first verso page «01-ZLf:'S' � :sr L W MORE MATH PUZZLES ANDGAMES � \:) 0' S 001t by Michael Holt ILLUSTRATIONS BY PAT HICKMAN �w"" �!! 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 76 43 77·75319 CONTENTS I ntroduction v Flat and Solid Shapes Routes, Knots, and Topology 17 Vanishing-Line and Vanishing-Square Puzzles 33 Match Puzzles 41 Coin and Shunting Problems 49 Reasoning and Logical Problems 56 Mathematical Games 66 Answers 88 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 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: � 500 ds MAIN STREET j THIS VALUABLE SITE I DEAL FOR STORES OR OFFICES Sale on A pril I Why y ou think there were no buyers? 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 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 Patio and Well A 08 B Get A cross th e Po ol Spiral in to Sq uares r=� I L9 L��- -� 09 More Triangle Trickery (a) M ove m atches in the square corner as shown to form a step Its area is sq uare match units less than that of the original triangle ( square m atch units) ; thus, it is square m atch units �t II ,I ,I = =':-_":l1.L I ( b ) M ove more m atches to m ake another step Its area is then square m atch units less- that is, its are a is squ are m atch units r -� _r ���= = � " " " " " , ' I I , II " " " " = = = = :: =.::'� Triangle Trio IV\ Triangle Quarte t The answer is a triangular pyramid , a tetrah edron 10 Times th e A rea Tr iangles Cherry in th e Glass Slide one m atch across and move the other like this : Coin and Shunting Problems Coin Sorting in Pairs We have num bere d the coins to explain the answer The coins can be re grouped in three m oves: Move coins and t wo places to the left Fill the gap by and Jump and over to the far left I 1�1®�01�1 Rats i n a Tunnel The eight m oves are as shown There are two general rules: ( ) Shift a coin forward into a free sp ace , then jump another c oin over the coin j ust shifted (2) Alway s m ake shift s or jumps into the center of the tunnel first before m aking j u m p s or shifts away from it A wrong m ove is shown to indicate how you can get block e d The 111 rats sh ould end up exactly ex changed an d not with two spaces between each of th e black rats or each of the wh ite ones • • Start Move 0 :::==I:::::!!=t:==t:==*==t:�:.j ( I nstead Blocked ' (Move inwards before mov ing outwards I this) Done l Th ree-Coin Tric k (a) U sing H for heads an d T for t ails , the moves are : Begin T H T Move I H T T Move H H H � � Done! ( b ) No, it cannot be done Each move is not going to alter whether there is an even or an odd number of tails (or heads) As y ou see above , at each stage there is always an odd number of heads an d an even number of tails So y ou cannot get three tail s because is an odd number Triangle of Coins The trick is to m ove the coins in the opp osite way to which you want the final triangle to p oint 112 Five-Coin Trick Five-Coin Puzzle The general plan is as follows Y ou can shorten the number of m oves but this de scrip tion is easy to rem em ber Slide all five coin s around clock wise till the half-d ollar is in the top right corner (p ic ture I ) You note there is now a space be tween the half-d ollar and the penny He re we break the flow of coins This is the cun n ing bit Sh ift aroun d jus t the p enny , the d imes and the nickel in a clock wise d irection un til the pen ny is in the bottom left c orner ( p icture 2) Shift aroun d just the dime, the half-d ollar, and the qu arter in a counter clockwise direction until the half-dollar is next to the penny ( p icture ) All y ou have to n ow is slide all five coins around clock wise un til the p enny is just above the half dollar , on the left The trick was to split into two the flow of coins and reverse the direc tion of flow of three of them � @ Q '- / G0 G @ t;,\ '-� @ (0 ® Coin Changeovers The pennies and n ickels can ch ange p laces in b oth c ases 0) 0@ Mission Impossible ? Writing F for Dr Flinf and S for Dr Sieben , an d and for the agents, here is one way of com pleting the mission F, 5, S, and start on the Slobodian bank First , F an d row across the river ; F stay s on far bank ro ws back , p icks up fellow agent an d rows him over, leaving S alone on the Slobod ian bank Then rows back alone and p ick s u p S an d t akes him across to j oin the other two Mission possible! 