Math Puzzles and Games

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MORE MATH PUZZLES ANDGAMES

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by Michael Holt

ILLUSTRATIONS BY PAT HICKMAN

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WALKER AND COMPANY New York

Copyright

©1978

by Michael Holt

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electric or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the Publisher. First published in the United States of America in

1978

by the Walker Publishing

Company, Inc. Published simultaneously in Canada by Beaverbooks, Limited, Pickering, Ontario Cloth ISBN: Paper ISBN:

0-8027-0561-8 0·8027·7114·9

Library of Congress Catalog Card Number: Printed in the United States of America

10 9 8 76 5 43 2 1

77·75319

CONTENTS I ntroduction

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1. Flat and Solid Shapes 2 . Routes, Knots, and Topology

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3. Vanishing-Line and Vanishing-Square Puzzles

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4. Match Puzzles

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5. Coin and Shunting Problems

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6. Reasoning and Logical Problems

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7. Mathematical Games

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Answers

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INT RODUCTION Here is my second book of mathematical puzzles and games. In it I have put together more brainteasers for your amusement and, perhaps, for your instruction. Most of the puzzles in this book call for practical handiwork rather than for paper and pencil calculations-and there is no harm, of course, in trying to solve them in your head. I should add that none call for prac­ ticed skill; all you need is patience and some thought. For good measure I have included an example of most types of puzzles, from the classical crossing rivers kind to the zany inventions of Lewis Carroll. As with the first book of mathe­ matical puzzles, I am much indebted to two great puzzlists, the American Sam Loyd and his English rival Henry Dudeney. Whatever the type, however, none call for special knowledge; they simply requ ire powers of deduction, logical detective work, in fact. The book ends with a goodly assortment of mathematical games. One of the simplest, "Mancalla," dates back to the mists of time and is still played in African villages to this day, as I have myself seen in Kenya. "Sipu" comes from the Sudan and is just as simple. Yet both games have intriguing subtleties you will discover when you play them. There is also a diverse selec­ tion of match puzzles, many of which are drawn from Boris A. Kordemsky's delightful Mosco w Puzzles: Three Hun dred Fifty­

Nine Ma thema tical Recrea tions (trans. by AIbert Parry, New York: Charles Scribner's Sons,

1972); the most original, how­

ever, the one on splitting a triangle's area into three, was given me by a Japanese student while playing with youngsters in a playground in a park in London.

v

A word on solving hard puzzles. As I said before, don't give up and peek at the answer if you get stuck. That will only spoil the fun. I've usually given generous hints to set you on the right lines. If the hints don't help, put the puzzle aside; later, a new line of attack may occur to you. You can often try to solve an easier puzzle similar to the sticky one. Another way is to guess trial answers just to see if they make sense. With luck you might hit on the right answer. But I agree, lucky hits are not as satisfy­ ing as reasoning puzzles out step by step. If you are really stuck then look up the answer, but only glance at the first few lines. This may give you the clue you need without giving the game away. As you will see, I have written very full answers to the harder problems or those need­ ing several steps to solve, for I used to find it baffling to be greeted with just the answer and no hint as to how to reach it. However you solve these puzzles and whichever game takes your fancy, I hope you have great fun with them.

-Michael Holt

VI

1. Flat and Solid Shapes All these puzzles are about either flat shapes drawn on paper or solid shapes. They involve very little knowledge of school geometry and can mostly be solved by common sense or by experiment. Some, for example, are about paper folding. The easiest way to solve these is by taking a sheet of paper and fold­ ing and cutting it. Others demand a little imagination: You have to visualize, say, a solid cube or how odd-looking solid shapes fit together. One or two look, at first glance, as if they are going to demand heavy geometry. If so, take second thoughts. There may be a perfectly simple solution. Only one of the puzzles is

a/most a trick. Many of the puzzles involve rearranging shapes or cutting them up.

Real Estate ! K .O . Properties Universal , the sharp est realtors in the West, were putting on the m arket a triangular p lot of land sm ack on Main Street in the p riciest part of the uptown sh opping area. K .O.P.U.'s razor-sharp assistant put this ad in the local p aper:

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THIS VALUABLE SITE I DEAL FOR STORES OR OFFICES Sale on A pril I

Why do y ou think there were no buyers?

1

Three-Piece Pie How can you cut up a triangular cranberry p ie this shape into three equal p ieces, each the same size and shape? You can do it easily . First cut off the c rust with a straigh t cut and ignore i t .

How Many Rectangles? How m any rectangles can y ou see?

Squaring Up How many squ ares can y ou find here? Remember, some squares are p art of o ther b igger squares.

2

Triangle Tripling Copy the blank triangle shown here. Divide it into sm aller ones by drawing another shaded triangle in the m iddle ; this m akes 4 triangles in all. Then repeat by drawing a triangle in the m iddle of each of the blank triangles, m aking 13 triangle s altogether. Repeat the process. Now how m any shaded and blank triangles will y ou get? And can y ou see a p at tern to the num bers of triangles? If y ou can, you will be able to say how m any triangles there will be in further d ivisions withou t actu ally drawing ;n the triangle s.

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13 triangles

The Four Shrubs Can y ou plant four shrubs at equal distances from each other? How d o you do it? HINT:

A square p attern won't do becau se opp osite corners are further apart than corners along one side of the square .

Triangle Teaser It's easy to p ick out the five triangles in the triangle on the left . But how m an y triangles can you see in triangle a and in triangle b ?

a

b

3

Triangle Trickery Cut a three-four-five triangle out of p aper. Or arrange 1 2 matches as a three- four-five triangle ( 3 + 4 + 5 1 2) . =

Those o f y ou w h o k n o w about Pyth agoras' s theorem will also know it must be right­ angled . The Egyp tian pyra m id builders used ropes with th ree-four-five knots to m ake righ t angles. They were called rope stretchers. The are a shut in by the triangle is (3 X 4)/ 2 . I f y ou d o n ' t know t h e form ula for t h e area of a triangle , think of it as half the area of a th ree-by-four rectangle. The puzzle is this:

U sing the same p iece of p aper ( or the sam e 1 2 m atches) , sh ow 1/ 3 of 6 2 . =

HINT :

This is a really difficult puzzle for adults! Think of the triangle divided into third s this way :

triangle If you are using paper, fold it along the dotted lines.

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Fold 'n Cut Fold a sheet of p aper once, then again the opp osite way . Cut the c orn er , as shown . Open the folded sheet out an d , as y ou see , there is one hole , in the m iddle .

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Now guess what happens when y ou fold three times and cut off the corner. How m any holes will there be n ow?

Four-Square Dance How m any d ifferent ways can y ou j oin four squares side to side? Here is one way . Don't count the same way in a different p ositio n , like the second one shown here , which is j ust the same as the first. Only count differe n t shapes.

Net fo r a Cube Each sh ap e here is m ade up of six squ ares j o ined side to side. Draw one, cut it out, and it will fold to form a cube. M athem aticians call a p lan like this a net. How m any d ifferent nets for a cube can you d raw? Only count differen t ones. For instance, the second net is the same as the first one turned round .

5

Stamp Stumper Phil A. Telist had a sheet of 24 stamp s , as sh own. He wants to tear out of the sheet j ust 3 stamp s but they must be all j o ined up. Can y ou find six differen t way s Phil can d o so? The shaded p arts sho w t wo ways.

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The Four Oaks A farm er had a sq uare field with four equally sp aced oak s in it stand ing in a row from the center to the m iddle of one sid e , as shown. In his will he left the square field to h is four sons "to be divided up into four identical p arts, e ach with its oak . " H o w did the sons divide u p the land?

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Box the Dots Copy this hexagon with its n ine dots. Can you d raw n ine lines of equal length to box off each dot in its own oblong? All oblongs must be the sam e siz e , and there must be no gaps between them .

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Cake Cutting Try to cut the cake sh own into the greatest num ber of p ieces with only five straight cuts of the bread k nife . HINT :

I t ' s m ore than the 1 3 p ieces sho wn here .

7

Four-Town Turnpike Four towns are placed at the corners of a ten-mile square . A turnpik e net­ work is needed to link all four of the towns. What is the shortest network y ou can p lan?

Obstinate Rectangles On a sheet of square d paper, m ark out a rectangle one square by two squares in siz e , lik e this :

[d/I J oin a pair of opp osite corners with a line , a diagonal. How m any squares d oes it slice through? As y ou see , two squares. Do the same for a b igger rectangle , two by three squares say . The d iagonal cuts four squares.

PUZZLE:

Can y ou say how m any squares will be cut by the d iagonal of a rectangle six by seven squares- withou t d rawing and counting? In short , can y ou work out a rule? Be careful to work only with rectangles, not squares. It's m uch harder to fin d a rule for squ ares. Stick to rectan gles ! HINT :

Add t h e le ngth a n d t h e width of each rectangle. Then look a t the num ber of sq uares cut.

One Over the Eight Here is an inte resting p attern of numbers y ou can get by drawing grids w ith an odd number of squares along each side . Begin with a three-by-three grid , as sh own in pictu re a. The central square is shaded, an d there are eight squares around it. We have , then , one square in the m iddle plus the other eigh t , or I + (8 X I ) = 9 squ ares in all. Now look at grid b: It has one

8

central square , shaded, and several step-shaped j igsaw p ieces, each m ade up of three squares. B y copying the grid and shading , c an y ou find how many j igsaw p ieces m ake up the complete grid? Then the number of squares in the complete grid sh ould be the number in each "j ig" times 8, plus I : I + ( 8 X 3 ) = 2 5 . Next , i n grid c see i f y ou can copy and finish off the j ig­ saw p iece s ; one has been drawn for y ou. Then com plete the num ber p at­ tern : I + 8 j igs = 49. You've got to find what num ber of squares there are in a j ig. Could y ou write the number p attern for a n ine-by-nine grid- with­ out even d rawing it? a

b

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9

G reek Cross into Square Out of some p ostcards cut several Greek crosses , like these shown here . Each , as y ou can see , is m ade up of five squares. What y ou have to do is cut up a Greek cross an d arrange the p ieces to form a perfe ct square . The cuts are indicated on drawings a, b, and c. In the last two puzzles, d and e, y ou need two Greek crosses to make up a squ are . See if y ou can do it. There is n o answer.

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I nside-out Co llar Take a strip of stiff p aper and m ake it into a square tube. A strip one inch wide and four inches long- with a tab for stick ing- will do nicely. Crease the edges and d raw or score the diagonals of each face before sticking the ends of the strip together ; scissors m ake a good scoring instrument . The trick is to turn the tub e inside out without tearing it. If y ou can't d o it , turn to the answer section.

10

Cocktails for Seven The picture shows how three cocktail st ick s can be connected with cherries to m ake an equilateral triangle. Can y ou form seven equilateral triangles with nine cocktail sticks? You can use m atchsticks and balls of p lasticine instead .

The Carpenter's Co lored Cubes A carpenter was m ak ing a child's game in which pictures are p asted on the six faces of wooden cubes. Suddenly he found he n eeded twice the surface area that he had on one big cube. How d id he double the area with out add­ ing another cube?

Painted Blocks The outside of this set of blocks is p ainted . How m any sq uare faces are painted?

11

I nstant I nsanity This is a p uzzle of putting four identically colored cubes together in a long block so no adjacent squ ares are the same color. You can m ake the cubes y ourself from the four nets sho wn in the p icture .

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3 In this p uzzle y ou have four cubes. Each cube's faces are p ainted with four d ifferent colors . Put the four cubes in a long rod so that no colors are re­ peated along each of the rod's fou r long sides. S ince there are over 40,000 d ifferent arrangements of the cubes in the rod , trying to solve the puzzle in a h it-or-miss fash ion is likely to drive y ou insane ! Y ou can m ake the cubes y ourself by cutting out the four cross-shaped nets sh own here. You can , of course , use red , green , blu e , and white , for instance , instead of our black , d otte d , h atched, and white. There is a l -in-3 chance of correctly p lacing the first cube , which has three like faces. The odds of correctly placing each of the other cubes is 1 in 24 : Each cu be can be sitting on any of its six faces ; and for each of these p ositions it can be facing the adjacent cube in four different ways-a total of 24 p ositions. Multip ly 3 X 24 X 2 4 X 24, and the answer is 4 1 ,47 2 -the t otal number of ways of arranging the cubes. S e e answer section for solution.

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The Steinhaus Cube This is a well-k nown puzzle invented by a m athem atician , H. Steinhaus ( say it S tine-h ouse ). The p roblem is to fit the six odd-shap e d p ieces to­ geth er to m ake the b ig three-by-three-by-three cube shown at top left of the p icture. As y ou can see , there are three p ieces of 4 little cubes and three p ieces of 5 little cubes, m aking 27 little cubes in all-j u st the right number to m ake the big cube. To solve the puzzles, the best thing is to m ake up the p ieces by gluing little wooden cubes together.

