Mathematical Teasers A distinctive collection of problems, puzzles, and tricks with complete explanations Download the full e-books 50+ sex guide ebooks 100+ ebooks about IQ, EQ, … teen21.tk ivankatrump.tk ebook999.wordpress.com Read Preview the book MATHEMATICAL TEASERS the text of this book is printed on 100% recycled paper MATHEMATICAL TEASERS the text of this book is printed on 100% recycled paper ABOUT THE AUTHOR Professor Mira holds the degree of bachelor of science from Pennsylvania Military College and the degree of master of arts from Columbia University After a few years in industry as an engineer, he began to teach mathematics at Manhattanville College, where he has been teaching for nearly 30 years Professor Mira is now chairman of the mathematics department at Fort Lauderdale University He has also been a lecturer in mathematics at Saint John's University and at the City University of New York Mr Mira is a fellow of the American Association for the Advancement of Science, and is author of Arith'Ynetic Clear and Simple and coauthor of Business Mathematics and Mathematics of Finance EVERYDAY HANDBOOKS MATHEMATICAL TEASERS Julio A Mira • IIIII BAR N E S & NOBLE BOOKS A DIVISION OF HARPER & ROW, PUBLISHERS New York, Evanston, San Francisco, London © Copyright, 1970 By BARNES & NOBLE, Inc All rights reserved No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage and retrieval system, without permission in writing from the publisher L C Catalogue Card Number: 74-101122 SBN 389 00261 Printed in the United States of America PREFACE Solving mathematical puzzles has been a popular pastime for men since antiquity From the cryptic utterances of the oracle of Apollo at Delphi in ancient Greece, six centuries before Christ, up to the present time, the solution of puzzles has contributed much to the development of modern mathematics Three famous problems of antiquity: the squaring of the circle, the duplication of the cube, and the trisection of certain angles, have occupied Greek geometricians for centuries and have led to the discovery of many theorems and mathematical processes In fact, only in recent times, with the development of the theory of groups, has it been proved that these problems cannot be solved by the formal methods of euclidean geometry The problem of settling stakes in a game of chance, which was proposed to Pascal (1623-1662) by de Mere, led Pascal and Fermat (1601-1655) to formulate the fundamental principles of probability, a branch of mathematics, which is the foundation of modern statistical methods, and upon which the insurance business is based Laplace (1749-1827) once said of probability, "A science which began with the consideration of play has risen to the most important objects of human knowledge." Great thinkers like Anaxagoras (c.500-c.428 B.C.), Kepler (1571-1630), Leibnitz (1646-1716), Euler (1707-1783), Lagrange (1736-1813), and many others have devoted much of their time to solving mathematical puzzles It was Leibnitz who remarked, "Men v vi Preface are never so ingenious as when they are inventing games." This book contains a number of interesting, and often challenging, mathematical puzzles, which have been very carefully selected and put together especially for enjoyment by the person who may be neither a mathematical genius nor even a mathematician It is the express hope of the author that the reader will enjoy these "mathematical teasers" as much as he has The solution to each of the problems is found at the end of the chapter containing that particular problem Some of the theory used in solving these problems can be found in Arithmetic Clear and Simple, also by the author The author wishes to express his gratitude to the heirs of Samuel Jones for permission to use his outstanding and unusual collection of mathematical puzzles, to Jonatha Foster for typing the manuscript, and to Pamela Singleton for her excellent cartoons TABLE OF CONTENTS Chapter Teasers for All Page More Teasers for All 34 Teasers for the Mathematicians 69 Teasers for the Wizard 115 Teasers for the Thinker 178 vii 20 Mathematical Teasers solutions how far is near Both trains are at the same distance from New York the ba n k teller's problem If the final remainder were zero, the sum of the amounts subtracted would equal the original amount However, the sum of the remainders does not equal the sum of the amounts subtracted, therefore the sum of the remainders does not yield the original amount yes, we have no apples Two apples She took one apple and left one apple the new a