13 R ailroad Switch Here are the six m ove s : (1 ) (2) (3) (4) (5) (6) The engine driver m oves p ast B, backs up into BC, and couples on the black car He p ulls the black car p ast B, then he b acks up into A B and un c ouples the black car Then he m oves p ast B and b acks into BC again He b acks p ast C and then shunts forward into A C and couples up the white car He p ushes the white car onto the m ain line out p ast A Still coupled to it , he backs up along AB and couples on the black car; he is now san dwiche d between the two cars S an d wiched between the two cars, he backs down past B Then he m oves up BC, where he uncouples the white car He now b acks p ast B and then m oves forward past A , still towing the b lack car He then backs up A C and uncouples it He m oves out of A C p ast A ; then he backs up into the stretch A B He is now facing the other way I ��t1tt1 1 11 111111 II II 14 I �Ofi?:t-I IIII R estacking Coins Just seven m oves are neede d , so they shouldn't have t aken too long to d o ! Here is t h e relationsh ip between the n u m ber of m oves and the number o f coins: I Coins Moves I 15 31 The number of m oves is times itself the same n u m ber of times as coins use d , less l Thus for three coins, it is (2 X X 2) - I , or - which is , - � � 15 R iver Crossing First , the two b oy s cross in the b oat : Now one soldie r is across the river and the t wo boys an d the boat are on the first bank with the soldiers Repe at the operation as m any times as there are soldie rs ! You note that the number of soldiers wasn't given ; it doesn't m atter Collision Course ? (1 ) (2) (3) (4) (5) (6) ( 7) (8) (9) (1 0) (1 ) ( 2) (1 3) (1 4) (1 5) (1 6) ( 7) 116 W ( white e ngine) w ith its car backs far out to the right (one reversal) W run s onto the switch , leaving its car on m ain track B (black engine) with its car runs out to the righ t W b acks onto m ain track ( t wo reversals) W couples with black car and m oves forward to left of switch B b acks onto switch (three reversals) W and black c ar back off to right and couple with white car ( four reversals) W pulls two cars to left of switch B run s onto m ain track B backs to train ( five reversals) B p icks up two cars and pulls them to right B backs rear ( b lack) car onto switch ( six reversals) B p ulls one car to right W b acks p ast switch and picks up white car from engine ( seven reversals) W pulls its car off to left and a way B backs up to switch and p icks up its o wn car ( eight reversals) B engine pulls its car from switch onto track and goes on its way Reasoning and Logical Puzzles Th in king Blocks A B a �-I I �-I , - - - - - I I I L _ c D I I I I - f t - -' I I i I - I _ _ _� LW � - - , � I L i _ _ - I � _ _ -, I J Martian Orders! (a) Thalia , Zane , Xeron ; (b) Shere e , Thalia, Zane, X eron Wha t Shape Nex t? (A ) Shape c , (B) shape e IQ Puzzle Shape Odd Shape Out a, ; b , ; c, ; d , I The Same Sh ape (A ) Shape d, (B) sh ap e c Nex t Shape, Please 17 Th e A p t House H ouse Wh o Is Telling th e Tru th ? Con is Th e Colored Ch em icals Puzzle O range , yello w , and green are p oison Mr Black, Mr Gray , and Mr Wh ite The key is that the m an in white is talking to Mr Black and so cannot be h im Nor can he be Mr White, since nobody is wearing his own color So the m an in white must be Mr Gray We can show what we know like this: Mr White black The straigh t line sh ows what m ust be true ; the wiggly line shows what can n ot be true Mr White can not be wearing white ; so he 's in black That leaves Mr Black wearing gray Hairdresser or Sh op A ssis tan t? Amy and B abs are sh op assistants ; Carol is a hairdre sser Th e Zo okeeper 's Puzzle A rt and Cora Wh o s Guilty ? Alf and B ert are guilty Wh o 's in th e Play ? Charle s an d Alice Tea, Coffee, or Malted Milk? M alted m ilk Soda o r Milksh ake ? Suppose Alan h as a soda Then (a) say s Bet has a m ilkshake But (c) tells you Cis cannot then have a m ilkshake and m ust h ave a soda But (b) says b oth Alan and Cis cannot b oth h ave a sod a Alan cannot have a sod a ; so Alan has a m ilkshake From (b ) that means Cis h as a soda Which , from (c) , leaves Bet free to choose either a soda or a m ilksh ake Thus there are two p ossible orders : ( I ) Alan , m ilksh ake ; Cis, soda ; and Bet , sod a ( 2) Alan, m ilksh ake ; Cis, sod a ; Bet , milkshake 118 Ne wton s Kittens Obviously the k ittens could have gotten in and out by the same hole as the m other cat ! March Hare 's Party Sylvie had tea u n der the tree at table I because she wouldn't go near water Al and Barbra sat at table : He couldn't take her in the b oat to table Gary j o ined the m at 3, roller-skating over the bridge : He couldn't go to table because of the "no boys" rule That leaves Don , who wouldn't sit with Gary at table and also couldn't take the p ath to table Don rowed to table and sat by h im self Answer: table I , Sylvie ; 2, n ob ody ; , AI, Barbra , and Gary ; and , Don Marriage Mix-up Ted is m arried to Barbra with daughter Ruth , Pete to Sue with daughter Wendy , and Charlie to Nicola with son Dick The reasoning goes like this: Ted's d aughter is not Wendy So his daughter must be Ruth So Pete is father of other girl , Wendy Which means Charlie is father of Dick S o his wife cannot be Barbra because she has a daughter ( Assume a girl plays Annie and Ophelia ! ) His wife is not Sue, so his wife has to be Nicola Now Pete's daugh ter is not Barbra's daughter because they have only one child each So Pete cannot be m arried to Barbra That m eans Ted is m arried to B arbra , and Pete there fore to Sue Wh o Does Wh ich Jo b ? Orville i s bartender and singer ; Virgil i s private ey e and racing d river; Homer is jockey and cardsharp This is how