Henry Ernest Dudeney - Amusements in Mathematics

H >> Henry Ernest Dudeney >> Amusements in Mathematics

Pages:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38

[Transcribers note: Many of the puzzles in this book assume a
familiarity with the currency of Great Britain in the early 1900s. As
this is likely not common knowledge for those outside Britain (and
possibly many within,) I am including a chart of relative values.

The most common units used were:

the Penny, abbreviated: d. (from the Roman penny, denarius)
the Shilling, abbreviated: s.
the Pound, abbreviated: L

There was 12 Pennies to a Shilling and 20 Shillings to a Pound, so there
was 240 Pennies in a Pound.

To further complicate things, there were many coins which were various
fractional values of Pennies, Shillings or Pounds.

Farthing 1/4d.

Half-penny 1/2d.

Penny 1d.

Three-penny 3d.

Sixpence (or tanner) 6d.

Shilling (or bob) 1s.

Florin or two shilling piece 2s.

Half-crown (or half-dollar) 2s. 6d.

Double-florin 4s.

Crown (or dollar) 5s.

Half-Sovereign 10s.

Sovereign (or Pound) L1 or 20s.

This is by no means a comprehensive list, but it should be adequate to
solve the puzzles in this book.

Exponents are represented in this text by ^, e.g. '3 squared' is 3^2.

Numbers with fractional components (other than 1/4, 1/2 and 3/4) have a +
symbol separating the whole number component from the fraction. It makes
the fraction look odd, but yeilds correct solutions no matter how it is
interpreted. E.G., 4 and eleven twenty-thirds is 4+11/23, not 411/23 or
4-11/23.

]

AMUSEMENTS IN MATHEMATICS

by

HENRY ERNEST DUDENEY

In Mathematicks he was greater
Than Tycho Brahe or Erra Pater:
For he, by geometrick scale,
Could take the size of pots of ale;
Resolve, by sines and tangents, straight,
If bread or butter wanted weight;
And wisely tell what hour o' th' day
The clock does strike by algebra.

BUTLER'S _Hudibras_.

1917

PREFACE

In issuing this volume of my Mathematical Puzzles, of which some have
appeared in periodicals and others are given here for the first time, I
must acknowledge the encouragement that I have received from many
unknown correspondents, at home and abroad, who have expressed a desire
to have the problems in a collected form, with some of the solutions
given at greater length than is possible in magazines and newspapers.
Though I have included a few old puzzles that have interested the world
for generations, where I felt that there was something new to be said
about them, the problems are in the main original. It is true that some
of these have become widely known through the press, and it is possible
that the reader may be glad to know their source.

On the question of Mathematical Puzzles in general there is, perhaps,
little more to be said than I have written elsewhere. The history of the
subject entails nothing short of the actual story of the beginnings and
development of exact thinking in man. The historian must start from the
time when man first succeeded in counting his ten fingers and in
dividing an apple into two approximately equal parts. Every puzzle that
is worthy of consideration can be referred to mathematics and logic.
Every man, woman, and child who tries to "reason out" the answer to the
simplest puzzle is working, though not of necessity consciously, on
mathematical lines. Even those puzzles that we have no way of attacking
except by haphazard attempts can be brought under a method of what has
been called "glorified trial"--a system of shortening our labours by
avoiding or eliminating what our reason tells us is useless. It is, in
fact, not easy to say sometimes where the "empirical" begins and where
it ends.

When a man says, "I have never solved a puzzle in my life," it is
difficult to know exactly what he means, for every intelligent
individual is doing it every day. The unfortunate inmates of our lunatic
asylums are sent there expressly because they cannot solve
puzzles--because they have lost their powers of reason. If there were no
puzzles to solve, there would be no questions to ask; and if there were
no questions to be asked, what a world it would be! We should all be
equally omniscient, and conversation would be useless and idle.

It is possible that some few exceedingly sober-minded mathematicians,
who are impatient of any terminology in their favourite science but the
academic, and who object to the elusive x and y appearing under any
other names, will have wished that various problems had been presented
in a less popular dress and introduced with a less flippant phraseology.
I can only refer them to the first word of my title and remind them that
we are primarily out to be amused--not, it is true, without some hope of
picking up morsels of knowledge by the way. If the manner is light, I
can only say, in the words of Touchstone, that it is "an ill-favoured
thing, sir, but my own; a poor humour of mine, sir."

