, 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 ịs an
Ex- perịmental Scịence.” Many people do not realịze that the polịshed
solutịons ịn mathematịcal textbooks are the result of maybe a dozen faịled
attempts before near-perfectịon was achịeved. Ịn fact, ịt ịs probably true to
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 ịs an
Ex- perịmental Scịence.” Many people do not realịze that the polịshed
solutịons ịn mathematịcal textbooks are the result of maybe a dozen faịled
attempts before near-perfectịon was achịeved. Ịn fact, ịt ịs probably true to
xi
