- 1. The Helix Rosy's instant acceptance of our model at first amazed me. I had feared that her sharp, stubborn mind, caught in her self-made antihelical trap, might dig up irrelevant results that would foster uncertainty about the correctness of the double helix. Nonetheless, like almost everyone else, she saw the appeal of the base pairs and accepted the fact that the structure was too pretty not to be true. James D. Watson, The Double Helix A STRAIGHT SWORD will fit snugly into a cular helix. This is a curve that coils around straight scabbard. The same is true of a a circular cylinder in such a way that it sword that curves in the arc of a circle: it crosses the "elements" of the cylinder at a can be plunged smoothly into a scabbard constant angle. Figure 1 makes this clear. of the same curvature. Mathematicians The elements are the vertical lines that sometimes describe this property of straight parallel the cylinder's axis; A is the constant lines and circles by calling them "self- angle with which the helix crosses every congruent" curves; any segment of such a element. Because of the constant curvature curve can be slid along the curve, from one of the helix a helical sword would screw its end to the other, and it will always "fit." way easily in and out of a helical scabbard. Is it possible to design a sword and its Actually the straight line and the circle scabbard that are not either straight or can be regarded as limiting cases of the curved in a circular arc? Most people, after circular helix. Compress the curve until giving this careful consideration, will an- the coils are very close together and you swer no, but they are wrong. There is a get a tightly wound helix resembling a third curve that is self-congruent: the cir- Slinky toy; if angle A increases to 90 de- of projections produce the cycloid and other familiar curves. Every helix, circular or otherwise, is an asymmetric space curve that differs from its mirror image. We shall use the term "right-handed" for the helix that coils clock- wise as it "goes away," in the manner of an ordinary wood screw or a corkscrew. Hold such a corkscrew up to a mirror and you will see that its reflection, in the words of Lewis Carroll's Alice, "goes the other way." The reflection is a left-handed corkscrew. Such a corkscrew actually can be bought as a practical joke. So unaccustomed are we to left-handed screw threads that a victim may struggle for several minutes with such a corkscrew before he realizes that he has to turn it counterclockwise to make it work. Aside from screws, bolts, and nuts, which are (except for special purposes) standard- ized as right-handed helices, most man- made helical structures come in both right and left forms: candy canes, circular stair- cases, rope and cable made of twisted 1. Circular helix (colored) on cylinder strands, and so on. The same variations in handedness are found in conical helices (curves that spiral around cones), including bedsprings and spiral ramps such as the grees, the helix collapses into a circle. On inverted conical ramp in Frank Lloyd the other hand, if you stretch the helix Wright's Guggenheim Museum in New until angle A becomes zero, the helix is York City. transformed into a straight line. If parallel Not so in nature! Helical structures rays of light shine perpendicularly on a abound in living forms, from the simplest wall, a circular helix held before the wall virus to parts of the human body, and in with its axis parallel to the rays will cast almost every case the genetic code carries on the wall a shadow that is a single circle. information that tells each helix precisely If the helix is held at right angles to the "which way to go." The genetic code it- rays, the shadow is a sine curve. Other kinds self, as everyone now knows, is carried by 2. Helical horns of the Pamir sheep have opposite handedness a double-stranded helical molecule of DNA, the right. A curious exception is the tooth its two right-handed helices twining around of the narwhal, a small whale that flourishes each other like the two snakes on the staff in arctic waters. This whimsical creature of Hermes. Moreover, since Linus Pauling's is born with two teeth in its upper jaw. pioneer work on the helical structure of Both teeth remain permanently buried in protein molecules, there has been increas- the jaw of the female narwhal, and so does ing evidence that every giant protein mole- the right tooth of the male. But the ~nale's cule found in nature has a "backbone" that left tooth grows straight forward, like a coils in a right-handed helix. In both nu- javelin, to the ridiculous length of eight or cleic acid and protein, the molecule's back- nine feet - more than