13

How Large I s the Cube? Plato, the Greek philosopher, thought the cube was one of the most per­ fect sh apes. So it's quite possible he wondered about this proble m : What size cube has a surface are a equal ( in nu mber) to its volu me? You had better work in inches ; of course , Plato d idn't !

Plato's Cubes A p roblem that Plato really did d ream up is this one : The sketch shows a huge b lock of m arble in the shap e of a cube. The block was made out of a certain number of sm aller cubes and stood in the m iddle of a square plaza p aved with these smaller m arble cubes. There were j u st as m any cubes in the p laza as in the huge block , an d they are all p recisely the same size . Tell how m any cubes are in the huge block and in the square p laza it stands on. HINT : One way to solve this is by trial an d e rror. Suppose the huge block is 3 cubes h igh ; it the n has 3 X 3 X 3 , or 2 7 , cubes in it. But the p laz a has to be su rfaced w ith exactly this num ber of cubes. The nearest size p laza is S by S cubes, which has 2S cubes in it ; this is too few. A plaza of 6 by 6 cubes h as far too m any cubes in it. Try , in turn , a huge block 2 , then 4, then S block s h igh .

The Half-full Barrel Two farmers were staring into a large barrel partly filled with ale . One of them said : " It's over half full ! " But the other declared : "It's more than half em pty ." How could they tell with out using a ruler, string , b ottles , or other m easuring devices if it was m ore or less than exactly half full?

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Cake-Tin Puzz le The round cake fits snugly into the squ are tin sh own here . The cake's radius is 5 inches. S o how large must the tin be?

Animal Cubes Look at the p icture of the d inosaur and the gorilla m ad e out of little cubes. How m any cubes m ake up each anim al? That was easy enough , wasn't it? B ut can y ou say what the volu me of each animal is? The volume of one lit­ tle cube is a cubic centim eter. That wasn 't too hard , either, was it? All right the n , can you say what the surface area of each animal is? The surface area of the face of one little cube is I square centimeter.

15

Spider and Fly A sp ider is sitting on one corner of a large box, and a fly sits on the oppo­ site c orner. The sp ider h as to be quick if he is to catch the fly . What is his shortest way ? There are at least four shortest ways. How m any shortest lines can y ou fin d?

The Sly Slant Line The artist has d rawn a rectangle inside a c ircle. I can tell you that the cir­ cle's diameter is 1 0 inches long. Can y ou tell me how long the slant line, m arked w ith a question m ark , is? H INT :

Don't get tangle d up with Pythagoras's theorem. If you don't know it, all the better!

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2. Routes, Knots, and Topology In fact all these puzzles are about the math of topology, the geometry of stretchy surfaces. For a fuller description of what topology is about, see the puzzle "The Bridges of Konigsberg" on page

25.

The puzzles include problems about routes, mazes,

knots, and the celebrated Mobius band.

I n-to-out Fly Paths A fly settles inside each of the shapes sh own and tries to cross each side once only , always ending up outside the shap e . On which shapes can the fly trace an in-to-out p ath? The picture shows he can on the triangle . Is there , perhap s, a rule?

I n-to-in Fly Paths This time the fly begins and ends inside each shape. Can he cross each side once only ? The p ictu re shows he cannot do so on the triangle : He cannot cross the third side and end up inside . Is there a rule here?

No 17

A BC Maze Begin at the arrow and let y our finger take a walk through this m aze. Can y ou p ass along e ach p ath once only and come out at A ? at B? and at C?

Eternal Triangle? Can y ou d raw this sign in one unbroken line without crossing any lines or taking y our pencil off the p aper? The sign is often seen on Greek monu­ ments. Now go over the same sign in one unbroken line but making the fewest number of turn s. Can y ou draw it in fe wer th an ten tu rn s?

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The Four Posts Draw three straight lines to go through the four posts sh own here without retracing or lifting y our pencil off the p aper. And you m ust return finally to y our starting p oint.

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The Nine Trees Find four straight lines that touch all nine trees. I n this puzzle y ou don't have to return to your starting p oint ; indeed y ou cannot! Do the " Four Posts" puzzle an d y ou should be able to do this one.

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Salesman's Round Trip A traveling salesman starts from h is home at Anville (A ). He has to visit all three towns sh own on the sketch map -Beeburg (B) , Ceton (C) , and Dee C ity (D ) . B ut he w ants to save as m uch gas as he can . What is his shortest route? The m ap shows the d istances between each town. So A is eight m iles from C, and B is six m iles from D.

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Swiss Race The sk etch m ap here sh ows the roads on a race through the Swiss Alps from Anlaken (A ) to Edelweiss (E) through the checkp oints B, C, and D. An avalanche blocks the roads at three p oints, as y ou can see. You've got to clear j u st one road block to m ake the sh ortest way to get through from A nlaken t o Edelweiss. Which one is it? And how long is the route then?

A

20

Get Through the Mozmaze The m aze sh own here is called a m ozmaze because it is fu ll of awfu l , b iting dogs, calle d mozzles. Top Cat is at the top left-han d corner, an d he has to get through the m oz m aze to the lowe r right comer, where it say s E N D . B ut o n h i s way he h a s t o pass the biting m ozzles chained at t h e various corners of th e m oz m aze . The triangles m ark the p osition of the d ogs that give three bites as Top Cat p asses each of them ; the squares of the d ogs that give two bites ; an d the c ircles of the d ogs that give only one b ite . What is Top Cat's best way through the m oz m aze so that he gets bitten the fewest times? What ' s the fewest number of bites he can get by with? Can you do better than 40 b ites?

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END 21

Space-Station Map Here is a m ap of the ne wly built space stations ano th e shuttle service link ing them in A.D. 2000. Start at the station m arked T, in the south , and see if y ou can spell out a complete English sentence by m aking a round­ trip tour of all the stations. Visit each station only once, and return to the starting point. This p uzzle is b ased on a celebrated one by America's greatest puzzlist , S a m Loy d . When i t first appeared in a m agazine , m ore than fifty thousand readers reporte d , "There is n o possible way." Yet it is a really simple puzzle.

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Round-Trip Flight Tran s-Am Airway s offers flight links between these five c ities : Alban y , B altim ore , Chicago , Detroit , an d E l Paso. There are eight flights, a s follows : Baltimore to Chicago , Detroit to Chicago, Al b any to Baltimore , Chicago to El Paso , Chicago to Detroit , B altimore to Albany , Albany to El Paso , and Chicago to Albany . What is the shortest way to m ake a trip from Albany to Detroit and b ack again? HINT : Draw a sketch m ap of the flights , beginning : A � B � C. This will show y ou how to avoid making too m any flights or getting stu ck in a "trap ! "

Faces, Corners, and Edges Here is a surprising rule ab out shape s y ou should be able to puzzle out for y ourself. Find a box-a m atchbox , a b ook , or a candy box , say . N o w run y our finger along the edges and c ount the m ( 1 2) and add 2 to the number you found ( m aking 1 4) . Now count the number of faces ( 6) an d add to that num ber the num ber of corners ( 8 ) , m aking 1 4 in all. It seems that there is a rule here . Count faces and corners and edges of the sh ape s shown in our p icture ; the dotted lines indicate hidden edges that y ou cannot see from the head-on view. Can you find the rule? The great Swiss m athemati­ cian Leonhard Euler ( say it oiler) was the first to sp ot it . The names of the shap es are te trahedron (4 faces) , octah edron (8 faces) , dodecahedron ( 1 2 faces) , and icosahedron ( 2 0 faces) .

23

Five City Freeways A p lanner wants to link up five cities by freeways. Each city must be linked to every other one. What' s the least number of roads he must have? R oads can cross by m eans of overpasses, of course. The planner then decides that overp asses are very costly . Wh at is the fewest number of overpasses he needs?

The Bickering Neighbors There were three n eighb ors who shared the fen ced park shown in the p ic­ ture . Very soon they fell to b ickering with one another. The owner of the center house complained that his neighbor' s d og dug up his garden and p rom ptly built a fenced p ath way to the opening at the bottom of the p icture . Then the neigh b or on the right built a p ath from his house to the opening on the left , and the m an on the left built a path to the opening on the right. None of the paths crossed. Can y ou d raw the p aths?

24

The Bridges of Konigsberg This is one of the most fam ous p roblems in all math . It saw the start of a whole new bran ch of m ath called topology , the geometry of stretchy sur­ faces. The p roblem arose in the 1 700s in the north German town of Konigsberg , built on the River Pregel, wh ich , as the p icture sho ws, splits the town into four p arts.

In summer the townsfolk liked to take an evening stroll across the seven bridges. To their surp rise they d iscovered a strange thing. They found they could n ot cross all the bridges once and once only in a single stroll without retracing their steps. Copy the m ap of Konigsberg if this is not y our b ook , and see if y ou agree with the Konigsbergers . The p roblem reached the ears of the great Swiss m athematician Leon­ hard Eule r. He dre w a b asic network , a s m athematicians would say , of the routes link ing the four p arts of the town. This cut out all the unnecessary details. Now follow the strolls on the network . Do y ou think the Konigs­ bergers could m anage such a stroll or not?

25

Euler's Bridges Euler actu ally solved the last p ro blem in a slightly different way from the one we gave , which is the way m ost book s give . What he did was to sim­ plify the p roble m . He started off with the very simple p roblems we give below. He then went on from their solutions to arrive at the solution we gave to "The Bridges of Konigsberg." The little p roblems go like this: A straight river has a north bank an d a south b ank with three bridges crossing it. Starting on the north bank an d crossing each bridge once only in one stroll without retracing y our steps, y ou touch the north bank twice ( see pictu re a). For five bridges ( picture b ) y ou touch n orth three times. Can you find a rule for any odd number of bridges?

North Bank

South Bank

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3 Norths

5 Bridges

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Now look at picture c. You touch the north bank twice for two bridges; and as shown by picture d, y ou touch north th ree times for four bridges. Can y ou fi nd a rule for any eve n num ber of b ridges?

Mobius Band One of the most fam ous odd ities in topology is the one-edged , single­ surfaced ban d invente d by Augu st Mobius. He was a nineteenth-century Germ an p rofessor of m ath . Take a collar an d before joining it give it one half-twist . Now cut it all the way along its middle . How m any p arts do y ou think it will fall into? You can try this o n y ou r friends as a party trick . Then try cutti ng i t o n e th ird in from a n edge , all the way rou n d . How m any p arts d o y ou think it will fall i nto n o w?

26

Double Mobius Band Take two strip s of paper and place them together, as shown. Give them both a half-twist and then j oin their ends, as shown in the p icture. We now have what seems to be p air of nested Mobius bands. You can sh ow there are two bands by putting y our finger between the b ands and running it all the way around them till y ou co me back to where you started from. So a bug crawling between the bands could circle them for ever and ever. It would alway s walk along one strip with the other strip sliding along its back. Nowhere would he find the " floor" meeting the " ceiling." In fact , b oth floor an d ceiling are one and the same surface . What seem s to be two bands is actually . . . . Find out and then turn to the answer section to see if y ou were right. As an added twist , having un nested the band( s) , see if you can p ut it (them) b ack together again .

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Viennese Knot In the 1 8 8 0s in Vienna a wildly p op ular m agician's trick was to put a k n ot in a p aper strip simply by cutting it with scissors. This is how it was done : Take a strip of p aper, ab out an inch wide and a couple of feet long. J ust before j oining the ends, give one end a twist of one and a half turns. (If you have read about the Mobius ban d , you'll k n ow this is like making one with an extra twist in it.) Then tape the ends togeth er to form a band . That done, cut along the m iddle of the closed b and u ntil y ou come back to where you started. At the last snip you wil l be left with one long b an d , which y ou will find has a k not in it . Pull i t and y o u should see a knot in the shape of a p erfect pentagon .

27

Release the Prisoners Here is an other problem in top ology. Con nect y ou r wrists with a longish p iece of rope. Make sure the loops aroun d y our wrists are not too tight . Have a frien d do the sam e , but before co mpleting the ty ing u p , loop his rop e aroun d y ours , as sh own in the p icture . Can you sep arate y ourself from y our friend without unty ing the knots or cutting the rope? It can be d one!

Three- Ring Rope Trick This is a fam ous p roblem from topology that with a little trial and error I am sure y ou can solve for y ourself. First m ake three loops of rope or string and link them in a chain like a Christm as decoration. Cut the middle loop an d all three p ieces of rope will come unlinked . Cut either end loop and the other two stay linke d . The puzzle is this : Can you link three loops of rope so that all three will come unlinked if any one is cut? It can be done.