As for the question of difficulty, some of the puzzles, especially in
the Arithmetical and Algebraical category, are quite easy. Yet some of
those examples that look the simplest should not be passed over without
a little consideration, for now and again it will be found that there is
some more or less subtle pitfall or trap into which the reader may be
apt to fall. It is good exercise to cultivate the habit of being very
wary over the exact wording of a puzzle. It teaches exactitude and
caution. But some of the problems are very hard nuts indeed, and not
unworthy of the attention of the advanced mathematician. Readers will
doubtless select according to their individual tastes.

In many cases only the mere answers are given. This leaves the beginner
something to do on his own behalf in working out the method of solution,
and saves space that would be wasted from the point of view of the
advanced student. On the other hand, in particular cases where it seemed
likely to interest, I have given rather extensive solutions and treated
problems in a general manner. It will often be found that the notes on
one problem will serve to elucidate a good many others in the book; so
that the reader's difficulties will sometimes be found cleared up as he
advances. Where it is possible to say a thing in a manner that may be
"understanded of the people" generally, I prefer to use this simple
phraseology, and so engage the attention and interest of a larger
public. The mathematician will in such cases have no difficulty in
expressing the matter under consideration in terms of his familiar
symbols.

I have taken the greatest care in reading the proofs, and trust that any
errors that may have crept in are very few. If any such should occur, I
can only plead, in the words of Horace, that "good Homer sometimes
nods," or, as the bishop put it, "Not even the youngest curate in my
diocese is infallible."

I have to express my thanks in particular to the proprietors of _The
Strand Magazine_, _Cassell's Magazine_, _The Queen_, _Tit-Bits_, and
_The Weekly Dispatch_ for their courtesy in allowing me to reprint some
of the puzzles that have appeared in their pages.

THE AUTHORS' CLUB _March_ 25, 1917

CONTENTS

PREFACE v
ARITHMETICAL AND ALGEBRAICAL PROBLEMS 1
Money Puzzles 1
Age and Kinship Puzzles 6
Clock Puzzles 9
Locomotion and Speed Puzzles 11
Digital Puzzles 13
Various Arithmetical and Algebraical Problems 17
GEOMETRICAL PROBLEMS 27
Dissection Puzzles 27
Greek Cross Puzzles 28
Various Dissection Puzzles 35
Patchwork Puzzles 46
Various Geometrical Puzzles 49
POINTS AND LINES PROBLEMS 56
MOVING COUNTER PROBLEMS 58
UNICURSAL AND ROUTE PROBLEMS 68
COMBINATION AND GROUP PROBLEMS 76
CHESSBOARD PROBLEMS 85
The Chessboard 85
Statical Chess Puzzles 88
The Guarded Chessboard 95
Dynamical Chess Puzzles 96
Various Chess Puzzles 105
MEASURING, WEIGHING, AND PACKING PUZZLES 109
CROSSING RIVER PROBLEMS 112
PROBLEMS CONCERNING GAMES 114
PUZZLE GAMES 117
MAGIC SQUARE PROBLEMS 119
Subtracting, Multiplying, and Dividing Magics 124
Magic Squares of Primes 125
MAZES AND HOW TO THREAD THEM 127
THE PARADOX PARTY 137
UNCLASSIFIED PROBLEMS 142
SOLUTIONS 148
INDEX 253

AMUSEMENTS IN MATHEMATICS.

ARITHMETICAL AND ALGEBRAICAL PROBLEMS.

"And what was he?
Forsooth, a great arithmetician."
_Othello_, I. i.

The puzzles in this department are roughly thrown together in classes
for the convenience of the reader. Some are very easy, others quite
difficult. But they are not arranged in any order of difficulty--and
this is intentional, for it is well that the solver should not be warned
that a puzzle is just what it seems to be. It may, therefore, prove to
be quite as simple as it looks, or it may contain some pitfall into
which, through want of care or over-confidence, we may stumble.

Also, the arithmetical and algebraical puzzles are not separated in the
manner adopted by some authors, who arbitrarily require certain problems
to be solved by one method or the other. The reader is left to make his
own choice and determine which puzzles are capable of being solved by
him on purely arithmetical lines.