half the animal's bone is a chain made up of units each one length from snout to tail! iZround this giant of which is an asymmetric structure of the tooth are helical grooves that spiral forward same handedness. Each unit, so to speak, in a counterclockwise direction [see Figure gives an additional twist to the chain, in 31. On the rare occasions when both teeth the same direction, like the steps of a helical grow into tusks, one would expect the right staircase. tooth to spiral clockwise. But no, it too is Larger helical structures in animals that always left-handed. Zoologists disagree on have bilateral symmetry usually come in how this could come about. Sir D'Arcy mirror-image pairs, one on each side of the Thompson, in his book 011 Growth and body. The horns of rams, goats, antelopes, Form, defends his own theory that the and other mammals are spectacular ex- whale swims with a slight screw motion amples [see Figure 21. The cochlea of the to the right. The inertia of its huge tusk human ear is a conical helix that is left- would produce a torque at the base of the handed in the left ear and right-handed in tooth that might cause it to rotate counter- 3. Helical grooves of the narwhal tooth are always left-handed clocku~isea s it grows (see "The Horn of found in Nebraska and Wyoming. These the Unicorn," by John Tyler Bonner; Sci- huge spirals, six feet or more in length, entific ,4merican, hlarch, 1951). are sometimes right-handed and sometimes \$'henever a single helix is prominent in left-handed. Geologists argued for decades the structure of any living plant or animal, over whether they are fossils of extinct the species usually confines itself to a helix plants or helical burrows made by ancestors of a specific handedness. This is true of of the beaver. The beaver theory finally countless forms of helical bacteria as well prevailed after remains of sillall prehistoric as of the spermatozoa of all higher animals. beavers were found inside some of the The hurnan umbilical cord is a triple helix corkscrews. of one vein and two arteries that invariably In the plant world helices are con~illoil coil to the left. The most striking instances in the structure of stalks, stems, tendrils, are provided by the conical helices of the seeds, flowers, cones, leaves-even in the shells of snails and other n~ollusks.N ot spiral arrangement of leaves and branches all spiral shells have a handedness. The around a stalk. The number of turns made chambered nautilus, for instance, coils on along a helical path, as you move from one one plane; like a spiral nebula, it can be leaf to the leaf directly above it, tends to be sliced into identical left and right halves. a number in the familiar Fibonacci series: But there are thousands of beautiful mol- 1, 2, 3, 5, 8, 13 . . . (Each number is the luscan shells that are either left- or right- sum of the preceding two numbers.) A handed [see Figure 41. Some species are large literature in the field known as "phyl- always left-handed and some always right- lotaxy" (leaf arrangement) deals with the handed. Sorne go one way in one locality surprising appearance of the Fibonacci and the other way in another. Occasional numbers in botanical phenoillena of this "sports" that twist the wrong way are prized sort. by shell collectors. The helical stalks of climbing plants are A puzzling type of helical fossil known usually right-handed, but thousands of as the devil's corkscrew (Daemonelix) is species of twining plants go the other way. 4. Three molluscan shells that are right-handed conical helices The honeysuckle, for instance, is always the woodbine the sweet honeysuckle/ left-handed; the bindweed (a family that Gently entwist." In Shakespeare's day includes the morning glory) is always right- "woodbine" was a common term for bind- handed. IVhen the two plants tangle with weed. Because it later came to be applied each other, the result is a passionate, vio- exclusively to honeysuckle many comrnen- lent embrace that ha5 long fascinated En- tators reduced the passage to absurdity glish poets. "The blue bindweed," wrote by supposing that Titania was speaking of Ben Jonson in 1617, "130th itself enfold wit11 honeysuckle twined with honeysuckle. honeysuckle." And Shakespeare, in A Mid- A~vareness of the opposite handedness of sumrrter Wight's Dreain, has Queen Titania bindweed and hone~.suckleh eiglitens, of speak of her intention to embrace Bottom course, the nleaning of Titania's metaphor. the Weaver (who has been transforlned into hlore recently, a charming song called a donkey) by saying: "Sleep thou, and I "hlisalliance," celebrating the love of will wind thee in my arms./ . . . So dot11 the honeysuckle for the bindweed, has been