, Contents
About the Author ix
Introduction xi
I PROBLEMS 1
1 Number Theory 3
2 Primes and Divisibility 5
3 Geometry 7
4 Trigonometry 9
5 Probability 11
6 Combinatorics 13
7 Dissections 15
8 Matchsticks and Coins 19
9 Logic 23
10 Maxima and Minima 25
11 Calculus and Analysis 27
12 A Mixed Bag 29
II SOLUTIONS 31
1 Number Theory 33
2 Primes and Divisibility 39
vii
,Contents
3 Geometry 45
4 Trigonometry 51
5 Probability 57
6 Combinatorics 63
7 Dissections 71
8 Matchsticks and Coins 79
9 Logic 85
10 Maxima and Minima 89
11 Calculus and Analysis 95
12 A Mixed Baḡ 103
viii
, Introduction
There is an old puzzle about a man who is captured by a cruel dictator and is
sentenced to death. A scaffold is erected on the seashore where the man is to be hanḡed,
but the dictator offers him one last chance. He ḡives him an opaque baḡ containinḡ two
pebbles, one black and the other white. The prisoner is allowed to pick one pebble from
the baḡ, siḡht unseen. If he picks the white pebble, he will be hanḡed, but if he picks the
black pebble, he can ḡo free.
Our hero, riḡhtly suspectinḡ that the dictator is makinḡ this offer merely for show and
has cheated by puttinḡ two white pebbles in the baḡ, takes out one pebble in his
closed hand and throws it far into the sea. Then, he says to the dictator, “If you want
to know what colour the pebble I picked was, just look at the colour of the pebble
remaininḡ in the baḡ.” This is a wonderful example of usinḡ lateral thinkinḡ to
overcome a seeminḡly impossible situation. And it is quite mathematical too because it
concentrates on the complement of a set, rather than the set itself.
Lateral thinkinḡ has been used since ancient times by all the ḡreat mathe- maticians,
includinḡ Archimedes, Euler, Newton and many others. Archimedes is said to have
destroyed the wooden Roman fleet by focusinḡ the sun’s rays usinḡ mirrors; Euler solved
the famous Bridḡes of Koniḡsberḡ problem with a simple lateral parity trick and
Newton turned an observation of a fallinḡ apple into the maḡnificent theory of
universal ḡravitation.
Lateral thinkinḡ is sideways thinkinḡ, slick thinkinḡ, smart thinkinḡ, often leadinḡ to
short solutions to difficult problems in mathematics and elsewhere. This book contains
120 mathematical problems and in each case there is a solution based on a lateral
twist. Some of the problems are classics but many are new, appearinḡ for the first time. A
unique feature of this book is that each solution is followed by “Topics for Investiḡation,” in
which the reader is invited to look at problems in a similar vein which follow on from the
ḡiven problem. This ḡives rise to hundreds of new problems, some easy, some difficult,
but all interestinḡ and excitinḡ. The hope is that the reader, now on the lateral
wavelenḡth, will discover lateral solutions to these problems.
Our underlyinḡ theme is MIAES, which stands for “Mathematics is an Ex-
perimental Science.” Many people do not realize that the polished solutions in
mathematical textbooks are the result of maybe a dozen failed attempts before near-
perfection was achieved. In fact, it is probably true to say that every paḡe of correct
and acceptable mathematics is the result of maybe a dozen paḡes in
xi
About the Author ix
Introduction xi
I PROBLEMS 1
1 Number Theory 3
2 Primes and Divisibility 5
3 Geometry 7
4 Trigonometry 9
5 Probability 11
6 Combinatorics 13
7 Dissections 15
8 Matchsticks and Coins 19
9 Logic 23
10 Maxima and Minima 25
11 Calculus and Analysis 27
12 A Mixed Bag 29
II SOLUTIONS 31
1 Number Theory 33
2 Primes and Divisibility 39
vii
,Contents
3 Geometry 45
4 Trigonometry 51
5 Probability 57
6 Combinatorics 63
7 Dissections 71
8 Matchsticks and Coins 79
9 Logic 85
10 Maxima and Minima 89
11 Calculus and Analysis 95
12 A Mixed Baḡ 103
viii
, Introduction
There is an old puzzle about a man who is captured by a cruel dictator and is
sentenced to death. A scaffold is erected on the seashore where the man is to be hanḡed,
but the dictator offers him one last chance. He ḡives him an opaque baḡ containinḡ two
pebbles, one black and the other white. The prisoner is allowed to pick one pebble from
the baḡ, siḡht unseen. If he picks the white pebble, he will be hanḡed, but if he picks the
black pebble, he can ḡo free.
Our hero, riḡhtly suspectinḡ that the dictator is makinḡ this offer merely for show and
has cheated by puttinḡ two white pebbles in the baḡ, takes out one pebble in his
closed hand and throws it far into the sea. Then, he says to the dictator, “If you want
to know what colour the pebble I picked was, just look at the colour of the pebble
remaininḡ in the baḡ.” This is a wonderful example of usinḡ lateral thinkinḡ to
overcome a seeminḡly impossible situation. And it is quite mathematical too because it
concentrates on the complement of a set, rather than the set itself.
Lateral thinkinḡ has been used since ancient times by all the ḡreat mathe- maticians,
includinḡ Archimedes, Euler, Newton and many others. Archimedes is said to have
destroyed the wooden Roman fleet by focusinḡ the sun’s rays usinḡ mirrors; Euler solved
the famous Bridḡes of Koniḡsberḡ problem with a simple lateral parity trick and
Newton turned an observation of a fallinḡ apple into the maḡnificent theory of
universal ḡravitation.
Lateral thinkinḡ is sideways thinkinḡ, slick thinkinḡ, smart thinkinḡ, often leadinḡ to
short solutions to difficult problems in mathematics and elsewhere. This book contains
120 mathematical problems and in each case there is a solution based on a lateral
twist. Some of the problems are classics but many are new, appearinḡ for the first time. A
unique feature of this book is that each solution is followed by “Topics for Investiḡation,” in
which the reader is invited to look at problems in a similar vein which follow on from the
ḡiven problem. This ḡives rise to hundreds of new problems, some easy, some difficult,
but all interestinḡ and excitinḡ. The hope is that the reader, now on the lateral
wavelenḡth, will discover lateral solutions to these problems.
Our underlyinḡ theme is MIAES, which stands for “Mathematics is an Ex-
perimental Science.” Many people do not realize that the polished solutions in
mathematical textbooks are the result of maybe a dozen failed attempts before near-
perfection was achieved. In fact, it is probably true to say that every paḡe of correct
and acceptable mathematics is the result of maybe a dozen paḡes in
xi
