Fiendishly difficult math puzzles by ivan mascovich

CONTENTS Introduction 4 How to Solve Problems 5 Match Blocks 8 Finding the Key 10 Continuous Paths 12 Sliding Coins 14 Magic Numbers 16 Magic Numbers 2 18 Combi-cards 20 Money Problems 2

PUZZLES I

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I v a n M o s c o v i c h

Sterling Publishing Co., Inc N e w York

Edited, designed and produced by Eddison/Sadd Editions Ltd

Creative Director: Nick Eddison

Art Director: Gill Delia Casa

Designer: Amanda Barlow

Editorial Director: Ian Jackson Project Editor: Hal Robinson Proofreader: Christine Moffat Artists: Keith Duran (represented by Linden Artists) 18-19, 24-25, 40-41;

Andrew Farmer 22-23, 28-29, 32-35; Mick Gillah 8-13, 30-31, 46-47;

Kuo Kang Chen 6-7, 38-39, 44-45, 48-49; Andy Pearson (represented by

Ian Fleming & Associates) 20-21; Larry Rostant (represented by Artists

Partners) 14-17, 26-27, 36-37, 42-43

Solutions artwork: Anthony Duke and Dave Sexton 50-63

Acknowledgments

Eddison/Sadd would like to acknowledge the assistance and cooperation

received from Clark Robinson Limited during the production of this book

Library of Congress Cataloging-in-Publication Data

First U.S edition published in 1991 by

Sterling Publishing Company, Inc

387 Park Avenue South, New York, N.Y 10016

Originally published by Penguin Books 1986

Original concepts © 1986 by Ivan Moscovich

This edition © 1986 Eddison/Sadd Editions Ltd

CIP

CONTENTS

Introduction 4 How to Solve Problems 5 Match Blocks 8 Finding the Key 10 Continuous Paths 12 Sliding Coins 14 Magic Numbers 16 Magic Numbers 2 18 Combi-cards 20 Money Problems 22 The 18-point Problem 24 Jumping Coins 26 Life or Death 28 From Pillar to Post 30 Gridlock 32 Crossroads 33 Separate and Connect 34 The Tower of Brahma 36 Interplanetary Courier 38 Husbands and Wives 40 The Octopus Handshake 42 Calculating the Odds 44

Up in the Air 46 Lucky Spinner, Lucky Dice 48 The Solutions 50

INTRODUCTION

I have always b e e n fascinated by puzzles and games for the mind

I enjoy brain g a m e s of all types - and like particularly those with some special aspect or feature Those I like best are not in fact always the hardest: sometimes a puzzle that is quite easy to solve has an elegance or a 'meaning' behind it that m a k e s it especially satisfying I have tried to provide a good selection in this book:

some are easy and some are fiendishly difficult but they are all tremendous FUN! Above all, I have tried to provide something for everyone, in order to share my delight in such puzzles and games

as widely as possible

Solving puzzles has as much to do with the way you think about them

as with natural ability or any impersonal measure of intelligence Most people really should be able to solve nearly all the puzzles in this book, although of course some will seem easier than others All

it takes is a commonsense, practical approach, with a bit of logic and - occasionally - a little persistence or a flash of insight

Thinking is what it's all about: comprehension is at least as important as visual perception or mathematical knowledge After all, it is our different ways of thinking that set us apart as

individuals and m a k e each of us unique

Although some of us feel we are better at solving problems mathematically, and others prefer to tackle problems involving similarities and dissimilarities, and others again simply proceed by trial-and-error persistence, w e all have a very good chance of solving a broad selection of puzzles, as I'm sure you will find as you tackle those in this book

From long and h a p p y experience, however, I can tell you one secret, one golden rule: w h e n you look at a puzzle, no matter how puzzling it seems, simply BELIEVE YOU CAN DO IT, and sure enough, you will!

HOW TO SOLVE PROBLEMS

To start things going, let's look at the

different approaches that can be useful in

solving puzzles

First, the logical approach Logic is always

valuable, as it helps you work things out

sequentially, using information received

to progress step by step to the answer

This is especially true when puzzles tend

to be oriented toward mathematics and

concentrate on using numbers for simple

calculations, or on ordering arrangements

of objects or figures Examples of this can

be found in the games Magic Numbers

In problem solving, there may also be a

need for an indirect' approach, whereby

you arrive at an answer by perceiving a n d

thinking about a subject in a way you have

never done before This d e p e n d s on how

you think normally, of course, a n d so for

some people it may be helpful for certain

puzzles, a n d for others for different ones

The first part of Match Blocks is solved most

simply, quickly, easily and elegantly' using

an 'indirect' approach of this kind

The visual approach is also important, especially in this book because all the puzzles are presented in visual terms and require initial visual comprehension (or conceptualization) to be combined with understanding the text of the problem This

is particularly the case with the tricky

puzzle set as The 18-point Problem

In general, the math puzzlers in this book are of four types They are concerned with:

1 simple calculation using patterns, objects

or symbols;

2 spotting serial links a n d connections;

3 the laws of chance a n d probability particularly in assessing the odds for or against specific events or results occurring;

-4 ordering, combining or grouping objects

or figures, following a defined rule, to achieve a stated target

Examples of all four types are given on the following pages, together with the answers See if you can solve them first without looking at the answers — then go on to enjoy the rest of the book!

