STUDY NOTES & REFERENCE GUIDE
courseroomsity - Stuvia | Page 1 of 121
, STUDY NOTES & REFERENCE GUIDE
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
v
i
i
courseroomsity - Stuvia | Page 2 of 121
, STUDY NOTES & REFERENCE GUIDE
Conten
ts
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
vi
ii
courseroomsity - Stuvia | Page 3 of 121
, STUDY NOTES & REFERENCE GUIDE
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 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 page
of correct and acceptable mathematics is the result of maybe a dozen pages in
x
i
courseroomsity - Stuvia | Page 4 of 121
courseroomsity - Stuvia | Page 1 of 121
, STUDY NOTES & REFERENCE GUIDE
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
v
i
i
courseroomsity - Stuvia | Page 2 of 121
, STUDY NOTES & REFERENCE GUIDE
Conten
ts
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
vi
ii
courseroomsity - Stuvia | Page 3 of 121
, STUDY NOTES & REFERENCE GUIDE
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 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 page
of correct and acceptable mathematics is the result of maybe a dozen pages in
x
i
courseroomsity - Stuvia | Page 4 of 121