rith metic = Write 12 X II , remove the lower half you have VII=7 /\ II , and 21 Teasers for All the calculating cook minutes, if all the eggs are boiled at the same time when larger is smaller Any proper fraction For example: i -;- 1, but i X i =! square geometry Remove matches numbered and the ca reless clerk 18 inches Only inches were lost on each yard in measuring 18 feet, or yards, with the 33 inch stick, but nothing was lost in measuring the last two feet Therefore, the total loss to the customer was X = 18 inches the tramp's cigarette Nine The 29 butts yield seven cigarettes with one butt left over When the tramp smokes these seven cigarettes, he has seven new butts, and, with the butt he had left over, he now has eight butts from which he can make two cigarettes The total number of cigarettes made by the tramp is seven plus two, or nine cigarettes, with two butts left over 10 discomposing Discs numbered 1, 7, and 10 22 Mathematical Teasers 11 a melon drama A ,~~ J: I- MELON PATCH i5 12 even from odd It = I + I = 2, 3~ = + I = 4, 5~ = + I = 6, etc 13 the land of ifthen 81 since X = 20, and i of 20 is -1 of 33, or 8-1 14 a bottle full equals a bottle em pty If the problem refers to the capacity of the bottle, the conclusion is correct; but, if the problem refers to the amount of liquid in the bottle, then twice the amount of 23 Teasers for All liquid in a bottle half empty is equal to twice the amount of liquid in a bottle half full Therefore, you would have a full bottle and not an empty bottle 15 a dirt full None There is no dirt in a hole 16 tent station The larger tent was pitched so that its base formed a square with E, F, G, H as the corners and A, B, C, and D as midpoints on the sides E F H G 17 how far does the horsefly fly? mile Horsemen approach each other at + 10 miles per hour The original distance between them is t mile Hence, they meet in t -:- 10 = of an hour Therefore, the horsefly flies for :to of an hour at 20 miles per hour, or X 20 mile = to = 18 m i lady's bracelet 25¢ Cut each of the five links of one of the pieces of chain Use four links to connect the five remaining Mathematical Teasers 24 pieces of chain and the fifth link to connect the ends Since there are a total of cuts and welds, it costs X 2¢ + X 3¢ = 25¢ 19 nail iuggling II I V,=6=3 20 identity not lost 21 the lazy mon key The lazy monkey rises 22 the rook's swindle $75 and the shoes 23 iohnson's cat The 14th day The cat's daily upward gain is 11 - = feet In 13 days she climbs 13 X = 52 feet up the tree, and on the 14th day, she climbs 11 feet Since 52 + 11 = 63 feet, the cat reaches the top on the 14th day 24 a nother identity not lost X or~ l! Teasers for All 25 25 the match rna ker LO v E 26 hotel stretch The manager did not give the eighth man a room 27 the road to st ives One I am the only one going to St I ves 28 a good match VI+IV • • x 29 take it away and it's there 9+8+7+6+5+4+3+2+1=45 1+2+3+4+5+6+7+8+9=45 8+6+4+1+9+7+5+3+2=45 30 square eggs A multiplier is always an abstract number In finding areas, for example, the statement, "feet times feet equals square feet," is misleading The fact is that in 26 Mathematical Teasers finding the area of a rectangle, we are not multiplying feet by feet, which is absurd, but the number of square feet in the base by the number of rows, which is an abstract number 31 some more sq ua res Remove either of the following pairs of matches: numbers and 6, and 7,6 and 9, or and 32 things are not what they seem AB = CD; EF is parallel to GH, therefore these lines will never meet; lines JK and NO are at the same level 33 travel in flatland 34 of cats, dogs, and mice First, he takes the cat across He returns and takes the mouse across and brings back the cat; then he takes the dog across and returns and takes the cat across Teasers for All 27 35 when twice is th rice Zero 36 the light-fingered porter Each man paid $10 and should have gotten back a third of $25 Thus, $10 - ~ X $10 - ¥$;5 $8t is the actual cost of the room per man Therefore, the calculation should be 81 X $3 = $25 Then, $25 + $3, which the porter returned, plus $2, the porter's profit, is $25 + $5 = $30 = = = 37 simple addition 11 o'clock plus hours is o'clock 38 for the postmaster 2¢ The cost of thirteen stamps is 1¢ and a quarter, or 26¢; hence one stamp costs 2¢ 39 for the english scholar Neither Since + grammatically correct = 15; but both statements are 40 old bi lis for new ones An old $10 bill is usually preferable to a new $1 bill 28 Mathematical Teasers 41 the mathematica I bookworm i day Since