you get the answer Draw a table to show the men and the jobs, and fill it in as follows: Orville Facts used I I Hom er Priv ate eye Racing driver S inger Virgil J ockey B artender X X X X X Cardsh arp First look at the jo bs Fact tells us the b artender is not the same m an as the racing driver Put a beside them ( as sho wn) Similarly the racing d river ( ) is not the singer (2) And so on Now look at the men Fact 19 tells us Homer is neither the racing d river nor the singe r ; so p ut an X in the table under Hom er opp osite those two job s , as shown Fact says Virgil is not th e singer; put an X under Virgil op posite Singer Fact tells y ou Homer is the j o ckey as Virgil and Orville are not ; opp osite Jockey put X's under V and and a check under H Now for the reasoning Orville m ust be the s inger- since neither Virgil nor Homer is-so put a check under opp osite Singer Here is the table so far in brief: H V o P X R S J x x X X J B C To fill the Singer line put a check under Orville ( 0) Then in the Jockey line put a check un der Homer (H) And so on Fact tells us to put a cross under H opp osite R Fact gives an X under H opp osite P and B That leaves o nly C for H's second j o b : put a check there The t able looks like this : o V H P X R X S J X X X X J B X C J Now we can put a check under V op p osite R So Fact gives an X under V opp osite B -thus forcing a J under on th at line ( that is , Orville's second job is bartender) Fin ally , the bottom line with two X' s means V irgil's second job is P ( p rivate eye) Birds and Insects A nswer A alone follows logically 20 Wonderland Golf Five shots : DDDS D or S DDSD There is a pattern To see it , turn back to the map of the golf course Working b ackward s from h ole , a D shot gets you back to hole 9, then an S sh ot to 8, followed by three D's to the first tee (From to I could be an S as well.) You can also work it out by arith metic Divide the hole number by over and over again , n oting if there is a remain der of l For y ou get : , , 18 r Count up the number of answers and rem ainders : , , , , , which m akes five numbers; that's how many shots it takes This rule w ork s for any hole Mad Hatte r 's Tea Party Set G to table , M to , and B to Table stay s e m p ty Mathematical Games Nim The way to calculate a winning position is best shown with the starting p osition of Nim It has , 4, an d m atches We rewrite the rows in binary that is, in powers of , or in "doublings." The numbers 0 , , an d in b inary are , , and in every d ay nu mbers While 1 in usual counting num bers means ten an d l one, in binary it m eans two and l one We can m ake out of + and in binary write it as 1 Then in binary is 00 , m eaning four and no twos and no ones, an d in binary is 1 , mean ing four , no twos an d l one We set the rows out in b inary as follows: Matches Fours Twos Ones I Top ro w Middle row 0 B ottom row 1 2 Totals As y ou see , we added each colu mn up in ordinary num bers ; but we did not 21 " carry " num bers over from one column to the next Two column totals are eve n , and one, the m iddle colu m n , is odd To m ake the p osition safe for y ourself, all y ou is make the totals of each column even So your first m ove is to take m atches from the top row , as explained This changes the t op binary num ber to I The colu mns then become : Matches Fours Twos Ones T op row I Mid dle row I 0 B ottom ro w I I 2 Totals I Now e ach column adds up to an even number The p osition is safe Daisy Here is the second player's winning strategy : Say the first player takes one p etal ; then the second p layer takes t wo pe tals, which m ust be n ex t to each other, d irectly opp osite the one taken by the first p layer If the first player takes two adj acent petals, the sec ond player takes one petal, again directly opp osite Either way this leaves two sets of five petals , sy m metrically ar range d ab out the two spaces All the second player has to now is to keep the p attern sy mmetric al, taking special note of the spaces Th e Cop and th e R o b ber The cop h as first to go around the triangular block at the top left corner Then he is an odd number of corners away from the robber and can cat ch him - p rovided the ro bber does not go arou nd the triangle! Remember, there are only three corners in the triangular block , and you can get right around it in three m oves Morra Morra' s Winnings Other sh ows I finger M orra shows fingers I finger +2 - fingers - I +2 M orra's best strategy , to reduce his losses, is to sh ow two fingers all the tim e ; then he never loses more than one penny 22 ... and Solid Shapes Routes, Knots, and Topology 17 Vanishing-Line and Vanishing-Square Puzzles 33 Match Puzzles 41 Coin and Shunting Problems 49 Reasoning and Logical Problems 56 Mathematical Games... 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...first verso page «01-ZLf:'S' � :sr L W MORE MATH PUZZLES ANDGAMES � :) 0' S 001t by Michael Holt ILLUSTRATIONS BY PAT HICKMAN �w"" �!! WALKER AND COMPANY New York Copyright ©1978 by Michael