Wedding Knots Russian girls use straws to foretell whether they will be m arried during the y ear. A girl will take six straws and fold each of them in half, keeping the folds hidden in her fist. Then she ask s another girl to tie the 1 2 straw ends together in pairs ; if a complete circle of straws is formed , she will be m alTied within the year. You can m ake a closed loop with four straws in two way s , as shown . String will do i n ste ad of straws. Can you join the loose ends of six straws to m ak e a single close d loop in three d ifferent way s?

28

Amaze Your Friends Ask a frie nd to d raw a m aze with a pencil on a large sheet of p aper. He c an m ake it as twisty as he likes, but none of the lines m ay cross and the e n d s m ust j oin to m ake a closed loop . N o w ne wsp apers a r e p laced around the edges as shown here so that only the m iddle part of the m aze shows. The

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friend now p laces h is finger any where in this still exposed area. Is h is finger inside or outside the m aze? The m aze is so com plicated it m ust be impos­ sible to say which p oints are inside the closed loop and which p oints are outside. All the same y ou state r; orrectly whether h is finger is inside or outside th e m aze. An other way to p resent th e trick is with string or rop e . Take a good length and tie the ends to form a long loop. Then ask the friend to m ake a closed-loop m aze with it . Put newsp apers d o wn to hide the outside of the m aze. The friend puts his finger on some spot in the m az e . Take o ne news­ paper away and pull an ou tside p art of the string across the floor. Will the string catch o n the friend's finger or not? Again y ou p redict c orrectly each time th e trick is perform ed. How is it d o ne? The secret is this : Take two p oints in the m aze an d j oin them with an imaginary line. I f the p o ints are b o th inside the loop , then the line will cross an even number of strings. If b oth p oints are ou tside , the same rule holds. B ut if one p oint is inside and the o ther outsid e , then the line con­ necting them will cross an odd num ber of p o ints. Th e easiest way to rem ember the rule is to th ink o f the sim plest m aze p o ssible , a circle. If b oth p oints are inside the circle ( o r b oth out side it) , then the line connect­ ing them will cross either n o strings or two strings ; b oth 0 an d 2 are even num bers. If one p o int is inside the o ther outside , then the line will cross the circle once ; I is an odd number. To do the stunt, as the newsp apers are being place d, let y ou r eye m ove through the maze fro m the ou tside until y ou reach a sp ot n ear the center that is easy to re mem ber. You know that sp ot is outside the m aze. When y our friend places his finge r, y ou have only to draw mentally a line from y our "outside" spot to his finger and n ote whether y ou cross an even number o f strings (then his finger i s outside) or an odd (his finger is inside) . A little p ractice will sh ow that the trick is easier to d o than to describe.

29

Tied in Knots? Pull the ends of each rope sh own here and fin d out which will tie itself in a k not. Knot h is very interesting ; it is o ften u sed by m agicians. It is k nown as the Chefalo k not. It is made from the reef knot shown in g.

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The Bridges of Paris In 1 6 1 8 the plan of Paris and its bridges over the River Seine looked like the sketch m ap here . The fam ous No tre Dame Cathedral is shown by the t on the islan d . Could the Parisian s then take a stroll over the bridges and cross each one only o nce with out retracing their steps? Draw a net work as was done for "The B ridges of Ko n igsberg. "

30

Tou r of the Castle The idea here is that y ou have to visit each room in the castle only once on a tour of it , starting at the in arro w and leaving by the out arrow. With the exit placed as in the first of the lit tle 4-roomed castles sho wn here y ou can do it ; in the second y ou can not. Try your hand at ( 1 ) the 9-roomed castles , and (2) the l 6-roomed castles.

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31

The Cuban Gunrunners Problem The Cuban gunrun ners plan to tran sp ort a trainload of guns and bombs from Havana to Santiago. There are several rail routes they could take, as y ou can see on the map of the rail sy stem shown . How can they be stopped from getting through? The easiest way is t o blow up a few bridges. What is the fewest number of b ridges y ou must blow up? And which ones are they?

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32

3. Vanishing-Line and Vanishing-Square Puzzles To Martin Gardner, America's leading popularizer of math and purveyor of puzzles, l owe ideas for this next lot of puzzles, all of which, and more, are to be found in his excellent Ma the­

ma tics, Magic and Mystery ( New York: Dover,

1956).

These puzzles all depend on a strange quirk of geometry. All but the first involve cutting and rearranging parts of a figure. When that is done, a part of the figure or a I ine apparently vanishes. Where has it gone? is the question. Before I describe some of these puzzles and explain them, look at the following puzzle, about counting, not about cutting up figures; it gives the clue to the puzzles of the vanished lines. There are no answers except to the next puzzle.

Mr. Mad and the Mandarins Mr. Mad was having three children to tea. Four p laces were laid , each with three m an darin oranges on a p late. B ut one of the children did n ' t turn up. So how should the others d ivide up the spare p lateful of m andarins? Mr. M ad sugge ste d this way , as shown :

All three m andarins on the first plate went to the second p late , from which two m andarins were put on the next p late , from which one m andarin was placed on the last plate , Mr. Mad's. "There ! " exclaimed Mr. Mad . "Fair shares for all. B ut I bet y ou can't tell m e which plateful has van ishe d ? " N one of t h e ch ildren c ould give an answer. Can y ou sugge st one?

33

The Vanishing-Line Trick Mightily simplified th ough this puzzle is, it form s the heart of m any bril­ liant p uzzles created by Sam Loy d , the great puzzlist. Draw on a card three equal lines , as shown here . Make certain that b oth the first and the third line touch the d iagonal of the card ( the broken l ine) , each with one of its ends. Cut the card along the d iagonal. Slide the top half to the right until the lines coincide again , as in the sec ond p icture . There are now only two lines where before there were three. What has happened to the third line? Which line vanished and where did it go? Slide the top part back and the third line returns.

I t is lik e the vanishing group of m andarins in " Mr. Mad and the Mand a­ rins" p uzzle. What happens is that the m iddle line is broken into t wo parts - one going to lengthen the first line, the other lengthening the third line. W ith m ore lines the d ist ribution of the lines is less obvious and the disap­ pearance of the center line becomes even m ore p uzzling.

The Vanishing-Face Trick We can d oll up "The Vanishing-Line Trick " by d rawing p ictures instead of lines. Our top p icture sh ows six cartoon faces divided by a broken line into two strip s.

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34

card . Shift the upper strip to the righ t , and-as the b ottom p icture sh ows­ all the hats rem ain , b ut one face vanishes. You cannot say which face has vanished. Four of them have been split into two p arts and the p arts redis­ tributed so that each new face has gained a small bit.

The Vanishing-Square Trick Conjurers perform m iraculous-seem ing tricks where a rectangular or square figure is cut up an d rearranged and in the process a whole square is lost to view. The sim plest and oldest example exp lains how it is done. The follow­ ing explanation is b ased on Martin Gardner's ex cellent book Ma th ematics, Magic and Mystery:

Start with a 4-by-4 square ; its area is 1 6. This squ are is cut along the slant line. This line is not a d iagonal, since it p asses through only one of the corners. This is the secret of the trick . Now shift the lower p art of the b oard to the left, as sh own in the right-h and picture. Snip off the shaded triangle stick ing out at the top right corner and fit it into the space at the lower left corner, as sh own by the arrow . This produces a 3-by-S rectangle ; its area is I S. Yet we started with a b ig square of area 1 6 . Where has the m issing lit tle sq uare gone? As we said , the secret lies in the way the slant line was drawn. B ecause that line is not a diagonal, the snipped-off triangle is talle r than I : It is 1 113 in heigh t . So the rectangle's h eight is actually 5 113 , n ot S. Its actual area i s then 3 X 5 113 = 1 6 . So, y ou see ( or rather y ou didn't " see" ) , we have n't lost a square . I t j u st looks like it. The trick is n ot very b affling with su ch a sm all b oard. But a larger num­ ber of squares will conceal the secret. You can see that this puzzle is like "The Vanish ing-Line Trick" when y ou look at the squ ares cut by the slant line. As y ou m ove up the line , you find that the p arts of the cut squares above the line get sm aller an d sm aller while those below get larger and larger-j ust like the vertical lines in the e arlier puzzle .

35

Sleight of Square In "The V anishing-Square Trick" all the trickery is confined to the squares e ither side of the slant line. The rest of the square plays no part in the trick at all ; it is there merely for disguise. Now instead of cutting the square board into two pieces, suppose we chop it into four. The trick w ould become eve n m ore mysterious. One way to do th is is shown in the picture of the 8 -by-8 b oard . ......

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When the four p ieces are rearranged , there is a gain of I square- from 64 t o 6 5 squares. You'll find there is a long, thin , diamond-sh aped gap along the d iagonal of the 5 -by - 1 3 rectangle. This is hardly noticeable. But it is where the "extra squ are" has come from . If you were to begin with the 5 -by - 1 3 rect angle , d rawing an accurate diagonal , then in the 8-by-8 square the upper rectangle would be a shade higher than it sh ould be an d the lower rectangle a b it wider. This bad fit is m ore n oticeable than the slight gap along the diagonal. So the first m ethod is better. Sam Loy d , J r. , d iscovered how to p u t the four p ieces together to get an area of only 6 3 -that is, to lose a square . This picture shows how it is done.

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36

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The Secret Fibonacci Lengths You can make a square come an d go at will with other size b oards-p ro­ vided y ou know the secret lengths of the p erp end icular sides (ex cluding the slant lines) of all pieces-b oth the inner cutou t s and the b oards made out of them . In "Sleight of Square " these length s were 3 , 5 , 8 , and 1 3 . These numbers are part of a fam ous n u m ber serie s , the Fibonacci (say it fib-o-NA H-ch ee ) num bers. It goe s I , 1 , 2 , 3 , 5 , 8 , 1 3 , 2 1 , 3 4 , and so o n . hach number from 2 on ward I S the su m of t h e t wo p revious numbers : 3 = I + 2 , 5 = 3 + 2 , and so on. Fibonacci, an I talian , was the first great European mathematician ; he lived in the 1 200s . I doubt if he ever foresaw this curious use of his number series for geom etrical trickery ! So we started with an 8-by-8 square with an area of 64 an d ended up with a 5 -by- 1 3 rectangle with an area of 65 . And y ou notice 8 lines be­ tween 5 and 1 3 in the Fibonacci series . The trick works with higher numbers in the serie s ; the higher the better because the "extra square" is m ore easily lost in a longer d iagonal. For exam ple , we can choose a 1 3-by - 1 3 squ are , with an are a of 1 69 , and d ivide its sides into lengths of 5 an d 8, as sh own .

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Cutting along the lines, we can rearrange the p ieces into an 8 -by-2 1 rec­ t angle , with are a 1 68 . A lit tle square has been lost, not gained , this time. The Fibonacci numbers used are 5, 8, 13, and 2 1 . There is a loss of a square because the p ieces along the d iagonal overlap instead of having a gap between the m . An odd fact emerges. A board usmg the lengths 3 , 8 , ., 2 1 , and so on- that is, every other Fibonacci number- gives a gain of a square . A board using the lengths 5 , 1 3 , 3 4 , and so on results in a loss of a little square. If y ou cut up a 2-by-2 board , m ak ing a 3-by - l re ctangle , the overlap ( re­ sult ing in a loss of a quarter of the b oard) is too obvious. And all the m ystery is lost.

37

Langman 's Rectangle A rectangle can also be cut up and the p ieces fitted together to m ake a larger rectangle. Dr. Harry Langm an of New York City has devised a way of cutting up a re ctangle. His method , shown belo w , m akes use of the Fib onacci numbers 2, 3 , 5, 8 , 1 3 , an d 2 1 . .,i-'

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Curry's Paradox A p aradox is an absurd trick that on the face of it looks flawless. An I I -by­ I I square is cut into five p ieces, as sho wn here. The p aradox is: When the pieces are put t ogether in another way , a hole appears. Two squares have seem ingly been lost. One of the L-shaped p ieces must be shifted to p ro­ duce the e ffect. I....... ......

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This paradox was invented by a New York City am ateur m agician Paul Curry in 1 9 5 3 . He also devised a version using a I 3-by - 1 3 square where a still larger hole ap pears an d three squares are lost . As you see , Curry 's p aradox uses Fibonacci num bers. I

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Gardner's Triangle It's possible to m ake part of a triangle ·disap pear. Martin Gardner has ap­ plied Curry 's paradox to a triangle. His m ethod is sho wn in the p icture of the triangles with two equal sides. B y rearranging the six piece s , t wo squares are lost .

The deception is increased by having t h e points P fall exactly on t h e cross­ ings of the grid , since the sides will slightly cave in or out .

