# Mathematical Amusements

## Plaiting Polyhedrons

In Plato's dialogue Phaedo, Socrates tells a story in which the earth, viewed from outer space, appears many-colored like the balls that are made of 12 pieces of leather. Historians take this to mean that the Greeks made balls by stitching together 12 leather pentagons stained with different colors and stuffing the interior to make the surface spherical. Rigid pentagons that are regular and identical would of course make a regular dodecahedron, one of the five Platonic solids. There are all...

## The Blank Column

A secretary, eager to try out a new typewriter, thought of a sentence shorter than one typed line, set the controls for the two margins and then, starting at the left and near the top of a sheet of paper, proceeded to type the sentence repeatedly. She typed the sentence exactly the same way each time, with a period at the end followed by the usual two spaces. She did not, however, hyphenate any words at the end of a line When she saw that the next word (including whatever punctuation marks may...

## Math Trick For Amusements

How to form a loop with the Rolamite band while end A is taped to a tabletop is shown in Figure 109. Robert Neale, whom we encountered in the chapter on paper folding, suggested applying this to a playing card, say the joker. Use a razor blade to cut along the lines shown on the card at the left of Figure 110. Discard the shaded cut-out re- gion. By carefully executing the trick bend with the little square loop, taking care not to crease or tear its sides, you can produce the structure shown...

## Two-gliders Fuse

. 4*. simplest i l l r i . a spark a dirty j. Fuse b in Figure 151 oscillates with a period of 4, giving off sparks that fade quickly. A dirty fuse, like the one shown in c in Figure 151, leaves clouds of debris behind as it burns. At one point it shoots off a glider. Fuse d in Figure 151, named the baker by its discoverer, McClelland, is a confused fuse that bakes a string of stable loaves while it burns. The last three fuses all oscillate with periods of 4, and all four burn with the speed...

Solutions to the six graphs that readers were asked to number gracefully are shown in Figure 103. None of these number-ings is unique. Readers were also asked to improve on the rows of Go-lomb's triangle, each row giving the shortest-known rulers of n marks (including end points) such that every distance between a pair of marks is a distinct integer. Walter Penney of Greenbelt, Md., was the first to lower the eight-mark ruler to length 34. The same ruler was also found by hand by Daniel A....

## A A 1 fiSAA

That only one man the white queen can move see Figure 116, No. 7 . No one has yet found a way to place legally all 32 men so that no move is possible. There are many legal ways to place the 16 nonpawns to achieve a maximum of 46 captures, and all 32 men can be legally placed to allow 88 captures. How about illegal positions If 32 black knights go on black cells and 32 white knights on white cells, 336 captures are possible. This was considered the maximum for many decades until 1967, when T....

The answers to the 36 quickie type problems brought more surprises by mail than any previous collection of short problems. Readers caught ambiguous phrasings, indulged in amusing quibbles, found alternate and sometimes better answers, spotted some errors, and argued that the last problem is meaningless. I shall comment on this correspondence, taking the problems in numerical order, and add some further observations of my own. (4) C. C. Cousins, Charles W. Bostick, and others noticed that four...

## Set Of Quickies

The following problems are of the quickie type in the sense that they are quickly stated and, at least so I believed when I first gave them, not hard to solve if properly approached. Some are joke questions, and others contain booby traps to catch the unwary. Problem 1 You want to construct a rigid wire skeleton of a one-inch cube by using 12 one-inch wire segments for the cube's 12 edges. These you intend to solder together at the cube's eight corners. Why not cut down the number of soldering...

Inexpensive advertising premiums are popular in all countries where businesses compete for consumer attention, and frequently such premiums are based on mathematical puzzles. Many premiums of this kind have been discussed in columns that are reprinted in my earlier book collections, and one involving a map fold will be found in this book in the chapter on paper folding. Now I shall consider some classic puzzle premiums that I have not previously discussed. One of the oldest and best is the...

Probability theory abounds in paradoxes that wrench common sense and trap the unwary. In this chapter we consider a startling new paradox involving the relation called transitivity and a group of paradoxes stemming from the careless application of what is called the principle of indifference. Transitivity is a binary relation such that if it holds between A and B and between B and C, it must also hold between A and C. A common example is the relation heavier than. If A is heavier than B and B...

