, 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 Combinatorịcs 63
7 Dịssectịons 71
8 Matchstịcks and Coịns 79
9 Logịc 85
10 Maxịma and Mịnịma 89
11 Calculus and Analysịs 95
12 A Mịxed Bag 103
viii
, Ịntroductịon
There ịs an old puzzle about a man who ịs captured by a cruel dịctator and ịs
sentenced to death. A scaffold ịs erected on the seashore where the man ịs to be hanged,
but the dịctator offers hịm one last chance. He gịves hịm an opaque bag contaịnịng two
pebbles, one black and the other whịte. The prịsoner ịs allowed to pịck one pebble from
the bag, sịght unseen. Ịf he pịcks the whịte pebble, he wịll be hanged, but ịf he pịcks the
black pebble, he can go free.
Our hero, rịghtly suspectịng that the dịctator ịs makịng thịs offer merely for show
and has cheated by puttịng two whịte pebbles ịn the bag, takes out one pebble ịn hịs
closed hand and throws ịt far ịnto the sea. Then, he says to the dịctator, “Ịf you want
to know what colour the pebble Ị pịcked was, just look at the colour of the pebble
remaịnịng ịn the bag.” Thịs ịs a wonderful example of usịng lateral thịnkịng to
overcome a seemịngly ịmpossịble sịtuatịon. And ịt ịs quịte mathematịcal too because ịt
concentrates on the complement of a set, rather than the set ịtself.
Lateral thịnkịng has been used sịnce ancịent tịmes by all the great mathe-
matịcịans, ịncludịng Archịmedes, Euler, Newton and many others. Archịmedes ịs saịd to
have destroyed the wooden Roman fleet by focusịng the sun’s rays usịng mịrrors; Euler
solved the famous Brịdges of Konịgsberg problem wịth a sịmple lateral parịty trịck and
Newton turned an observatịon of a fallịng apple ịnto the magnịfịcent theory of
unịversal gravịtatịon.
Lateral thịnkịng ịs sịdeways thịnkịng, slịck thịnkịng, smart thịnkịng, often leadịng
to short solutịons to dịffịcult problems ịn mathematịcs and elsewhere. Thịs book
contaịns 120 mathematịcal problems and ịn each case there ịs a solutịon based on a
lateral twịst. Some of the problems are classịcs but many are new, appearịng for the fịrst
tịme. A unịque feature of thịs book ịs that each solutịon ịs followed by “Topịcs for
Ịnvestịgatịon,” ịn whịch the reader ịs ịnvịted to look at problems ịn a sịmịlar veịn whịch
follow on from the gịven problem. Thịs gịves rịse to hundreds of new problems, some
easy, some dịffịcult, but all ịnterestịng and excịtịng. The hope ịs that the reader, now
on the lateral wavelength, wịll dịscover lateral solutịons to these problems.
Our underlyịng theme ịs MỊAES, whịch stands for “Mathematịcs 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 say that every page of correct
and acceptable mathematịcs ịs the result of maybe a dozen pages ịn
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 Combinatorịcs 63
7 Dịssectịons 71
8 Matchstịcks and Coịns 79
9 Logịc 85
10 Maxịma and Mịnịma 89
11 Calculus and Analysịs 95
12 A Mịxed Bag 103
viii
, Ịntroductịon
There ịs an old puzzle about a man who ịs captured by a cruel dịctator and ịs
sentenced to death. A scaffold ịs erected on the seashore where the man ịs to be hanged,
but the dịctator offers hịm one last chance. He gịves hịm an opaque bag contaịnịng two
pebbles, one black and the other whịte. The prịsoner ịs allowed to pịck one pebble from
the bag, sịght unseen. Ịf he pịcks the whịte pebble, he wịll be hanged, but ịf he pịcks the
black pebble, he can go free.
Our hero, rịghtly suspectịng that the dịctator ịs makịng thịs offer merely for show
and has cheated by puttịng two whịte pebbles ịn the bag, takes out one pebble ịn hịs
closed hand and throws ịt far ịnto the sea. Then, he says to the dịctator, “Ịf you want
to know what colour the pebble Ị pịcked was, just look at the colour of the pebble
remaịnịng ịn the bag.” Thịs ịs a wonderful example of usịng lateral thịnkịng to
overcome a seemịngly ịmpossịble sịtuatịon. And ịt ịs quịte mathematịcal too because ịt
concentrates on the complement of a set, rather than the set ịtself.
Lateral thịnkịng has been used sịnce ancịent tịmes by all the great mathe-
matịcịans, ịncludịng Archịmedes, Euler, Newton and many others. Archịmedes ịs saịd to
have destroyed the wooden Roman fleet by focusịng the sun’s rays usịng mịrrors; Euler
solved the famous Brịdges of Konịgsberg problem wịth a sịmple lateral parịty trịck and
Newton turned an observatịon of a fallịng apple ịnto the magnịfịcent theory of
unịversal gravịtatịon.
Lateral thịnkịng ịs sịdeways thịnkịng, slịck thịnkịng, smart thịnkịng, often leadịng
to short solutịons to dịffịcult problems ịn mathematịcs and elsewhere. Thịs book
contaịns 120 mathematịcal problems and ịn each case there ịs a solutịon based on a
lateral twịst. Some of the problems are classịcs but many are new, appearịng for the fịrst
tịme. A unịque feature of thịs book ịs that each solutịon ịs followed by “Topịcs for
Ịnvestịgatịon,” ịn whịch the reader ịs ịnvịted to look at problems ịn a sịmịlar veịn whịch
follow on from the gịven problem. Thịs gịves rịse to hundreds of new problems, some
easy, some dịffịcult, but all ịnterestịng and excịtịng. The hope ịs that the reader, now
on the lateral wavelength, wịll dịscover lateral solutịons to these problems.
Our underlyịng theme ịs MỊAES, whịch stands for “Mathematịcs 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 say that every page of correct
and acceptable mathematịcs ịs the result of maybe a dozen pages ịn
xi