39

Hole in the Square An other quite d iffere nt way of losing area is to cut a square into four ex actly eq ual pieces, as sh own , by two crosscuts.

Rearrange the p ieces, an d a sq uare hole appears in the center. The size of the hole varies with the angle of the cuts. The area of the hole is spread around the sides of the square . This trick suffe rs from the fact that it is fairly obvious that the sides of the square with the hole are a bit longer than th e sides of the first square. A m ore mysterious way of cutting a square into four pieces to form a h ole is sh own in th is p icture .

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The effect is base d on "The Vanishing Square Trick , " described earlier. Two of the p ie ces m ust be shifted to p roduce the effect-the long strip of the lower edge an d the L-shaped p iece. If y ou remove the largest piece ( t op le ft) , y ou are left with another Gardner triangle.

40

4. Match Puzzles The next batch of puzzles all make use of matches. Toothpicks or orange sticks all the same length are equally good .

Squares from 24 Matches Take 24 m atches. How m any squares the same size can y ou get with them? With 6 matches to a side y ou get one square. You can't come out even with squares with 5 and 4 m atches to a side. With 3 m atches to a side , y ou get two squares, as shown .

With 2 matches to a side , y ou get three squares :

S uppose we allow squares of differen t sizes. (a) With 3 m atches to a side , how m any extra, sm aller squares can y ou get n ow? ( Clue: Squares can overlap .) (b) With 2 m atches to a side sho w how y ou can get a total of seven squares. With I m atch on a side y ou can m ake six identical squares, as shown.

(c) With I match on a side how d o y ou make seven identical squares? Eight identical squares? And nine identical squares? There'll be some extra , b igger square s too. With the nine squ ares there are five extra squ ares.

41

PA R T-MATC H SQU A R ES Th e nex t th ree puzzles need 24 ma tch es. You make half-ma tch squares and o th er part-m atch squares by crossing one m a tch over an oth er.

Half-Match Squares U se half a m atch as the side of a squ are . Can y ou get 1 6 small squares? H o w m any larger squares can y ou see?

Third-Match Squares Can y ou get 27 small squares, one th ird of a match on each side? How m an y larger squares can y ou see?

Fifth-Match Squares Can y ou get 5 0 small squares in 2 m atch-stick size squares? How many larger squares of all sizes can y ou see?

Move-or- Remove Puzzles I B egin with 1 2 m atches, m ak ing four sm all squ ares as sh own.

(a) (b) (c) (d) (e) (f)

R e m ove 2 m atches, leaving two squ ares o f d ifferent sizes R e m ove 4 m atches, leaving two equal squares M ove 3 m atches to m ake three squares the same size M ove 4 m atches to m ake three squares the same size M ove 2 matches to m ake seven squ ares of various sizes (you'll have to cross one m atch over another) M ove 4 matches to m ake 1 0 square s, n ot all the same size (you'll have t o cross one m atch over an other m ore than once)

Move-or- Remove Puzzles I I B egin with 24 m atches, m ak ing n ine sm all squ ares as sh own.

42

(a) (b) (c) (d)

(e) (f)

(g) (h)

Move 1 2 matches to m ake two squares the same size Remove 4 m atches, leaving four sm all squ ares and one large sq uare Remove 6 m atches, leaving three squares Remove 8 m atches, leaving four squares, each I m atch to a side ( t wo answers) Remove 8 m atches, leaving two squ ares ( two an swers) Remove 8 m atches, leaving three squares Remove 6 m atches, leaving two squares and t wo L-sh aped figures Remove 4, 6 , then 8 m atches to m ake five squ ares, each I m atch to a side

Windows Make six squares-not all the same size- with nine matches. The an swer looks l ike two w in d ows.

G reek Temple

The temple shown is m ade out of I I matches.

(A ) M ove 2 m atches and get I I squares (B ) M ove 4 matches and ge t I S sq uares

An Arrow This arrow is made of 1 6 matches.

_i� / (A ) Move 1 0 m atches in this arro w to form e igh t equal triangles (B) M ove 7 matches to m ake five equal four-sided figures

43

Vanishing Trick There are 1 6 squares here with one m atch on a side. But how m any squares in all?

Take away n ine m atches and m ake every square - of any size -vanish .

Take Two The eight m atches here form , as y ou see , 1 4 squares.

Take two m atches and le ave only 3 squares.

Six Triangles Three m atches m ake an equ al-sided , or equilateral, triangle . Use 1 2 matches to m ake six equilateral triangles, all the same size. That done, m ove 4 of the m atches to make three equilateral triangles n o t all the same size.

Squares and Diamonds Form three squares out of ten m atches. Remove one match . Leaving one of the sq uares, arrange the other five m atches around it to m ake two d iam onds.

Stars and Squares Put d own e ight m atches to m ake two squares, eight triangles, and an eight­ p ointed star. The m atches m ay overlap.

44

A G rille In the grille shown here m ove 1 4 matches to m ak e three squares.

The Five Corrals Here is a field , four m atches square. In it there is a barn one m atch square. The farm er wishes to fe nce off the field into five equal L-shaped c orrals. How d oes he do it? (Use ten m ore m atches for the fencing.)

Patio and Well In the m id dle of this patio, five matches square , is a square well.

o (a) Use 1 8 more m atches to split the patio into six L-shaped tiles all the same size and shape (b) Use 2 0 m ore m atches to split the p atio into eight equal L-shaped tiles

45

Four Equal Plots Here is a square build ing site 4 m atches on a side. We will call its area 1 6 square m atch unit s ( 4 X 4 = 1 6 ) .

Add 1 1 m atches to fence o ff the site i n t o four plots, each with a n area o f 4 square m atch units. B u t y ou m ust do it s o that each plot borders on the other three. One of the plots is a square , two are L-shaped, and one is a rect angle.

Get Across the Pool Here is a g ard en pool with a square island in the middle.

A d d two " plank s" ( m atches) and step across the water onto the isl and .

Spiral into Squares M ove four m atches in this sp iral in order to form three squares.

46

More Triangle Trickery M ake a th ree-four-five triangle out of 1 2 matches. The m atches shut in an area of 6 sq uare m atch units. (This is easy to see because the triangle is exactly half of a three-by-four rectangle , whose area would be 1 2 square m atch units.)

(a) Move 3 matches to form a sh ape with are a 4 square m atch units (b) Move 4 matches to form a shape with are a 3 square m atch units CLUE :

In b oth a and b m ove the m atches from the sh orter sides of the

triangle.

Triangle Trio Can you m ake j ust three equal-sided triangles out of seven m atches?

Triangle Quartet With these six m atches can y ou m ake four equ al-sided triangles?

3 Times the Area L ook at the rectangle on the left. It has 3 times the area of the rectangle on the right , as the dotted lines show.

Add one m atch to the sm aller rec tangle so it has 7 m atch es. Make it into a b ox-girder shap e made up of three equal-sided triangles. Now add four m atches to the rectangle on the left and m ake it into a shape m ade up of 1 9 equal-sided t riangles-so it has an area 3 times as great as the box-girder shape.

47

Cherry in the Glass Arrange a penny and four m atches as shown. This is y our cherry in a glass. T ake the cherry out of the glass sim ply by m oving two m atches. You must n ot touch the cherry ( penny) , of course .

48

5. Coin and Shunting Problems The next section includes puzzles about shuffling coins and the classic puzzles of ferrying people across rivers in boats. Also there is a wide selection of railway shunting problems. The best way to solve these is by actually drawing a plan of railway tracks, making coins or bits of paper stand for the engines and their cars, and moving them about on the tracks. It is a good idea to j ot down the moves you make so you don 't forget them, particularly if you are successful and solve the problem. There is nothing more vexing than to solve such a puzzle and then not be able to remember your moves !

Coin Sorting in Pairs Arrange three pen nies and two dimes in a row , penny-d ime-penny-dime­ penny . Move the coins in p airs so that the three pennies are together and next to them the two dimes , as sh own in the second p icture.

'81.1810101 , ,

A m ove is like this : Y ou place the tips of y our first and second fingers on any two coins- the c oins don't h ave to be next to each oth er or of the sam e denomination-then y ou slide the p air to an other part of the row , b u t y o u must keep the same spacing between the coin s. I t helps to u se squares to keep the sp acing. Y ou must not m ake any p air of c oins m erely change p laces. When y ou finish , there must be no sp aces between any of the coins. You can m ove the coins as m any spaces as y ou like left or right. But ten sp aces should be enough . Can y ou do it in three m oves?

49

Rats in a Tunnel Two brown rats and two white rats met head on in a tunnel . How did they p ass one another and change ends of the tunnel? They could only move by m oving forward into a sp ace or by hop p ing over another rat ( o f their o wn or the o ther color) into a sp ace. Or they could m ove back . What is the fe w­ est number of m oves needed to change the rats over? Here are the k ind of m oves y ou can m ak e :

-­

_ c��� e To work it out , use two pennies ( for the bro wn rats) an d t wo d imes ( for the white rats) . Put them in a line with a gap bet ween , as shown in our sketch.

50

Three-Coin Trick Begin with three coins sho wing a head p laced between t wo tails. Each m ove in this puzzle consists of turning over t wo coins next to each other.

(a) Can y ou get all the coin s showing heads in j u st t wo m oves? (b) Can y ou make them sh ow all tails in any num ber of m oves?

Triangle of Coins Start w ith a triangle of ten coins p ointing up ward , as shown . Can you m ove three coins o nly and m ake the triangle p oint d ownward ?

o 0 0 0 0 0 0 0 0 0 Five-Coin Trick Take five coins, all the same k ind- say all dimes. Can y ou place them so that each coin touches the other four?

Five-Coin Puzzle Can y ou shift the c oins shown on this b oard so that the penny and the half­ dollar on the left swap places?

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Coin Changeovers Place three pen nies and th ree nickels as sho wn here . Can you m ake the pen nies an d nickels ch ange places? You m ay move o nly one coin at a tim e . M ove it d ire ctly to an empty place , or j u m p it over another coin to an empty sp ace . You can m ove or j u m p up and d own or across but not d iagonally .

GG G ® @®

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Now try this puzzle :

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X No

Mission I mpossible? Two secret agents, 005 and 007 , are each try ing to get this top scientist out of Slobodia. The only way out is across the Red River Danube. Agent 005 's m an is Dr. Fiinf and 007's m an is Dr. S ieben. N one of them can swim . A ro wing boat awaits them , h idden on the Slobodian side. It c arries only two pe ople at a tim e . Ne ither scientist dare be alone with the other agent unle ss h is own agent is also with them. Nor c an the two scientists be left alone together, in case they swap top secrets. It's a case o f two's n ot allowe d , three's company ! For instan ce , Dr. Funf cannot ro w across the river alone with the other agent, 007 , or be alone with him on either river b ank ; but he can be on either bank when b o th agents are with him. How d id the agents row the scientists across the river? HINT :

Five crossings from bank to b ank should complete the m ission.

Railroad Switch The driver of a shunting engine has a p ro blem : to switch over the black and white cars on the triangle-shaped siding shown here. That is , he m ust shunt the white car from the branch AC to the bran ch BC an d the black car fro m BC to A C. The siding beyond is only b ig enough to take the engine or one car. Th at's all. The engine can go from A to B, back up past C, and then forward along A C. But when it d oes so , it will end up facing the other way along the tracks A B. The driver isn't bothered about which way h is engine faces. Can y ou switch the cars in six m oves? Each coupling and un coupling counts as one m ove. Remem ber, the d river can couple up both cars to the engine and then uncouple j u st one of them . To solve the shun ting problem , draw a large m ap of the railroad and u se coins on it .

53

Restacking Co ins There use d to be a toy calle d the To wer of Hanoi. It was in the form of 6 4 wooden rings of graded sizes stacked on o n e of three pegs-largest at the b ottom , smallest at the top. The rings h ad to be restacked in the same order on a d iffe rent p eg by m oving them one at a time. A story goes that this p roblem was sen t to Buddh ist m onk s. Working at a m ove a secon d , they would have needed s o m e 5 8 5 billion y ears to finish it! Here is a new- and shorter! - version of this old puzzle . Place three saucers or table mats in a line. In the first saucer on the left stack a quarter, a pe nny , and a d im e ; the qu arter must be at the b ot tom , the penny on top , as sh own . Restack the coins in exactly the same way in the far righ t saucer. You m ust follow these rules: M ove only one coin at a time, from one saucer to another. A lways put a smaller coin on a larger coin. Never put a larger coin on top of a sm alle r one. Use all three saucers when m oving the coins. You can m ove to and fro .

R iver Crossing A p latoon of soldie rs m u st cross a river. The b ridge is down, the river wide. S uddenly the p latoon's officer spots two b oy s play ing in a tiny rowboat . The boat only holds two b oy s or one soldier-not a b oy an d a soldier, for instance. All the same, the p latoon su cceeds in c rossing the river in the b oat. H ow? Work it out with matches and a m atchbox on the t able across a m ake-believe river.