SAMPLE GAMES

GAME I

^Tn The Magic Square is possibly the

oldest mathematical puzzle in

C existence Examples have been found

dating back to before 2000 BC By AD 900

one Arab treatise was recommending that

pregnant women should wear a charm

marked with a Magic Square for a favorable

birth

Can you distribute the numbers 1 through

16 in this 4x4 square so that lines across,

lines down, and major diagonals all add up

to the same total?

Hint: M a k e each line a d d to 34

Tv puzzles that initiate a

V series and then require you to carry on when they leave o f f This means that you have to spot the links or connections between the figures or symbols that make up the series

What is the next entry in each of these series?

a)ABDEGH J ? b) 3628 21 15 106 ? c) ' I A • O ?

b) 3

itZltS' 1 ^

^ c t e d e a c T i

GAME 3 GAME 4

Two coins fail through the air turning as

they drop Fach coin has the usual two

C sides heads (h) and tails (t) In how many

combinations oi those sides can they end up

when they come to rest on a iiat surface?

Well, one way of looking at the possible

results is:

heads heads

heads tails

tails tails

- three possibilities, from an overall point oi

view Does that mean that there is a 1:3

chance oi any one result?

Suppose we number our coins, odd

numbers on the heads side, and evens on

the tails How does this help to prove that

the odds oi heads tails occurring is actually

2:4 Of 1:2?

In a darkened room there

is a box ot mixed gloves:

D 5 black pairs, 4 red

pairs, and 2 white pairs You

find the box by feeling lor it

I low many gloves must you take out-without being able to see them - to make sure you have two of the same color?

And how many must you take out to make sure you have both the left and right hand of the same color?

(Solutions p a g e fiOJ

MATCH BLOCKS

The blocks in columns on these two pages

can be a r r a n g e d in a 7 X 7 square formation

so that the horizontal rows are n u m b e r e d in

succession Irom top to bottom 1 through 7,

as shown in the diayram below

The columns of blocks shown below and right can b e used in two puzzles

You can m a k e your own columns of blocks if you like, but a pencil and some thought with the g n d should suffice

GAME 2

Arrange the columns again

so that no number appears more than once not only in a horizontal or vertical row but also in a large or small diagonal,

9

(Solutions p a g e fiOJ

FINDING THE KEY

Most of us carry a few keys around with us;

some, tike me, carry vast collections

weighing down their pockets It's not

surprising, really, in view of the number of

different things we now need to k e e p

locked, automobiles, suitcases and briefcases, office doors and safes, even desks and bureaus at home So here are

a couple of puzzles on the subject I hope you 11 find the key to solving them

KEYS TO THE

KEYS

On a circular key ring there

are 10 keys, all with round

handles, in a specitic order

that you have memorized

Each fits one ol 10 different

locks The trouble is, it 's

pitch dark, you can't see the

keyring, you can only feel

the keys with your fingers If

you had some way of telling

in the dark which key was

which, it wouldn t take you

long to find any particular

one you wanted So you

decide to give some keys

dilierent-shaped tops- but

do you need JO different tops ?

What is the least number

of different key tops you'll

need to be sure, once you ve

felt them, that you ve

identified where you are on

the ring? And would you put

all the new keys together or

give them some sort ol

COMBINATION

IOCK

A safe has ten locks in

combination, requiring ten

keys, each of which bears a

letter inscribed on its

handle Rut to confuse

thieves some of the letters

are the same

The safe opens only when

all the keys have been

inserted in the locks, the

handles then spelling out a

secret code word

Fortunately, you have a

diagram of the interior of

the locks, showing the

shapes of the appropriate

keys Otherwise you might

have to spend a lot of time

trying out all the possible

3.6 million combinations of

ten locks And of course you

also know the secret code

w o r d

What is the secret code

word?