MONEY PUZZLES.

"Put not your trust in money, but put your money in trust."

OLIVER WENDELL HOLMES.

1.--A POST-OFFICE PERPLEXITY.

In every business of life we are occasionally perplexed by some chance
question that for the moment staggers us. I quite pitied a young lady in
a branch post-office when a gentleman entered and deposited a crown on
the counter with this request: "Please give me some twopenny stamps, six
times as many penny stamps, and make up the rest of the money in
twopence-halfpenny stamps." For a moment she seemed bewildered, then her
brain cleared, and with a smile she handed over stamps in exact
fulfilment of the order. How long would it have taken you to think it
out?

2.--YOUTHFUL PRECOCITY.

The precocity of some youths is surprising. One is disposed to say on
occasion, "That boy of yours is a genius, and he is certain to do great
things when he grows up;" but past experience has taught us that he
invariably becomes quite an ordinary citizen. It is so often the case,
on the contrary, that the dull boy becomes a great man. You never can
tell. Nature loves to present to us these queer paradoxes. It is well
known that those wonderful "lightning calculators," who now and again
surprise the world by their feats, lose all their mysterious powers
directly they are taught the elementary rules of arithmetic.

A boy who was demolishing a choice banana was approached by a young
friend, who, regarding him with envious eyes, asked, "How much did you
pay for that banana, Fred?" The prompt answer was quite remarkable in
its way: "The man what I bought it of receives just half as many
sixpences for sixteen dozen dozen bananas as he gives bananas for a
fiver."

Now, how long will it take the reader to say correctly just how much
Fred paid for his rare and refreshing fruit?

3.--AT A CATTLE MARKET.

Three countrymen met at a cattle market. "Look here," said Hodge to
Jakes, "I'll give you six of my pigs for one of your horses, and then
you'll have twice as many animals here as I've got." "If that's your
way of doing business," said Durrant to Hodge, "I'll give you fourteen
of my sheep for a horse, and then you'll have three times as many
animals as I." "Well, I'll go better than that," said Jakes to Durrant;
"I'll give you four cows for a horse, and then you'll have six times as
many animals as I've got here."

No doubt this was a very primitive way of bartering animals, but it is
an interesting little puzzle to discover just how many animals Jakes,
Hodge, and Durrant must have taken to the cattle market.

4.--THE BEANFEAST PUZZLE.

A number of men went out together on a bean-feast. There were four
parties invited--namely, 25 cobblers, 20 tailors, 18 hatters, and 12
glovers. They spent altogether L6, 13s. It was found that five cobblers
spent as much as four tailors; that twelve tailors spent as much as nine
hatters; and that six hatters spent as much as eight glovers. The puzzle
is to find out how much each of the four parties spent.

5.--A QUEER COINCIDENCE.

Seven men, whose names were Adams, Baker, Carter, Dobson, Edwards,
Francis, and Gudgeon, were recently engaged in play. The name of the
particular game is of no consequence. They had agreed that whenever a
player won a game he should double the money of each of the other
players--that is, he was to give the players just as much money as they
had already in their pockets. They played seven games, and, strange to
say, each won a game in turn, in the order in which their names are
given. But a more curious coincidence is this--that when they had
finished play each of the seven men had exactly the same amount--two
shillings and eightpence--in his pocket. The puzzle is to find out how
much money each man had with him before he sat down to play.

6.--A CHARITABLE BEQUEST.

A man left instructions to his executors to distribute once a year
exactly fifty-five shillings among the poor of his parish; but they were
only to continue the gift so long as they could make it in different
ways, always giving eighteenpence each to a number of women and half a
crown each to men. During how many years could the charity be
administered? Of course, by "different ways" is meant a different number
of men and women every time.

7.--THE WIDOW'S LEGACY.

A gentleman who recently died left the sum of L8,000 to be divided among
his widow, five sons, and four daughters. He directed that every son
should receive three times as much as a daughter, and that every
daughter should have twice as much as their mother. What was the widow's
share?

8.--INDISCRIMINATE CHARITY.

A charitable gentleman, on his way home one night, was appealed to by
three needy persons in succession for assistance. To the first person he
gave one penny more than half the money he had in his pocket; to the
second person he gave twopence more than half the money he then had in
his pocket; and to the third person he handed over threepence more than
half of what he had left. On entering his house he had only one penny in
his pocket. Now, can you say exactly how much money that gentleman had
on him when he started for home?