54

Collision Course? Two train s have met head on on a single track in the desert. A b lack e ngine (B) an d car on the left ; a white engine ( W) and car on the righ t. There is a short switch j ust large enough for one engine or one car at a time. Using the switch , the e ngines and cars can be shunted so they can p ass each other. How m any times will the drivers h ave to b ack or reverse their engines? Count each reversal as a move . A car cannot be linked to the fro n t of an engine.

55

6. Reasoning and

Logical Problems In this section I 've included some novel thinking exercises with blocks. And there is also a selection from several types of I Q tests that are visual and mathematical in nature. The section continues with a sprin kling of some of the better known (and lesser known) logical puzzles that call for strict reasoning. I have concluded the section with some unusual logical puzzles not often seen in puzzle books.

Thinking Blocks The following are proble m s about placing six rectangu lar blocks so that they touch only so m any other blocks. They were originally included in a b ook by Ed ward de Bono, Five-Day Course in Th in king (New York : Basic Books, 1 9 6 7 ) . He used the problems as a cunning thinking exercise . You m ay fin d y ou simply cast y our six blocks on the table in rand om fashion and h ope for the best . Of course , y ou 've still got to check the p attern of b locks you get th is way . Or y ou can ad op t a less higgledy-p iggledy ap­ p roach an d carefu lly build up a pattern , block by block . One way is as good as the other. Any method will serve j u st so long as it gives y ou the correct answer. There is a sim ple lesso n to be learned from this puzzle . We often cannot solve a p roblem because we get a thinking block . We get blocked in our thinking, or rather, in one way of think ing. So the lesson is this : I f one way of th ink ing p roves unhelp ful, try another. Often the m ore ridiculous the new way of thinking see m s , the bet ter it m ay be. One other tip : Do n't d iscard ideas that didn't work . To know that a certain p attern of block s d oesn't give the righ t answer is in itself u seful. The trick is to remem ber all these " b lind alleys" so that you don ' t try them repeate dly and thus lose p atience. Matchboxes m ake good homely blocks. A. Place six blocks so that each touches only two other block s. They m ust touch flush along their side s , y ou can not h ave the p oint of one block " d igging into" another. To help y ou solve th is p uzzle , y ou can copy each

56

pattern y ou fo rm and j o t down the num ber of touching b locks on each b lock , as sh own here. This p attern won't d o b ecause two o f the b lock s touch th ree others.

B. C. D.

Place the six blocks s o that each touches only three other b lock s. Place the b locks so that each touches four others. Place them so that each touches five others.

Martian Orders ! On Mars y oung Mart ian s have to line up at school in order , according to two rules. First , girls come before boys. Secon d , where two girls come next to each other, the taller girl goes first ; an d the sam e goes for two boys together in a line. Zane is a Mart ian boy who is the sam e height as Thalia ( a girl) , but he is taller than his friend X eron ( another b oy ) . (a) How do the three line up, from left to right? (b) They are j oined b y Th alia's friend Sheree (an other girl) , who is taller than she is. Now how d o they line up?

What Shape N ext? Here are two picture puzzles of the kind y ou see in intelligence tests. Fol­ low the pattern of sh apes in each from left to righ t . Then work out which of the lettered shap es best fi ts onto the end o f the line.

A

a

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Cl dD el fl0TIl lillillJ

A A A 57

I Q Puzzle Another I Q-sty le p uzzle . Look at the four numbered shapes and say which one best fits the sp ace in the bottom right-hand corner of the p icture .

Odd Shape Out In each of the sets of sh apes shown here one of them ( 1 , 2, 3 , or 4) is the odd sh ape out : It is different in some way from the other three shapes. Can y ou p ick it out in each set? a

e

58

2 (])

EB 3 EB 4 E9 1

b

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The Same Shape Which of the sh ap es shown here is the same as the boxed one on the left?

Next Shape, Please Can you say which is the next shape in the pattern?

The Apt House Yet an other of those IQ-like tests, or rather "ap titud e " tests. Look at the num bered h ouses. Only one can be m ade by folding the p lan . Which h ouse is it? I f y ou can see this easily in y our m ind's eye , then the testers say y ou sh ould be good at being an engineer. Or m aybe y ou know y our ap t ( ap art­ m ent) houses!

59

Who Is Telling the Truth? The j u dge listened quizzically to the four well-k nown crook s . "You're ly ing y our heads off," he declared . "Still y ou'll look better that way ! " The officer laughed dead on cue and then said : "I happen to know one of this lot is telling the truth ." Th e j u dge snappe d : "Well , wh at've y ou got to say in defe n se , y ou lot ? " AI said : " O n e of us i s lying ! " Bob : " N o , I tell you, two o f u s i s lying. " C o n : " Look here , three of us is ly ing." Don : "Nope , not true ! Four of us is telling the truth ." S o who is telling the truth? The officer was quite right .

The Colored Chemicals Puzzle Mr. Mad the chem ist has six big bottles of colored liquids. There is one of each of th ese rainbow colors : re d , orange , yellow, green , blu e , and violet. Mr. M ad knows that some of the bottles contain p oison . B ut he can't rem em ber which ! H o wever, he can remem ber the facts from which y ou sh ould be able to work out which colored b ottles contain p oiso n . I n e ach of t h e following pairs of bottles o n e i s p oisonous, the other is n ot : the violet and the green b ottle ; the red and the yellow one ; the b lue and the o range one. And he also re mem bers that in each of the following p airs of b ottles there is o ne that contains a nonp oisonous liquid : the violet and th e y ellow one ; the red and the orange on e ; and the green and the blue one . "An d , I nearly fo rgot," add s Mr. Mad. "The re d b o t tle has a non­ p oisonous liquid in i t . " Wh ich bottles have p oison in them ?

Mr. Black, Mr. G ray, and Mr. White Three men met on the street-Mr. Black , Mr. Gray , an d Mr. Wh ite . "Do y ou k n o w ," asked Mr. B lack , "that between us we are wearing b lack , gray , an d white? Yet not one of us is wearing the color of his nam e ! " "Why , that 's righ t," said the m an in wh ite . Can y ou say who was wearing which color?

Hai rdresser or Shop Assistant? A m y , B abs, and Carol are either hairdressers or shop assistants. Amy and B ab s do the sam e job. Amy and Carol do different j obs. If Carol is a shop assistan t , then so is Babs. Who d oe s which j o b ?

The Zookeeper's Puzzle The Zookeeper wants to take two out of a p ossible three chimps to a TV stu d i o . The two m ale chimps are Art a n d Bic ; the third , a fe male , is called Cora. He daren't leave Art and Bic behind because they fight. And he can­ not take b oth with him either. But Cora doesn ' t get o n with Bic. So who can he take?

60

Who's GUilty? Alf, Bert and Cash are the suspects in a robbery case. Their trial shows up the following fact s : Either Cash is innocent or B ert is gu ilty . I f Bert is guilty , then Cash is innocent. Alf and Cash never work together and Alf never does a job on his own . Also , if Bert is guilty , so is Alf. Who is guilty?

Who's in the Play? Alice won't take part in the B uskin Players annual (am ateur) p lay if Betty is in it ! But Ch arles will only play if Alice is in it. The p oor producer in­ sists th at o n e o f the girls is in the play . Two people are neede d . Who is in the p lay?

Tea, Coffee, or Malted Milk? The professor had enjoyed h is usual after-lunch beverage so much he th ough t he'd have another. But he could not for the life of him re member what he had drunk . So he called the waiter ove r. And this is what h e said to h im : "Now , if this was coffe e , I want tea, and if this was tea , bring m e a m alted m ilk . But if this was m alted m ilk , bring m e a coffee." The waiter, wh o was logically m inded , then brought him coffee. Can y ou say what d rink- tea, coffe e , or m alted m ilk -the waiter had originally served the p rofessor?

Soda or Milkshake? Three friends- Alan , Bet, and Cis- often go to the same sod a fountain . Each either orders a soda or milksh ake. The soda jerk notices : (a) when Alan chooses a sod a, Bet has a m ilkshake ; (b) either Alan or Cis has a sod a , b u t never b oth ; a n d ( c ) B e t an d Cis never b oth h ave a m ilkshak e . There are only two p ossible ord ers they can m ake. What are they ? HINT :

Since this is a hard one, we'll tell y ou that o nly Bet has a choice of

d rinks.

Newton's Kittens Isaac Newto n , as you pro bably know, was one of the cleverest m e n the world has ever known . He was the great scientist and mathem atician who solved the riddle of gravity , of why things fall to the ground . Well , Newton had a cat and she used to come and go into his house near Cam bridge , Englan d , through a large hole bore d in the bottom of his kitchen door. One day the cat had three kittens. And so Newton had three sm all holes b ored in the door for the m . Why d o y ou think this was fun ny?

61

March Hare 's Party The M arch Hare was giving a p arty . His y oung guests had to get to their rightfu l tables- I , 2, 3 , or 4-by one of the four p aths shown in the p icture . As y ou see, he would n't l e t any b oy s go along o n e p ath , which would later fork into two p aths. Al wanted to h ave tea on an island with Barbra , but he refu se d to have tea with Silvie or Don. Sylvie said she would n't have tea near water. On top of this, Gary j u st had to roller-skate over one of the bridges. To m ake matters worse , Don and Gary would n't have tea with each other. By the way , only the boy s could ro w, and the single boat was only b ig enough for one child and could only travel to table 4. Where did each guest have tea?

62

Marriage Mix-up The absent-min ded p rofessor had j u st been to a p arty . His wife naturally wanted to know who was there . " Usual cro wd ," he rep lied . "And some new faces. Ted , Pete , and Charlie. And their wives- Barbra , Sue , and Nicola . Can't re member who's m arried to whom . Any way , each couple has one child : They 're calle d Ruth , Wendy , and Dick . Told me all about the m . Barbra said h e r child w a s playing Annie in A n nie Get Your Gun , the school play . Pete told m e his child was play ing Ophelia. I do remem ber Ted p oint­ ing out th at his daugh ter was not Wendy . And Charlie's wife is n ot Sue. I suppose we can work out the m arriage p artners from that ." See if y ou can work out wh o is m arried to whom and who their chil­ dren are .

Who Does Which Job? There are three men- Orville , Virgil , and Homer. Each h as two j o b s . The j obs are : private eye, racing d river, singer, j ockey , bartender , and card­ sharp . Try to find each man's t wo jobs from these facts: (1) (2) (3) (4)

The b artender took the racing driver's girl frien d to a party Both the racing d river and the singer like playing card s with Homer The j o ckey often had a drink with the bartender V irgil o wes the singer a buck ( 5 ) Orville beat both Virgil an d the j o ckey at cards

Birds and Insects Here's an easy logical p oser-or is it? Think about these statements: N o birds are insects. All swallows are birds. Which of the next sentences follows logically from the above two state­ ments?

(A ) No swallows are insects (B) Some b irds are not swallows (C) All b irds are swallows (D) No insects are birds

Wonderland Golf The Am erican m athem atician Paul Rosenbloom specially devised this z any golf game for y oungsters. He set it as a p ie ce of m athem atical research , actu ally . On the Wonderlan d Golf Course the holes are num bered I , 2 , 3 , and s o o n u p t o 1 8 . The link s are laid out i n a sp iral , a s sh own , t o m ake the shots easier! You have two special clubs. One of them holes out in one

63

for y ou ! This is the Single-Shot Iron , or S iron for short . The other club even lets y ou skip h oles! It hits your b all from any hole to the one double its number; so it hit� your b all from , say , h ole I t o hole 2, hole 3 to hole 6 , and hole 9 t o hole 1 8 . Call i t the D ( for d ouble) iron. PUZZLE :

What is the sm allest num ber of shots, u sing e ither iron , to get from h ole I to hole 1 8 ? Th at is , what is p ar for the course? Strangely , it is the sam e as for h oles I I , 1 3 , 1 4 , and 1 7 ! Don't supp ose y ou can spot a p attern , can y ou?

64

Mad Hatter's Tea Party The Mad Hatter had p lanned a special children's tea p arty . He had laid out the tables in the garden in the way our picture shows. He h ad sp lit his guests into three sets- G: all girls; B: all b oy s ; and M: b oy s and girls, mixed. You can see them in their sets , on the left of the p icture , waiting to have tea. He told the m : "Every b ody in each set has to get to his table b y taking t h e correct path through t h e garden. Y o u c a n see which way to go by the word s se t in the p aths." Can y ou work out which of the tables- I , 2, 3 , or 4 - e ach set should get to? One of the tables rem ains empty . Which one?