1 1

(Solutions p a g e fiOJ

CONTINUOUS

PATHS

Fifteen lines join the six points, or nodes, of

a regular hexagon Where each line crosses

another there is a further node, giving a

total of 19 nodes in all Every line also

carries an arrow: no matter w h e r e the arrow

is located on this line, it m a k e s the whole

line directional

The object of the g a m e is to try to find

a continuous path connecting all 19

nodes, starting anywhere (which

becomes n o d e n u m b e r 1) You must always

travel down lines - or parts of lines - in the

direction of the arrow, and you may visit

each n o d e only once

SAMPLE GAME

T h e sample ^ S E ^ E S S ' "

i, is not so ^ V - " - ™ w h ic h h e or

player may because

she is then unable to.move^ l h a l

X S ^ ^ - n / o n e - h a s

f o x e d the player

The first hexagon A (above) has arrows that point in the same directions as on the sample game Can you complete the puzzle? Is there more than one node

DEVISE YOUR OWN

This version of the g a m e can also b e played by two people, each taking turns

to s h a d e an arrow (until there are no more arrows) and m a k e a move; the last

to move is the winner

The decision about which way each arrow points can also be determined by chance: toss a coin for each arrow -heads points left, tails points right

Arrows should point in only one

direction In the hexagons below (E and

F), however, all the arrows are

two-h e a d e d , because I'm giving you a ctwo-hance

- before you start playing - to make up

your own mind which direction you want

the arrows to point S h a d e off lightly in

pencil the unwanted end of each arrow

Then play the game as usual

13

(Solutions p a g e fiOJ

SLIDING COINS

the games O n e move involves moving a

piece from its position to a free space; this

need not b e an adjacent space, but it must

b e reached without any other piece being

disturbed

In these g a m e s I challenge you to reverse

the positions of sets of coins within a

confined space Cash-flow problems, you

might say! If you can't find coins of the right size, counters will do Small circles in the

g a m e bases show the centers of the possible positions of coins or counters; the miniature diagrams indicate the starting positions tor

GAME I GAME 2

This game requires only eight coins: four one w ay up (heads) and four the other (toils) But it

is not necessarily easier-fewer coins are compensated tor by less space in which to move What is the smallest number

of moves in which you can reverse the positions of the two sets of four coins? Can you do better than 30?

GAME 3

In this game it is trie starting and ending space that is the linear element and it is nil too easy to block everything with coins all trying to get past each other

What is the smallest number

of moves in which you can reverse the positions of the two sets of three coins successfully? Can you beat lb?

Start with nine coins, four one

way up (heads), four the other

(tails), and one com altogether

different If you use counters,

choose difterent colors By

moving pieces one at a time w to

available tree spaces, can you

rearrange ull pieces to reverse

the starting pattern ? What is the

least number of moves required

to complete the reversal? Can

you do better than 36

(Solutions p a g e 53)

'Magic Squares' - in which lines of numbers

add u p to the same total whether read

horizontally or vertically, or sometimes

e v e n diagonally have b e e n the delight of

magicians (and mathematicians)

throughout history Yet many other s h a p e s

can b e used equally well, if not better Some

are actually simpler - like the Magic Cross

In most puzzles on these two pages, 1 have

given you the total all the lines should add

u p to - the 'magic number With or wiihout

the magic number, can you fill in the

required spaces in each line?

v l o

t °wo p p0 t o

MAGIC NUMBERS

SEVEN-POINT STAR

In the seven-point Stai, can you distribute

the numbers 1 through 14 around the nodes

so that each of the seven lines adds up to

the same total? No rnagic number is given

Hint Find a relationship between the

highest number inserted m the six-point

Star and its magic number , and you may

be able to calculate the 'magic number for

the seven -point Star

(Solutions page fiOJ

V - - *

MAGIC NUMBERS 2 •i »*

T h e s e Magic S q u a r e s are all slightly m o r e just addition to h a v e to worry about., or complex t h a n t h e other m a g i c s h a p e s in t h e t h e re is s o m e other restriction or condition book, e v e n though they ore merely squares affecting your choice that I h a v e p u t in to

T h a t is b e c a u s e f i t h e r t h e r e is more than • perplex you

so that lines across, lines

down and the 2 main

diagonals all total zero?

Continuing this theme, can

you distribute the numbers

12 11 10 9 8 7 6 5

Now let's turn to a 3 x 3

Magic Square First, can you

distribute the numbers 1

through 9 in such a way that

by subtracting the central

number in any line oi three

from the sum of the outer

two, all total the same,

Third, can you distribute those identical numbers

Finally, here's a 5 x 5 Macric

Square with some interna/

squares shaded Can you

distribute the numbers 1

through 25 m such a way

that Unes across, lines down

and the two ma in diagonals

all add to the s a m e

total-a n d only odd numbers

appear in the shaded

squares?

Jt

19