9.--THE TWO AEROPLANES.

A man recently bought two aeroplanes, but afterwards found that they
would not answer the purpose for which he wanted them. So he sold them
for L600 each, making a loss of 20 per cent. on one machine and a profit
of 20 per cent. on the other. Did he make a profit on the whole
transaction, or a loss? And how much?

10.--BUYING PRESENTS.

"Whom do you think I met in town last week, Brother William?" said Uncle
Benjamin. "That old skinflint Jorkins. His family had been taking him
around buying Christmas presents. He said to me, 'Why cannot the
government abolish Christmas, and make the giving of presents punishable
by law? I came out this morning with a certain amount of money in my
pocket, and I find I have spent just half of it. In fact, if you will
believe me, I take home just as many shillings as I had pounds, and half
as many pounds as I had shillings. It is monstrous!'" Can you say
exactly how much money Jorkins had spent on those presents?

11.--THE CYCLISTS' FEAST.

'Twas last Bank Holiday, so I've been told,
Some cyclists rode abroad in glorious weather.
Resting at noon within a tavern old,
They all agreed to have a feast together.
"Put it all in one bill, mine host," they said,
"For every man an equal share will pay."
The bill was promptly on the table laid,
And four pounds was the reckoning that day.
But, sad to state, when they prepared to square,
'Twas found that two had sneaked outside and fled.
So, for two shillings more than his due share
Each honest man who had remained was bled.
They settled later with those rogues, no doubt.
How many were they when they first set out?

12.--A QUEER THING IN MONEY.

It will be found that L66, 6s. 6d. equals 15,918 pence. Now, the four
6's added together make 24, and the figures in 15,918 also add to 24. It
is a curious fact that there is only one other sum of money, in pounds,
shillings, and pence (all similarly repetitions of one figure), of which
the digits shall add up the same as the digits of the amount in pence.
What is the other sum of money?

13.--A NEW MONEY PUZZLE.

The largest sum of money that can be written in pounds, shillings,
pence, and farthings, using each of the nine digits once and only once,
is L98,765, 4s. 31/2d. Now, try to discover the smallest sum of money
that can be written down under precisely the same conditions. There must
be some value given for each denomination--pounds, shillings, pence,
and farthings--and the nought may not be used. It requires just a little
judgment and thought.

14.--SQUARE MONEY.

"This is queer," said McCrank to his friend. "Twopence added to twopence
is fourpence, and twopence multiplied by twopence is also fourpence." Of
course, he was wrong in thinking you can multiply money by money. The
multiplier must be regarded as an abstract number. It is true that two
feet multiplied by two feet will make four square feet. Similarly, two
pence multiplied by two pence will produce four square pence! And it
will perplex the reader to say what a "square penny" is. But we will
assume for the purposes of our puzzle that twopence multiplied by
twopence is fourpence. Now, what two amounts of money will produce the
next smallest possible result, the same in both cases, when added or
multiplied in this manner? The two amounts need not be alike, but they
must be those that can be paid in current coins of the realm.

15.--POCKET MONEY.

What is the largest sum of money--all in current silver coins and no
four-shilling piece--that I could have in my pocket without being able
to give change for a half-sovereign?

16.--THE MILLIONAIRE'S PERPLEXITY.

Mr. Morgan G. Bloomgarten, the millionaire, known in the States as the
Clam King, had, for his sins, more money than he knew what to do with.
It bored him. So he determined to persecute some of his poor but happy
friends with it. They had never done him any harm, but he resolved to
inoculate them with the "source of all evil." He therefore proposed to
distribute a million dollars among them and watch them go rapidly to the
bad. But he was a man of strange fancies and superstitions, and it was
an inviolable rule with him never to make a gift that was not either one
dollar or some power of seven--such as 7, 49, 343, 2,401, which numbers
of dollars are produced by simply multiplying sevens together. Another
rule of his was that he would never give more than six persons exactly
the same sum. Now, how was he to distribute the 1,000,000 dollars? You
may distribute the money among as many people as you like, under the
conditions given.

17.--THE PUZZLING MONEY-BOXES.