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65

7. Mathematical

Games

All but two of these games are mathematical ; I have included two word games, "Coincidences " and "The Crossword Game," because they are so good and so popular. Do not be put off by the word mathema tical : the games can be played without the foggiest notion of math. Indeed, "Mancala " is a very ancient African game and has been played since time immemorial by people without the slightest inkling of what we know as school math. First and foremost, the games are meant to be played to win and for fun. Any educational spin-off is purely coincidental, as they say.

Nim A Game for Two This is one of the oldest and most enj oy able of m athemat ical games for two. The word nim p robably comes from the Shakespearean word mean­ ing to steal. Possibly it was first played in China. Nim is play e d with m atches or c oins. In the m ost p opular version 1 2 m atches are placed in three rows- 3 m atches, 4 m atches, and 5 matches , as sh own.

The rules are sim ple. The players take turns in rem oving one or more m atches, but they must all come from the same ro w. The one who takes

66

the last m atch w ins. (You can also play the other way : The one to take the last match loses .) Playing a fe w games will soon show y ou how y ou can always win : (a) Your m ove must leave two ro ws with more than I m atch in a row and the same number in each ; (b) y our m ove leaves I m atch in one row , 2 m atches in the second ro w, and 3 in the third ; or (c) if you p lay first, on your first m ove y ou take 2 matches from the t op row and after that play accord ing to the first two winning strategies j ust given. You can p lay Nim with any n u m ber of m atches or pennies in each ro w , an d with any num ber of rows. A s it happens, there is a way of working out how to take the right number of m atches to get into a winning p ositio n . Y o u simply use " computer counting ," or b inary . This method w a s first given in 1 90 1 . A descrip tion of it is given in the answers section .

Tac Tix A Game for Two Tac Tix is an exciting version of "Nim ," invented by the Dan ish puzzlist Piet Hein . Hein is the inventor of " Hex ," page 8 6 . In Tac Tix pennies or counters are placed in a sq uare , as sh own in the p icture . Players in turn rem ove one to four pennies from the b oard ; they m ay be taken from any row or column. But they must alway s be adj acent pen nies with no gaps be­ tween the m . For example , say the first p layer took the two middle pennies in the top ro w ; the other player could not take the other two pennies in that ro w in one move . Tac Tix has to be played with the p layer taking the last penny losing. This is because a simple tactic makes playing the usual way uninteresting­ perhap s the reason fo r the game's name-for it allows the second p layer to alway s win . All he has to do is play sym metrically -that is, he takes the " m irror" penny or pennies to the one(s) the first p lay er rem oved . The game can also be p lay ed on a three-by-three b oard , and there , when play­ ing the usual way , the first player can win by taking the center penny , a corn er one , all of a central ro w , or all of a central colu m n .

0 0 0 0 ® ® 0 ® ® ® @ @ @ ® @) ® 67

Battleships A Game for Two One of the m ost p opular of all paper-an d-p enci! games, Battlesh ip s can also be a serious exercise in math ! Each player has a fleet of ship s, which he m arks on a grid ; he fire s salvos at named enemy squares, and the enemy tells him if he has hit a ship or not and , if so , what k ind of ship is hit. From this he tries to work out wh ere the enemy ship s lie. To sink an enemy ship , he m ust hit every squ are of that ship . First to sink the enemy's fleet wins. Each p layer needs two ten-by-ten grid s m arked A, B, C, . , J along the top an d 1 , 2, 3 , . , 1 0 along the left sid e , as sh own in the picture . On one sh eet he m ark s the p ositions of h is fleet ; the other sheet is for marking his own shots at the enemy fleet . (The second sheet represents a different area from the first ; otherwise it would be p ossible for a player' s ship and his enem y's to occupy the same spot.) The p icture shows the p osition and size of each kind of ship . .

.

A

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One battlesh ip Two cruisers Three destroyers Four subm arines

( four squares each) ( three squ ares each) ( two squares each) ( one square each)

All sh ip s ex cept su b m arines, must be rectangles one squ are wid e . No L-sh aped or cro oked ship s are allowed ; otherwise a player could not work out how ships lie from enemy rep orts on his salvos without tremen dous

68

d ifficulty . Two ship s m ay n ot touch , even at corners . And a sh ip can have at m ost o ne side of a sq uare on the border of the "sea." So a su b m arine cannot be placed in a corner. When the two fleets have been m arked i n position on the grid s , one of the p layers fires a salvo of three shots: He tells the enemy p lay er where he wants the three shots to lan d ; they d on't h ave to land on adj acent squares. His enemy must then tell him h ow m any shells fell in the sea and how m any h it which types of sh ip s , but he d oes not have t o say which shot did what . For example , he might say , "Two in the sea and one on a destroyer." No matter what order these results were gotten. The second player now fires a salvo, and the first player tell him what happened. Each player keeps a record of his hits and m isses on his ch art of "enemy waters" to work out where the enemy fleet is m oored. Play continues until one of the play ers sinks the enemy's entire fleet and announces the fact .

Boxes A Game for Two This is a game of drawing boxes on a grid of dots. It is very much like "S nake" ( p age 7 7) an d can be p lay ed on th e sam e sort of grid. Play ers take turns d rawing a line across or down to link adj acent dots not yet link e d . A play er wins a box when he draws the fourth an d last side of a square ; he then writes his initial into the box to show he m ade the box. And he can then take another tu rn . I f he's lucky , he m ay be able to m ake several boxes without his opponent hav ing a turn . BUT after m ak ing a b ox he m ust d raw one more line im m ediately. One line m ay m ake two boxes at once, but the p lay er takes only one further turn for that line. A p layer d oes not have to m ake a box even th ough there m ay be a squ are with three sides draw n . POINT A BOUT STRATEGY :

Near t h e end o f t h e g a m e y ou usually g e t open "corridors" of lines, l ike two uprigh ts of a " ladder." Once one player h as closed off one end of the corridor ( or indeed put in a rung anywhere on the ladder) , the other player can m ake all the boxes in the corridor during his turn . The winner is the one wh o h as m ade more boxes. It is best to play on a grid with an even number of dots on each side -eight by ten , say -so that there will be an od d number of b oxes in the completed grid .

Mastermind A Reasoning Game for Two This game is m arkete d , alth ough the p rinciple is simple e nough for y ou to m ake y our own version. The basic idea is this: One p lay er sets a problem by inserting five colored pegs, ou t of a possible eight colors , in a row. His pegs are then covere d , and his opponen t , the "m astermind ," has to work out wh at the colors and their correct p laces are by forming t rial rows. The proble m setter ind icates by the use of b lack and white pegs whether or

69

n ot , first, the opp onent has the righ t colored pegs and , second , they are in the righ t p lace. The comm ercial board is m ade of plastic and has rows of five holes with a sq uare grid at the end of each ro w. Th e version sh own here has only four holes with a two-by-two square grid at the end of each row. A sample game will serve to indicate the ru les and method of p lay . To simplify things, we will p lay with four colors and white only . S up p ose the first player puts up these four peg s : green ( G) , blue ( B) , red ( R ) , and yellow ( Y ) . H e then covers them with a little hood , o r cloche, so th at his opp onent cannot see the pegs. The op p onent puts in the top row : re d , green , white (W) , and green. As the sketch shows , he has two colors , right , but they are not in the right place ; to sh ow this , the first p layer puts in two white pegs. The opponent's second try is the line green , blue , black ( Bk ) , an d white. Because this row has two colors and two places righ t-the green and the blue pegs-the first p layer puts in two black p egs. The opponent's th ird ro w is gre e n , blue , wh ite , an d red , which has two colors in the righ t p lace ( t wo black pegs) an d one color right (red) but in the wrong place ( one white peg) . The game ends when he has formed a row exactly the same as the one o riginally set . The opp onent work s out by reasoning which pegs to change. The shorter the number of rows he can solve the p roblem i n , the better he is at reasoning. The game can also be played on paper, w ith colored pencils or felt-tip pens substituting for pegs.

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Coincidences A Word Game Any Number Can Play This is a word game rather lik e "Masterm ind" but played with letters in­ stead of c olore d pegs. One player acts as the "accountan t ," who thinks of a five- or six-letter word . He n otes this secretly on a sheet of p aper, which he keeps. He calls out the num ber of letters it has. The other players try to d iscover h is word by calling out a line of the same num ber of let ters. The accountant then tells each p layer how m any of the letters in his line m atch in p osition those in his word . Say the accountant's word is "C ENTS " and a p lay er calls "A-A-A-E-E" ; then the accountant announces " N one" be­ cause alth ough the player has gotten one letter right (E) , n either E is in the correct p osition . A good strategy for play ers is to c all out all vowels, since m ost words contain them (A , E, I, 0, U). I f the accountant's word is d is­ . covered in fewer tries than there are letters in his word ( four tries, say ) , then he scores n othing. H e scores o n e p oint for eve ry try over the number of letters in his word . Here is how a sample game began . Accoun tant's word : SHIRT

Play er 's lines AAEEE 11000 1II00 THITH TlUTl THILl

A ccoun tan t calls None None One Two Three Three

To p lay well , one should know the letter frequences of E n glish , as follows : E , T , A , 0 , N , I , S , R, H , L, D, C, U T H , I N , E R , R E , A N , H E , A R , E N , n , T E , AT, O N , HA Three-le tter gro ups: THE, lNG, AND, I ON , E N T , FOR, TI O , E R E , H E R , ATE , V E R , TE R , T H A

Single le tters: Two-le tter groups:

Eleusis A Reasoning Game for Four Here is a game with a really novel twist , invented by a New Yorker, Robert Abbot (taken from More Ma th ematical Puzzles and Diversions, by Martin Gardner, New York : Penguin , 1 9 6 1 ) . He originally devised it as a card gam e , but it can also be p lay ed with p aper an d pencil. Its n ovel twist is this : M ost games have rules you learn and use in order to decide y our best m ove, but in Eleusis y ou p lay to discover the rule ! The gam e is rather like discovering a scien tific law, except that in science there is nobody to tell y ou if your law is the right one (or indeed if there even is su ch a law) . In Eleusis an "umpire" secretly sets the rule , which o ther p lay ers have to d iscover. The re are several games based on Eleusis ; ours is the p aper-an d-

71

pencil version for four players-an umpire and players A, B , and C. (It c ould also be p layed by just two players.) Our game consists o f four sets­ one for each p lay er. When every body has h ad a go at being umpire , the game ends, and the p lay er with the lowest num ber of penalty p oints is the winner.

U M PI RE c c c c

C C C C C C

A A A "" 8:'C�A

t l� � �t.'fl'�t ..

c-a·A

A A A

c1-6'

fA

A A A A A A

"/,c e;t--

�-

B 8 B B B B B B B B Two sim p le u m p ire's rules for the p layers to discover are shown in the p icture of the b oard . On the left the ru le is: The letters A -B-C-A -B-C-A . . . go in a sp iral ; the letters can m ove in any d irection so long as their position is correct. On the right the ru le is : The first letter A can go any where ; the next letters m ust follow a "one square up , one square to the side" rule­ that is, on the diagonal. The umpire keep s a drawing of his rule to show afterward in case of d isp ute. Each p lay er arm s h im self with a sheet of squared paper to keep a record of b oth h is correct m oves an d incorrect attem pts ; about ten-by-ten squares should be big enough ( see p icture) . Play er A sits on the u m p ire's left , then B , then C. Each player has ten of his own letters , which he tries to correctly place on the board . Player A t akes a tum by p ointing to an empty square an d asking if he may write a lette r A in it. If the umpire say s yes, he p uts an A in that square and crosses off one of the A 's o n his side of the sheet. I f the u mpire says n o , t h e turn p asses to player B , and player A cannot cross o u t o n e of h is let­ ters. When A, B, and C have each had a turn, a round is com pleted ; the next roun d begins when A t akes h is nex t tu m . The idea is that the umpire shou ld se t rules that are neither too easy n or too hard fo r the players to d iscover ; ideally the p layers shou ld be rid

72

of their letters in the fifteenth rou n d . The umpire is penalized if either h is ru le is so easy that the players are ou t before the fifteenth round or if the rule is so hard that the game continues after that rou nd. The ump ire keeps a tally of his "score" ( penalty ) by c ircling numbers on the t ally card shown here , one number fo r each round :

o 0 8 1

0 0 5 2

0 0 2 5

0 o 1 8

0 15 0 15

For the first nine rounds the ump ire circles O's and loses no points , since it would be impossible for the p layers to get rid of their letters before the tenth round . At the tenth round he is penalized 1 5 p oints ; were the game t o be over then , the play ers would have k nown the rules from the begin­ ning. The penalty is p rogressively re duced until the fifteenth round , when it is again O. After that round the p enalty is p rogressively increased un til the twentieth roun d , when it is back up to- the m aximum, 1 5 .