## Dodecahedronquintomino Puzzle

John Horton Conway defines a quintomino as a regular pentagon whose edges (or triangular segments) are colored with five different colors, one color to an edge. Not counting rotations and reflections as being different, there are 12 distinct quintominoes. Letting 1, 2, 3, 4, 5 represent the five colors, the 12 quintominoes can be symbolized as follows The numbers indicate the cyclic order of colors going either clockwise or counterclockwise around the pentagon see Figure 12, left . In 1958...

## Golombs Graceful Graphs

One of the least explored areas of modern mathematics is a class of problems that combine graph theory and arithmetic. Recreational problems of this type have been discussed before in my earlier book collections for example, in the chapter on Magic Stars and Polyhedrons in Mathematical Carnival. In this chapter we take up a family of numbered-graph problems that has recently been defined and developed by Solomon W. Go-lomb, professor of engineering and mathematics at the University of Southern...

## Salmon On Austins

In Chapter 8 one of the short problems, posed by A. K. Austin of the University of Sheffield, England, aroused considerable controversy among readers. Indeed, the problem proved to be an amusing new variant of Zeno's famous paradox of Achilles and tire Tortoise, and one that, so far as I know, had never been Formulated before. Here is how I phrased the problem and its answer A boy, a girl and a dog are at the same spot on a straight road. The boy and the girl walk forward the boy at four miles...

## Mathematical Tricks With Cards

Do you like card tricks No, I hate card tricks, I answered. Well, I'll just show you this one. f e showed me three. Maugham's experience with card magicians is all too familiar. I don't really like people who do card tricks, Elsa Maxwell once wrote (I quote from an autobiography of a lady magician, You Don't Have to Be Crazy, by Frances Ireland). They never stop at one or two, but go on and on and on, and always make you take cards, or turn up cards, or cover cards, until you are worn out....

## The Game Of Life Part

So much has been discovered about Conway's Life since I first wrote the last two chapters, that it was impossible to summarize the highlights in an addendum. A book could and should be written about the game, an Encyclopedia of Life, or a Handbook of Life, that would put all the important known Life forms on record and thereby save Lifenthusiasts the labor of rediscovering them. The eleven issues that appeared of Robert Wainwright's periodical Lifeline continue to be the main repository of such...

## The Game Of Life Part Ii

Cellular automata theory began in the mid-fifties when John von Neumann set himself the task of proving that self-replicating machines were possible. Such a machine, given proper instructions, would build an exact duplicate of itself. Each of the two machines would then build another, the four would become eight, and so on. This proliferation of self-replicating automata is the basis of Lord Dunsany's amusing 1951 novel The Last Revolution. Von Neumann first proved his case with kinematic...

## The Game Of Life Part I

Most of the work of John Horton Conway, a distinguished mathematician at the University of Cambridge, has been in pure mathematics. For instance, in 1967 he discovered a new group some call it Conway's constellation that includes all but two of the then known sporadic groups. (They are called sporadic because they fail to fit any classification scheme.) It is a breakthrough that has had exciting repercussions in both group theory and number theory. It ties in closely with an earlier discovery...

## And Other Curious Questions

Pride in craftsmanship obligates the mathematicians of one generation to dispose of the unfinished business of their predecessors. Two familiar irrational numbers are it (3.141 . . .), the ratio of the circumference of a circle to its diameter, and e (2.718 . . .), the base of natural logarithms. Each has a nonrepeating decimal fraction. Both tt and e are also transcendental numbers, that is, numbers that are not algebraic. Specifically, a transcendental number is an irrational number that is...

Everyone who calls a chess problem beautiful is applauding mathematical beauty, even if it is beauty of a comparatively lowly kind. Chess problems are the hymn-tunes of mathematics. G. H. Hardy, A Mathematician's Apology It has been my policy to avoid chess problems of the type Mate in n moves on the assumption (perhaps a mistaken one) that too few readers play chess and that, even among those who do, too few like chess problems. In this chapter, however, I shall consider a variety of what are...

## Ninedigit Problem

One of the satisfactions of recreational mathematics comes from finding better solutions for problems thought to have been already solved in the best possible way. Consider the following digital problem that appears as Number 81 in Henry Ernest Dudeney's Amusements in Mathematics. (There is a Dover reprint of this 1917 book.) Nine digits (0 is excluded) are arranged in two groups. On the left a three-digit number is to be multiplied by a two-digit number. On the right both numbers have two...