Four brothers--named John, William, Charles, and Thomas--had each a
money-box. The boxes were all given to them on the same day, and they at
once put what money they had into them; only, as the boxes were not very
large, they first changed the money into as few coins as possible. After
they had done this, they told one another how much money they had saved,
and it was found that if John had had 2s. more in his box than at
present, if William had had 2s. less, if Charles had had twice as much,
and if Thomas had had half as much, they would all have had exactly the
same amount.

Now, when I add that all four boxes together contained 45s., and that
there were only six coins in all in them, it becomes an entertaining
puzzle to discover just what coins were in each box.

18.--THE MARKET WOMEN.

A number of market women sold their various products at a certain price
per pound (different in every case), and each received the same
amount--2s. 21/2d. What is the greatest number of women there could
have been? The price per pound in every case must be such as could be
paid in current money.

19.--THE NEW YEAR'S EVE SUPPERS.

The proprietor of a small London cafe has given me some interesting
figures. He says that the ladies who come alone to his place for
refreshment spend each on an average eighteenpence, that the
unaccompanied men spend half a crown each, and that when a gentleman
brings in a lady he spends half a guinea. On New Year's Eve he supplied
suppers to twenty-five persons, and took five pounds in all. Now,
assuming his averages to have held good in every case, how was his
company made up on that occasion? Of course, only single gentlemen,
single ladies, and pairs (a lady and gentleman) can be supposed to have
been present, as we are not considering larger parties.

20.--BEEF AND SAUSAGES.

"A neighbour of mine," said Aunt Jane, "bought a certain quantity of
beef at two shillings a pound, and the same quantity of sausages at
eighteenpence a pound. I pointed out to her that if she had divided the
same money equally between beef and sausages she would have gained two
pounds in the total weight. Can you tell me exactly how much she spent?"

"Of course, it is no business of mine," said Mrs. Sunniborne; "but a
lady who could pay such prices must be somewhat inexperienced in
domestic economy."

"I quite agree, my dear," Aunt Jane replied, "but you see that is not
the precise point under discussion, any more than the name and morals of
the tradesman."

21.--A DEAL IN APPLES.

I paid a man a shilling for some apples, but they were so small that I
made him throw in two extra apples. I find that made them cost just a
penny a dozen less than the first price he asked. How many apples did I
get for my shilling?

22.--A DEAL IN EGGS.

A man went recently into a dairyman's shop to buy eggs. He wanted them
of various qualities. The salesman had new-laid eggs at the high price
of fivepence each, fresh eggs at one penny each, eggs at a halfpenny
each, and eggs for electioneering purposes at a greatly reduced figure,
but as there was no election on at the time the buyer had no use for the
last. However, he bought some of each of the three other kinds and
obtained exactly one hundred eggs for eight and fourpence. Now, as he
brought away exactly the same number of eggs of two of the three
qualities, it is an interesting puzzle to determine just how many he
bought at each price.

23.--THE CHRISTMAS-BOXES.

Some years ago a man told me he had spent one hundred English silver
coins in Christmas-boxes, giving every person the same amount, and it
cost him exactly L1, 10s. 1d. Can you tell just how many persons
received the present, and how he could have managed the distribution?
That odd penny looks queer, but it is all right.

24.--A SHOPPING PERPLEXITY.

Two ladies went into a shop where, through some curious eccentricity, no
change was given, and made purchases amounting together to less than
five shillings. "Do you know," said one lady, "I find I shall require no
fewer than six current coins of the realm to pay for what I have
bought." The other lady considered a moment, and then exclaimed: "By a
peculiar coincidence, I am exactly in the same dilemma." "Then we will
pay the two bills together." But, to their astonishment, they still
required six coins. What is the smallest possible amount of their
purchases--both different?

25.--CHINESE MONEY.

The Chinese are a curious people, and have strange inverted ways of
doing things. It is said that they use a saw with an upward pressure
instead of a downward one, that they plane a deal board by pulling the
tool toward them instead of pushing it, and that in building a house
they first construct the roof and, having raised that into position,
proceed to work downwards. In money the currency of the country consists
of taels of fluctuating value. The tael became thinner and thinner until
2,000 of them piled together made less than three inches in height. The
common cash consists of brass coins of varying thicknesses, with a
round, square, or triangular hole in the centre, as in our illustration.

Pages:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38