Ticktactoe A Game for Two T icktacktoe, or Noughts an d Crosses, has to be the oldest battle of wits known to children and adults alike. The object is for one player to com­ plete a line-horiz on tal , vert ical , or diagonal-by himself. Any astute play er will learn how to play t o a d raw in only a fe w hours' p ractice . The gam e m ust end in a draw unless one p layer m akes a slip . There are j u st three possible opening plays shown by the X's in the picture-into a corner, the center, or a side b ox . Th e second p layer replies with O's. He can save himself from being trapped by one of the e igh t p os­ sible choices ; he can m ark the center. The side opening (third picture) offers trap s to b oth players ; it must be met by m ark ing one of four cells. I

\

-,

... ...

X

-,

,, - ,

\

/

, -, I J , - "

I

I

'- /

'-�

X

,-, , I ,

t

\

I

' -, , I , - ' ,, - , ,

I

,_/

,. '

\

,

I

... ...

X

,-, , , , - "

73

Ticktacktoe with Coins A Game for Two A m ore exciting variant of "Ticktacktoe" is where y ou p lay with coins or counters that can be m oved after being place d . The game was p layed a lot in E nglan d in the 1 3 00s, when it was called "Three Men's Morris ," the forerunner of "Nine Men's Morris," page 7 5 . It was also p opular in ancient China, Greece, and Rome . You use six coins in all , three silver d im es , say , for one player and three pennies fo r the other, on a three-by-three b oard . You take turns placing a coin on the board un til all six coins are down. By this stage e ither p layer could have won by having three of his o wn coins in a ro w-horiz ontal , vertical , o r d iagonal. I f neither player h a s w o n , they con tinue playing b y m oving a single coin o n e square , across or down , to any empty square. Diag onal m o ves are not allowed.

00 � � 0 � Teeko , o r Five-by-Five Ticktacktoe A Game for Two A m odern variant of "Ticktacktoe with Coins" called Teek o was invented by the Am erican m agician J ohn Scarne. It is p layed on a five-by-five b oard . Each p lay er takes turn s placing four coins. Then e ach m oves in turn one squ are horiz ontally , vertically , or diagonally . A player wins by getting his four coins in a square pattern on four adj acent squares or four coins in a horiz ontal , vertical , or d iagonal ro w.

74

Nine Men's Morris A Game for Two This old English game was kn own to Shakespeare and has been play e d by y oung and old ever since. It is really a varian t of "Tick tacktoe ." It used to b e p layed in the village green on a pat tern like this one scratched in the earth . Nowaday s it is played on a b o ard . You can copy this d iagram onto a sheet of stiff paper.

Y ou need eighteen "men," nine b lack and nine white . Tiddledy wink discs in any two colors will d o well. Each p lay er has his o wn color men. Each play er in turn p uts one man d own on one of the 24 d ots. He tries to form a three-in-a-row , or m ill as it is calle d. A m ill m ust be in a straigh t line ; it cannot bend around a corner. This is a m ill :

,., This is not a mill :

A play er who h as put down a mill can , on that move , re m ove an opp osing m an , p roviding it is not already p art of a m ill. The loser of the game is the player who loses all his men first. The shape I sh ow here is the com m onest form used ; but other shapes are possible. In fact , why not invent one of y our own?

75

Peggity A Game fo r Two This is an ancient game of p osition played on squared paper. It is also k nown as Peg Five and Sp oil Five. Thousands of years ago in China the game went by the name of Go-Moku. It is like the famous Japanese game Go, except that Peggity does not involve capture of e nemy pieces and so can be p layed o n p aper with pencil . T h e b oard i s 1 9-by- 1 9 squ are s b i g ; t h e p ieces, X ' s an d O's , are play ed one at a time into any of the sqau res (not on the c orners , as in Go.) X m oves first . The aim is to be first to get a straight line of exactly five adj acent X's or O's all in a line along a row , column , or either d iagonal . E ach p lay er has a s m any X's an d O ' s a s he needs. A p lay er with four X's (or O's) in a line-known as an open-four- must win on his next m ove becau se his opponent cannot block both ends in one m ove. But when he has another X one sp ace away at the end of the line ( see picture) , all the opponent has to do is play an 0 at the other end of the open-four. I f the open-four player plays into the sp ace , he gets a line with six X' s in a row ; this d oes n o t qualify as a line of five sym bols. After form ing an ope n-three- three O's or three X's in a line- it is u sual to call "three." This is because it can become an open-four on the next turn and thus a p otential winning p osition . Calling three avoids the likelihood of a play er losing by an oversight , which is fun for neither. A j o ined p air of lines of th ree X's (or three O 's) is called a d ouble-three ( see p icture) .

x 0 0 X 0 X

o pe n - t ) ree

X 0 0 0 0 X )(

open- f our

0 0 X X X )C X 0 0

_�Ren - four w i t h .-?Roce ��es not win

76

X

0

X 0 )(

X )( 0

0

ao u b l e - t hree for x

Snake A Game for Two This game is played on a five-by-six b oard of d ot s , like this one. Players take turns at j oining two dots by a line to m ake one long snake. No d iago­ nal lines are allowe d . You cannot leave any breaks in the snake. Each player adds to the snake at either end ; a player can o nly add to h is oppo­ nent's segment, not to his own. The first to m ake the snake close on itself loses. Here is an actual game. I n it straight lines began and lost.

l oses -- w i n s

--

Daisy A Game for Two The two players take turns to pluck fro m the daisy either one petal or t wo adj acent p etals. The player taking the last petal is the winner. This is a game invented by the great puzzlist Sam Loy d . M ake a daisy with 1 3 petals out of m atches, like this. On a p ostcard mark little c ircles where the petals ( m atches) gro w fro m . You need to know whether y ou have left a sp ace between petals or whether petals are next to each other. The second p lay er can always win- if he k nows how. See the answer section for this winning strategy . Remember y ou cannot take two petals if there is a space between them. That 's why we recommend marking the petals' p ositions.

77

Sipu A Game for Two S ipu is an old folk game from the Sudan. It is like "Ticktack toe," but it is d ifferent in two way s : It hasn't got an obvious strategy for not losing , and you can m ove y our X' s and O's as counters after they 've been put down. It is p lay ed on a square board with any odd number o f squares along each side ; the odd number ensures that there is a center square . ( Actually , Sipu is the name of the game played on a five-by-five board . The three-by -three b oard gam e , described here, is called Safragat .) You need counters or p ebbles, k nown as "dogs," of two colors, or two kinds of coins (pennies and d im es, say ) . We'll call them Black s and Whites. I n order to see how p lay goes, we'll start with a th ree-by -th ree board , for which y ou need four B lacks and four Whites. Play goes in two stages : First placing the counters and then making the m oves an d taking the opponent's counters. The best way to place your coun ters w ill become clear after play ing a few games. To see who starts the placing, toss for it or conceal a different coin in each fist and let your opp onent guess. Say Black begins the placing. Then you p lace the counters in tum - a Black , then a Wh ite , and so on-un til the b oard is fille d , leaving the center sq uare e m p ty . Le t's say y ou have filled the board like this :

0�0 0 � �0� You are ready to begin m oving counters. Toss to see who moves first. Le t's say it is White's first m ove. Coun ters are m oved either up or down or side to sid e , but not d iagonally . They can only be move d into an empty square .

( 1 ) White ( we'll say) moves into the e m p ty center squ are .

78

(2) Now Black m oves up to the square White has just left empty. In m oving, Black has flanked White's coin in a line "trap ," along the m iddle ro w . B ecause he m ade the trap , \ he can take White's counter on the center , square .

�---+----��r-� I

, b l ac k t a k es w h i t e

o�o � � 3

NOTE :

0 � 0

(3) White's next m ove forms another trap ,

which allows him t o take Black's counter, m arked X in the p icture .

wh i t e takes black

B ut supp ose Black had m oved into the trap him self like this : Then he could not have been taken because White hadn't m ade the trap by m oving himself. The play er who trap s the m ost c oins is the winner of the game .

Consider the starting p ositions on this board . White can m ove the center counter in the bottom ro w up, but Black still would have trapped him on the second m ove. In this starting p osition Black cannot m ove at all. Clearly , a position to be avoide d ! Longer-p laying and harder gam es can be play ed on a five-by-five board ( with 1 2 players ap ie ce) or even a seven-by-seven b oard ( with 24 players ap iece). The sam e rules ap ply .

79

Mancala A Game for Two M ancala i s an old African game now coming into fashion in America and E urope . Two p lay ers- we'll call them Al and Fey - sit on either side of a b oard , which is usu ally about a foot long with six h ollows on e ach side. (The h ollo ws can also be scratched in the groun d .) At the start of the game each hollo w is filled with four stones, balls, beads, or pebbles. The aim is for one player to capture all the others' stones ; these are the loot.

A

B

E

C 0 A I 's s i d e

F

A p layer m oves by taking all the stones out of one of the hollows on his own side and dealing them out in order, coun terclock wise around the b oard , I stone into each h ollo w. The p layers m ove in turn . We've lettered the h ollows simply to sh ow how the game goes. AI ' s are A, B, C, D, E, and F; Fey's are a, b , c, d, e, and f Say Al empties hollow E : He deals I ball each in F, a, b, and c. Then Fey empties, say , b ( which now holds 5 stones) . She deals them out into hollows c, d, e, t, an d A . The board then looks like this :

t e d c b a

( 5 5 5 6 0 5" �5 4 4 4 0 5 1 A B C D E F H ow d oes a p layer take loot? By placing the last stone in his opponent's last h ollow (F or f) so that there are 2 or 3 stones there . Here are three possible m oves to illustrate the point. 1 . This is the setu p . It is AI's m ove.

t e d c b (1 2 2 3 1 �O 0 0 0 0 A B C D E

80

a 2 1t

61

F

AI m oves all 6 stones from F (his only m ove ) , giving :

f e d (2 3 3 -lfo 0 0 A B C

c 4 0 D

b 2 0 E

a 3 0 F

1

AI's last stone went into f. which now has 2 stones. He takes these , to­ gether with the 3 stones in hollo w e an d the 3 in d. His loot d oes not skip back over c to collect b and a. He wins a total of 8 stones. 2 . Here's another setup. Again it is AI's m ove.

f e d 0 ( 2 3 .\II 0 0 A B C

c 0 0 D

b 3 7 E

a 1 8 F

,.­

M oving from F. AI would win no loot, since his last stone would go in B, on his own side of the b oard . Moving from E. he also would win nothing ; his last stone would go in f. which it m ust do to collect the loot, but d oes not result in 2 or 3 stones in that h ollow. 3. Empty hollows aren't necessarily safe . Here all but one of the hol­ lows on AI's side are empty . But the game still goes on.

f (18 � 0 A

e 0 I B

d 0 0 C

c 0 0 D

b a

I 01) 0 oj E F

Fey deals fro m hollow f:

f e d c b a

�� ; � � ; ;' A B C D E F

The last stone goes in a. She takes all the stones fro m AI's side. Why are there n ow no stones in f? Why didn't Fey put a stone in f? Because she took 1 2 or m ore stones out of it. When you take 1 2 or m ore stones out of a hollow , you skip that hollo w when you come to it ; thus the twelfth stone in Fey 's han d goes in the next hollow . The game ends when the p layers agree there are not enough stones left to form loot or when a player can not m ake a m ove .

81

The Crossword Game A Game for Two to Five This is su ch a p op ular game that though it is not a m athem atical game , I have put it in. Each p lay er has his own five-by-five grid . After it is decided who p lay s first , the first player c alls ou t a letter an d writes it in one of the 25 squ are s on h is grid. Each of the other players writes the same letter in som e square on his o wn grid . The next p lay er then calls out a letter-the sam e or another one- which the other players enter on their grids. Each player must write his letter before the next letter is called . The aim is to form words reading across or d own . If you c annot form a word , you can call an "unuseful" letter such as Z or Q so that nob ody else is likely to be able to use it . Only word s from an agreed-upon dictionary count, not p roper names or slang words. When 2 5 let ters have been called out and each p layer's grid is filled , scoring begins. Two-letter word s score 2; three-letter words score 3 ; four-letter, 4 ; and five-le tter, 6 (an extra p oint) . Totals for across and d o wn are added to­ gether to get the final score . Highest score wins. Two word s in the same row, or colu m n , m ay not share the same letters . For example , the letters H-E-A -R - T score 6 p oints ; whereas H-E scores 2 and A -R - T scores 3 p oints, totaling only 5 p oints. You could not ad d the two together to m ake I I . Also y ou could not sc ore 2 for H-E as well as 4 for H-E-A -R. H ere is a comple te grid with its score s and a grand total of 3 6 .

