More math puzzles and game

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MATH
PUZZLES
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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

v

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

17

3. Vanishing-Line and Vanishing-Square Puzzles

33

4. Match Puzzles

41

5. Coin and Shunting Problems

49

6. Reasoning and Logical Problems

56

7. 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


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:

�
500 ds
MAIN STREET

..j

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.

I.. triangles

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.

4

3


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 .

--

¢-

--

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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6


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 .

•

•

•

•
•

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

c

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.

d

e

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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.......

1

2

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.

12


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?

14


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!

16


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