## Answers

The errors in the fallacious geometric proofs are briefly explained as follows Theorem 1. An obtuse angle is sometimes equal to a right angle. The mistake lies in the location of point K. When the figure is accurately drawn, K is so far below line DC that, when G and K are joined, the line falls entirely outside the original square ABCD. This renders the proof totally inapplicable. Theorem 2. Every triangle is isosceles. Again the error is one of construction. F is always outside the triangle...

## Scrambled Quotation

Letters in the sentence Roses are red, violets are blue are scrambled by the following procedure. The words are written one below the other and flush at the left The columns are taken from left to right and their letters from the top down, skipping all blank spaces, to produce this ordering The task is to find the line of poetry that, when scrambled by this procedure, becomes Walter Penney of Greenbelt, Md., contributed this novel word problem to the February 1970 issue of Word Ways The Journal...

## Xl

And the window since both would fall to the ground, but removing the edge joining the spider to the window removes only the spider. Taking edge B chops down the entire apple tree. If one edge of the streetlight's base is taken, the structure still stands, but taking the second edge on a later move topples the entire structure. The person who takes the picture's last edge is the winner. As in nim, every picture is either safe (second-player win) or unsafe (first-player win), and the winner's...

## Problem

ABCD are four consecutive digits in increasing order. DCBA are the same ffo ut in decreasing order. The four dots represent the same four digits in an unknown order. If the sum is 12,300 what rrumber is represented by the four dots (W. T. Williams and G-. H. Savage, The Strand Problems Book.) Problem 21 A primeval snake is formed by writing the positive integers consecutively along a snaky path see Figure 38 . If contimaed upward to infinity, every prime number will fall on the same diagonal...

## Kobon Triangles

Kobon Fujimura, a Japanese puzzle expert, recently invented a problem in combinatorial geometry. It is simple to state, but no general solution has yet been found. What is the largest number of nonoverlapping triangles that can be produced by n straight line segments It is not hard to discover by trial and error that for n 3, 4, 5 and 6 the maximum number of triangles is respectively one, two, five and seven see Figure 108 . For seven lines the problem is no longer easy. The reader is asked to...

## Addendum

John Selfridge reports that a solution has been found for his 4 x 4 infinity ticktacktoe. This is played on a strip that is four cells high and infinitely wide, the winner being the first to get four of his marks in an orthogonal or diagonal row. Carlyle Lustenberger, in his master's thesis in computer science at Pennsylvania State University, developed a computer program with a winning strategy for the first player on a four-by-30 board. The actual lower bound for the width is a few cells...

## And Other Probability Paradoxes

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...

## 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 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...

## Alephs And Supertasks

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...

## 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...

## 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...

## 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...

## 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...

## Info

17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34, 17,____ Michael Beeler, William Gosper and Rich Schroeppel give these results in HAKMEM (short for Hacker's Memo), Memo 239, Artificial Intelligence Laboratory, M.I.T., 1972, page 64. No one has yet come up with good ideas about how to establish the general case for all nonzero integers. (Zero, of course, is already in a 0, 0, 0, . . . loop.) No one knows if there are other loops, or if there are integers that...

## 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...

## 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...

## 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...

## V5 x2

Solve for x and change the result to the nearest integer. It is 618,034. Because no other integer, when related to 1,000,000, gives a closer approximation of the golden ratio, 618,034 is the next-to-last term of the longest possible chain of positive integers in a generalized Fibonacci series ending in 1,000,000. One can now easily work backward along the chain to the first two terms. (This method is explained in Litton Industries' Problematical Recreations, edited by Angela Dunn, Booklet 10,...

## 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...

## Chess Tasks

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...

## The Twenty Bank Deposits

A Texas oilman who was an amateur number theorist opened a new bank account by depositing a certain integral number of dollars, which we shall call x. His second deposit, y, also was an integral number of dollars. Thereafter each deposit was the sum of the two previous deposits. (In other words, his deposits formed a generalized Fibonacci series.) His 20th deposit was exactly a million dollars. What are the values of x and y, his first two deposits (I am indebted to Leonard A. Monzert of West...

## S

The only pentomino that does not end quickly by vanishing, becoming stable or oscillating is the R pentomino a in Figure 129 . Conway has tracked it for 460 ticks. By then it has thrown off a number of gliders. Conway remarks It has left a lot of miscellaneous junk stagnating around, and has only a few small active regions, so it is not at all obvious that it will continue indefinitely. Its fate is revealed in the addendum to this chapter. .' ' , r i b...

## Wheels

The miraculous paradox of smooth 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. Transportation aside, if we consider wheels as simple machines pulleys, gears, gyroscopes and so on it is hard to imagine any advanced society without them. H. G. Wells, in The War of the Worlds, describes a Martian civilization far...

## 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...

## 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...

## Advertising Premiums

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...

## The Three Coins

Three coins are on the table a quarter, a half-dollar and a silver dollar. Smith owns one coin and Jones owns the other two. All three coins are tossed simultaneously. It is agreed that any coin falling tails counts zero for its owner. Any coin falling heads counts its value in cents. The tosser who gets the larger score wins all three coins. If all three come up tails, no one wins and the toss is repeated. What coin should Smith own so that the game is fair, that is, so that the expected...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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....

## 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...