usually volume I is to the left of volume II, the first page of volume I is separated from the last page of volume II by only two covers, which together total :l inch Since the bookworm bored through at the rate of i inch per day, it took him i -7- i = i day 42 two to two 43 a party with threes In a series of indicated operations, mUltiplications and divisions are to be performed first in the order in which they appear from left to right; additions and subtractions are to be performed after these operations have been completed Hence, 3 X - -7- - = -1-3=8 + 44 how fast is the cutter 59 minutes The last piece does not have to be cut 45 five sq ua res from th ree II~ [] I + 29 Teasers for All 46 a fast buck hour The width of the woods is 60 miles; hence the buck ran into the woods 30 miles, for after that he would be running out of the woods Since the buck travels 30 miles per hour, he runs for 30 miles -+- 30 miles per hour = hour 47 a perplexing choice Company A The young man would receive the following yearly amounts: Year 1st 2nd 3rd 4th $4,000 4,600 5,200 5,800 Company A + 4,300 = $8,300 + 4,900 = 9,500 + 5,500 = 10,700 + 6,100 = 11,900 Company B $8,000 9,200 10,400 11,600 Company A's offer is better by $300 a year 48 the eager beaver The company has a claim to the young man's working hours only, that is, to hours a day for a 5-day week, discounting all legal holidays 49 grandpa's watch chain 30¢ Cut each of three links of one of the pieces and connect the four other pieces Since there is a total of three cuts and three welds, it costs (3 X 2¢) + (3 X 8¢) = 30¢ Mathematical Teasers 30 50 triality Form an equilateral triangle with three matches With this triangle as the base, erect a pyramid with the three remaining matches The three sides of the pyramid and the base should form four equilateral triangles 51 the pigpens 52 post time 44 posts 53 rhythmic roman One hundred one = CI Six = VI Fifty =L Thus, we obtain CIVIL 54 daft arithmetic = Write IX have IV=4 , remove the lower half 1/\ , and you Teasers for All 31 55 square quarters 16 quarter-inch squares; 64 quarter-inch cubes 56 dozens of dozens Six dozen dozen = X 12 X 12 = 864 ; one half a dozen dozen = t X 12 X 12 = 72 57 show biz He takes the goat across He returns and takes the basket of cabbages across and brings back the goat; then he takes the wolf across and returns and takes the goat across 58 match wit 59 post man 40 posts 60 more tria lity Note that the perimeter of the figure also forms an equilateral triangle 32 Mathematical Teasers 61 iuicy fruits Give one of the boys the box with an orange in it 62 duck a duck Three ducks 63 lend an ear 21 days Two of the ears are the squirrel's 64 ya rd tract Since t yard square equals tXt = ! square yard and ! square yard equals half of t square yard, t square yard equals half of t yard square 65 common places 99~= 99~= 99~ = 99 + = 100 66 more daft a rith metic Write 6= VI , turn VI upside down, 1\1 , put it at the bottom of V I , and you have XI=II Teasers for All 33 67 slippery snail days At the beginning of the eighth day the snail is feet from the top, and it reaches the top at the end of that day 68 rolling around One revolution 69 the millionaire Zero area These dimensions not form a triangle but a line, since 51! + 49! = 101 Therefore the lot has no area 70 a count up 150 triangles; 30 squares chapter more teasers for all discotheque go-go The outer track of a long-playing record is lli inches in diameter and the label at the center is 5i inches in diameter If the record has twenty grooves per inch, how far does the needle travel to play the whole record? instant product Find the product of the digits 1,2,3,4,5,6,7,8,9,0 as fast as possible, and time yourself an L of a design A lady asked her son Henry, an architect, to design a house for her in the form of an "L" with windows in each wall and with all the windows having a southern exposure How did the architect solve this problem? 34 ... York Mr Mira is a fellow of the American Association for the Advancement of Science, and is author of Arith'Ynetic Clear and Simple and coauthor of Business Mathematics and Mathematics of Finance... 34 Teasers for the Mathematicians 69 Teasers for the Wizard 115 Teasers for the Thinker 178 vii chapter teasers for all how for is near A train leaves New York for Chicago traveling at the rate... problem A man having $50 in a bank withdrew it as follows: $20 15 $50 leaving leaving leaving leaving $30 15 $51 Where did the extra dollar come from? Mathematical Teasers yes, we have no apples A