## The Rotating Disk

A frieze is a pattern that endlessly repeats itself along an infinite strip. Such patterns can exhibit different kinds of basic symmetry, but here we shall be concerned only with what is called glide symmetry. A glide consists of a slide (more technically a translation) combined with mirror reflection and a half-turn. For example, repeatedly gliding the letter R to the right along a strip generates the following frieze H. S. M. Coxeter, a geometer at the University of Toronto, recently...

## The Flexible Band

Simmons, in charge of research and development at Rolamite Inc., Albuquerque, N.M., sent this curious topological problem. Work at Rolamite involves complex banded rolling systems. One of the Rolamite engineers, Virgil Erbert, was confronted in the course of his work with the problem shown in Figure 105. End A of a flexible band was fastened to an object that was too large to pass through the slot at end B. It was essential that the band be formed into the loopedconfiguration shown...

## Nimand Hackenbush

William Shakespeare, Corporal Nym in The Merry Wives of Windsor In recent decades a great deal of significant theoretical work has been done on a type of two-person game that so far has no agreed-on name. Sometimes these games are called nim-like games, take-away games or disjunctive games. All begin with a finite set of elements that can be almost anything counters, pebbles, empty cells of a board, lines on a graph, and so on. Players alternately remove a...

## The Game Of Halma

An admirable place for playing halma, said Chelifer, as they entered the Teatro Metastasio. Aldous Huxley, Those Barren Leaves Two new families of puzzles based on a long-neglected counter-moving game have recently come to light. Each family offers a series of unsolved problems and the opportunity to devise ingenious proofs that some solutions are impossible. The puzzles stem from Dialogue on Puzzles, a splendid collection of unusual problems by Kobon Fujimura and Michio Matsuda published in...

## Ticktacktoe Games

It's as simple as tit-tat-toe, three-in-a-row, and as easy as playing hooky. I should hope we can find a way that's a little more complicated than that, Huck Finn. Mark Twain, The Adventures of Huckleberry Finn Ticktacktoe (the spelling varies widely) is not nearly so simple as Tom Sawyer thought. When Charles Sanders Peirce wrote his Elements of Mathematics, a textbook that was not published until 1976, he included a 17-page analysis of only the side opening of this ancient game. It was one of...

## Of Paper Folding

The easiest way to refold a road map is differently. One of the most unusual and frustrating unsolved problems in modern combinatorial theory, proposed many years ago by Stanislaw M. Ulam, is the problem of determining the number of different ways to fold a rectangular map. The map is pre-creased along vertical and horizontal lines to form a matrix of identical rectangles. The folds are confined to the creases, and the final result must be a packet with any rectangle on top and all the others...

How then can they combine To form a line Every finite set of n elements has 2 subsets if one includes the original set and the null, or empty, set. For example, a set of three elements, ABC, has 23 8 subsets ABC, AB, BC, AC, A, B, C, and the null set. As the philosopher Charles Sanders Peirce once observed (Collected Papers 4. 181), the null set has obvious logical peculiarities. You can't make any false statement about its members because it has no members. Put another...

## And Fermats Last Theorem

The methods of Diophantus May cease to enchant us After a life s pent trying to gear 'em To Fermat's Last Theorem. An old chestnut, common in puzzle books of the late 19th century when prices of farm animals were much lower than today , goes like this. A farmer spent 100 to buy 100 animals of three different kinds. Each cow cost 10, each pig 3 and each sheep 50 cents. Assuming that he bought at least one cow, one pig and one sheep, how many of each animal did the farmer buy At first glance this...

## Diophantine Analysis And Fermats Last Theorem

The methods of Diophantus May cease to enchant us After a life spent trying to gear 'em To Fermat's Last Theorem. An old chestnut, common in puzzle books of the late 19th century (when prices of farm animals were much lower than today), goes like this. A farmer spent 100 to buy 100 animals of three different kinds. Each cow cost 10, each pig 3 and each sheep 50 cents. Assuming that he bought at least one cow, one pig and one sheep, how many of each animal did the farmer buy At first glance this...

## Introduction

What is it Ennui, I said. The easiest of all. No rules, no boards, no equipment. What is Ennui Amanda asked. Ennui is the absence of games. Donald Barthelme, Guilty Pleasures. Unfortunately, as recent studies of education in this country have made clear, one of the chief characteristics of mathematical classes, especially on the lower levels of public education, is ennui. Some teachers may be poorly trained in mathematics and others not) trained at all. If...