C H- E 5 5 C:> (c hess) H E A R T fa (heart) V

J

-r

0

0

p

2

2

N E R 4- (v i ne) -r 0 y 4- (toto) 0 L 0 Lt- ( p olo)

.3 2 .3 (to) (he) (ant) (re) (try) 82

®

The Cop and the Robber A Game for Two Here is a single-board game for two. You can p lay on the city plan sho wn here or draw a larger version for y ourself.

Y ou need two coins, one for the cop , the o ther for the ro bber. St art with each coin on its p icture . The rules are simple : The cop alway s m oves first. After that , the players take turn s to move. You move a coin one block only , left or right, up or down- that is, fro m one comer to the nex t . T h e a i m is fo r t h e c o p to catch t h e robbe r , which is d one b y t h e c o p land­ ing on the robber on his move. To m ake the game interesting , the cop m ust catch the robber in 20 m ove s, or he loses. HINT :

There is a way for the cop to nab the robber. The secre t lies in the top left comer of the city plan .

83

Sprouts A Game for Two S p routs is one of the best of the really new p aper-and-pencil games. A C am bridge ( England) m athem atician invented it in the 1 9 60s. Its name comes from the sh apes y ou end up with . I t is a game of topology . a branch of m athem atics which is very briefly explained in "The Bridges of Konigs­ berg " ( page 2 5 ) . Top ology is the geom etry of flop py rulers . wiggly lines , an d stretchy sheets of paper! This is how Sprouts is played. On a clean 0 0 sheet of p aper begin by drawing three or four sp ots. We'll work with three spots.

o

O--...O H--O

E ach of the two p lay ers takes turns at j oining the sp ots with lines, which can be as wiggly as y ou like. You must put a new spot som ewhere along that line.

o

No lines m ay cross.

Y ou can d raw a line fro m a spot back onto itself to m ake a loop - with , of course, a new sp ot on it .

A sp ot is "dead " when it has three lines lead­ ing to it ; no m ore lines can connect to it. To sh ow it is dead , put a stroke through it or shade it in. The winner is the one wh o d raws the last line . A good way to win is to trap "live" sp ots inside loops so that y our opponent can n ot use them . M athem aticians h ave worked out how m any m oves the gam e can go on for : The number lies between twice an d 3 times the number of sp ots you start with. St arting with 3 sp ots , the game can go on for be­ t ween 6 and 9 m ove s ; with 4 start ing spots,

84

0 o

o

a

l o op

between 8 and 1 2 m oves. And so on. But nob ody h as pro ved th is yet ! Here is a sample game. In it player A wins in 7 m oves.

START I N G POI N TS

1

2

0

:7

0

0

0

4

5

B p la y s

85

Mo rra A Game fo r Two This very old finge r game comes from Italy . One player is called Morra . On a given signal- a nod of the head , for example- both p layers put up either o ne or two fingers both at the same time. The ru les are in su mm ary form : B oth p l ay ers show same number of fingers : M orra wins two pennies Morra two fingers, opp onent one finger: M orra loses one penny M orra one finger, opponent two fingers : M orra loses three pennies S ee if y ou can fin d a strategy that cuts Morra 's losses or even lets h im win. The best strategy is given in the answer section.

Hex A Game fo r Two O nly recently invented in Denmark , this is a m arvelous gam e , which is also called B lack and White. It seems ab surdly simple but is open to very cun­ n ing p lay or strategy . as it is called . The game is played on a d iamond­ shap ed b o ard m ad e up of either hexagons, hence the name, or triangles.

The board u sually has I I hex agons ( or triangles) on each side. Two oppo­ site sides of the d iamond are Black 's side ; the o ther two are White's . The hex agons at the comer of the board belong to either player. The players take turns m arking hex agons. White m arks his p oint with a circle , Black with a heavy blob . The aim is to connect opp osite sides of the board with

86

an unbroken line of dots or blobs. (On the triangle paper, t wo dots are adj acent if there is a link ing line between them . )

T h e first p layer to m ake a n un broken line is the winner. T w o lines c annot cross, so there can never be a draw. M athem aticians h ave proved that the first player can alway s win , but they d on't say how he is to do so! You can buy special p rinted paper with a grid of hexagons or triangle s p rinted on it. I f y ou draw up y our own board , as here , do so in ink and play in pencil lightly so y ou can rub out the circles after each gam e. To learn som e of the strategies of Hex , p lay a game on a 2-by-2 b oard with just four hexagons. The p layer who makes the first m ove obviously wins. On a 3 -by-3 board the first player wins by m ak ing his first m ove in the center of the b oard . This is because the first p lay er has a d ou ble p l ay on b oth sides of his opening cell , so his opponent h as no way to keep him from winning in the next two moves. On a 4-by-4 board ( see pictu re)

things are more comp licate d . The first p layer will win if he play s in o ne of the four num bered cells , but if he p lays in any other cell , he can alway s be defeated. For an I I -by-I l b oard , as sho wn , the p lay is far too complicated to be analyze d .

87

ANSWERS

1 . Flat and Solid Shapes R eal Esta te! The combined length of the two sh orter sides of the triangular p lot come to the sam e as the long sid e : 2 3 0 + 2 7 0 = 5 00. The plot is merely a straight line and covers no lan d ! Three-Piece Pie Find the m iddle of the crustless triangle and m ake cuts from each corner of the p ie to the middle . Otherwise , y ou could m easure the angle of the slice and d ivide it by 3 . How Many R ec tangles ? N ine. Squaring Up S even squares. Triangle Trip ling Coun ting the little triangles in each corner gives three lots of 1 3 plus the b ig black triangle in the middle , m aking 40 in all . So y ou have 1 , 4 , 1 3 , 40 triangles. Note the pattern of d ifferences bet ween adj acent numbers (4 - I = 3 , 1 3 - 4 = 9, 40 - 1 3 = 2 7 ) . Each difference is 3 times the previ­ ous o ne-as y ou would expect from triangles! Th e Four Sh ru bs Plant three of the sh rubs at corners of an equilateral triangle ; p lant the fourth shrub on top of a little hillock in the m iddle of the triangle so that all four shrubs are at the corners of a tetrah edron (t riangu lar pyramid). See answer to "Triangle Quartet" ( page 1 1 0) . Triangle Teaser a. 1 3 , b . 2 7 .

88

Triangle Trickery Fold the p aper over as shown here. The folded flap ( its un derside sh owing uppermost) will co nceal a third of the triangle's face-up area still flat on the table. It is now only two thirds of the original triangle 's face-up area. So you have one third taken from two third s , leaving one third of the original area. It therefore sh ows one third of the original triangle .

Fold 'n Cu t Two h oles. Fou r-8quare Dan ce Seven different ways.

EE � C§ � dF qn �

Ne t for a Cu be There are eleven n ets that fo rm a cube. The first six are , p erhap s , fairly obviou s ; the other five y ou m ight not have thought of.

§=rn Etfo Etw Etta cqp qto cg Cftp � Cftn I I

89

Stamp Stumper The other ways are : 3 stam ps j oined side to side in a row , and three other L-sh ap es, like th is I , I, and -.J . Th e Fo ur Oaks

• • • •

Box th e Dots

90

Cake Cutting 1 6 p ieces. The rule is shown by the t able. One cut p l ainly gives two p ieces. For two cuts y ou add 2 to that num ber to m ake 4 . For three cuts y ou add 3 to the 4 to m ake 7. For fou r cuts y ou add 4 to the 7 to m ake I I . When y ou d raw th e lines, the third line will cu t two lines already d rawn ; the fourth line cuts th ree lines already drawn .

This table shows the n u m ber of pieces m ade by v arious n u m bers of cuts. n o . of cuts

no. o f p ieces

0

I

I

2

2

4

3

7

4

II

5

16

Four-To wn Turnpike The sh ortest network is made up of two diagonal turnpikes ; each is J 2 times 1 0 m iles long, or 1 4 . 1 4 m iles. So the t otal length of turnp ike is 2 8 . 2 8 miles, or about 28 .3 m iles. The J 2 comes from Pythagoras's theore m . It sho ws that with a right-angled triangle ( t wo o f which are p ro­ duced by bisecting a squ are with a d iagonal) , where the shorter sides ( the sides of the squ are ) are each I unit long, the long side opp osite the right angle (the diagonal) is J 2 units long, or the square root of 2 units.

91

Obstinate Rectangles In a six-by-seven rectangle , the diagon al cu ts 1 2 squ ares. Rule : Add the length to the width and sub tract I . One Over th e Eigh t I + 8 j igs 8 1 . A j ig m ust have 1 0 squares in it : I + (8 X 1 0) =

=

81.

Inside-out Collar To follow these instructions, it's best to label the corner of the tube a, b, c, a n d d around the top edge and A , B, C, and D around the bottom edge , as sh own in p icture I . A s shown in picture 2 , push comer c down into the tube t o meet corner A ; this will p ull the comers b and d together. As shown in black in p icture 3 , the square CcdD is already inside out , as is the square CcbB. The tri­ angular part with the edge A a stiII has to be turned inside out. This is done by p ulling the corners B and D ap art and pushing the peak (a ) of the tri­ angle d own to meet corner c - like pushing someone's head (a ) down be­ tween their k nees (B and D ) . Pull the corners b and d outward to turn the "beak" BCD inside out ( picture 3 ) . You wiII now find that as the tube unfolds, it is insid e out. The trick n eeds p ractice to perform it well : The secret is to do it in two stages- first stage is up to p icture 2 ; second stage is p ushing the peak d own "between the knees." a

B

c

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b

Cocktails for Se ven

The Carpe n ter's Colored Cu bes He cut the cube into eigh t equ al blocks, as sho wn.

Pain ted Blocks 1 8 faces are p ainte d , as sh own.

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Instan t Insan ity Take the cube m arked I in the p icture in the p roblem ; it has three d otted faces. Place it so that two of these faces are n ot on any of the long sides of the rod . Nex t , t ake cube num ber 2 and p lace it so that the four different colors of it are on the long sides. Then p lace cube number 3 so that one of its white faces is hidden and both hatched faces are on the long sides. Place cube num ber 4 so that neither of the hatched faces appear on the long sides. All you h ave to do now is t wist the cubes around the rod's axis until the solu tion shows up.

Th e Steinhaus Cube S tart building y our cube by m aking this step ped sh ape . The rest should fit toge ther e asily .

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Ho w Large Is th e Cube ? The surface area of the cube is 6 times the are a of one of its six faces. Sup­ p ose the cube has an edge x inches. One of its faces h as an area of x2 square inches. So i t s total surface area is 6x 2 square inches. But this must be equal in num ber to its volume or x X x X x = x3 cubic inches. So 6x2 = x3 , which means 6 = x . So the cube has a side of 6 inches . If this reasoning is too hard to follow, go from the equation 6x2 = x 3 a n d then try x = I , x = 2 , an d so on. Plato 's Cu bes The problem calls for a number which when m ultiplied by itself twice over gives a square number. This work s with any number that is itself alre ad y a square . The sm allest sq uare ( aside from I ) is 4 ; so the huge block might have 4 X 4 X 4, or 6 4 , cubes in it , and this would stan d o n a 8 X 8 square . The picture suggests that a side of the p laza is twice the extent of a side of the block . So this is the correct answer. The nex t size for the cube is 9 X 9 X 9 = 7 29 ; this cube would be standing on a 27 X 27 square , which , ac­ cord ing to the p icture , is too large. The Half-fu ll Barrel All they h ad to do was tilt the barrel on its b ottom rim . S ay the barrel was exactly half full . Then when the water is just about to p our out , the water level at th e bottom of the barrel shou ld j ust cover all the rim . That way half the barrel is full of water; the other half is air space. Cake-Tin Puzzle 1 0 inches square-that is , twice the radius. A nimal Cu bes 27 cubes in each animal. Both v olumes are 27 cubic centimeters. Are as: din osaur 90, gorilla 8 6 . Spider a n d Fly S ix shortest way s ; each goes along only three sides. A typ ical way is shown by the solid lines on the cube here .

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Th e Sly Slan t L ine 5 inches. The slant line must be the same length as the radius because it is one of the two equal d iagon als in the re ctangle .

2 . Routes, Knots, and Topology In-to-o u t Fly Pa ths He can fo r sh apes with an odd number of sides-the triangle , the pentagon , an d the seve n-sided sh ap e (hep tagon) . As he begins insid e , he has to cross an odd num ber of sides in order to e.nd up outside. In -to-in Fly Paths He can for shap es with an even number of sides- the square , the hex agon, an d so on. A BC Maze Out at A only , because an odd number of paths ( 5 ) lead to A . An even num ber of p aths lead to B, or to C ; so y ou cannot leave by them. Eternal Triangle ?

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