, 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 Bag 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 hanged, but the dictator offers him one last
chance. He gives him an opaque bag containing two pebbles, one black
and the other white. The prisoner is allowed to pick one pebble from
the bag, sight unseen. If he picks the white pebble, he will be hanged,
but if he picks the black pebble, he can go free.
Our hero, rightly suspecting that the dictator is making this offer
merely for show and has cheated by putting two white pebbles in the
bag, 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
remaining in the bag.” This is a wonderful example of using lateral
thinking to overcome a seemingly impossible situation. And it is quite
mathematical too because it concentrates on the complement of a set,
rather than the set itself.
Lateral thinking has been used since ancient times by all the great
mathe- maticians, including Archimedes, Euler, Newton and many
others. Archimedes is said to have destroyed the wooden Roman fleet by
focusing the sun’s rays using mirrors; Euler solved the famous Bridges
of Konigsberg problem with a simple lateral parity trick and Newton
turned an observation of a falling apple into the magnificent theory of
universal gravitation.
Lateral thinking is sideways thinking, slick thinking, smart
thinking, often leading 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, appearing for the
first time. A unique feature of this book is that each solution is followed by
“Topics for Investigation,” in which the reader is invited to look at
problems in a similar vein which follow on from the given problem.
This gives rise to hundreds of new problems, some easy, some difficult,
but all interesting and exciting. The hope is that the reader, now on
the lateral wavelength, will discover lateral solutions to these
problems.
Our underlying theme is MIAES, which stands for “Mathematics
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 Bag 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 hanged, but the dictator offers him one last
chance. He gives him an opaque bag containing two pebbles, one black
and the other white. The prisoner is allowed to pick one pebble from
the bag, sight unseen. If he picks the white pebble, he will be hanged,
but if he picks the black pebble, he can go free.
Our hero, rightly suspecting that the dictator is making this offer
merely for show and has cheated by putting two white pebbles in the
bag, 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
remaining in the bag.” This is a wonderful example of using lateral
thinking to overcome a seemingly impossible situation. And it is quite
mathematical too because it concentrates on the complement of a set,
rather than the set itself.
Lateral thinking has been used since ancient times by all the great
mathe- maticians, including Archimedes, Euler, Newton and many
others. Archimedes is said to have destroyed the wooden Roman fleet by
focusing the sun’s rays using mirrors; Euler solved the famous Bridges
of Konigsberg problem with a simple lateral parity trick and Newton
turned an observation of a falling apple into the magnificent theory of
universal gravitation.
Lateral thinking is sideways thinking, slick thinking, smart
thinking, often leading 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, appearing for the
first time. A unique feature of this book is that each solution is followed by
“Topics for Investigation,” in which the reader is invited to look at
problems in a similar vein which follow on from the given problem.
This gives rise to hundreds of new problems, some easy, some difficult,
but all interesting and exciting. The hope is that the reader, now on
the lateral wavelength, will discover lateral solutions to these
problems.
Our underlying theme is MIAES, which stands for “Mathematics
xi
