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Amusements in Mathematics
[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.
[Illustration]
These are strung on wires like buttons. Supposing that eleven coins
with round holes are worth fifteen ching-changs, that eleven with
square holes are worth sixteen ching-changs, and that eleven with
triangular holes are worth seventeen ching-changs, how can a Chinaman
give me change for half a crown, using no coins other than the three
mentioned? A ching-chang is worth exactly twopence and four-fifteenths
of a ching-chang.
26.--THE JUNIOR CLERK'S PUZZLE.
Two youths, bearing the pleasant names of Moggs and Snoggs, were
employed as junior clerks by a merchant in Mincing Lane. They were
both engaged at the same salary--that is, commencing at the rate of
L50 a year, payable half-yearly. Moggs had a yearly rise of L10, and
Snoggs was offered the same, only he asked, for reasons that do not
concern our puzzle, that he might take his rise at L2, 10s.
half-yearly, to which his employer (not, perhaps, unnaturally!) had no
objection.
Now we come to the real point of the puzzle. Moggs put regularly into
the Post Office Savings Bank a certain proportion of his salary, while
Snoggs saved twice as great a proportion of his, and at the end of
five years they had together saved L268, 15s. How much had each saved?
The question of interest can be ignored.
27.--GIVING CHANGE.
Every one is familiar with the difficulties that frequently arise over
the giving of change, and how the assistance of a third person with a
few coins in his pocket will sometimes help us to set the matter
right. Here is an example. An Englishman went into a shop in New York
and bought goods at a cost of thirty-four cents. The only money he had
was a dollar, a three-cent piece, and a two-cent piece. The tradesman
had only a half-dollar and a quarter-dollar. But another customer
happened to be present, and when asked to help produced two dimes, a
five-cent piece, a two-cent piece, and a one-cent piece. How did the
tradesman manage to give change? For the benefit of those readers who
are not familiar with the American coinage, it is only necessary to
say that a dollar is a hundred cents and a dime ten cents. A puzzle of
this kind should rarely cause any difficulty if attacked in a proper
manner.
28.--DEFECTIVE OBSERVATION.
Our observation of little things is frequently defective, and our
memories very liable to lapse. A certain judge recently remarked in a
case that he had no recollection whatever of putting the wedding-ring
on his wife's finger. Can you correctly answer these questions without
having the coins in sight? On which side of a penny is the date given?
Some people are so unobservant that, although they are handling the
coin nearly every day of their lives, they are at a loss to answer
this simple question. If I lay a penny flat on the table, how many
other pennies can I place around it, every one also lying flat on the
table, so that they all touch the first one? The geometrician will, of
course, give the answer at once, and not need to make any experiment.
He will also know that, since all circles are similar, the same answer
will necessarily apply to any coin. The next question is a most
interesting one to ask a company, each person writing down his answer
on a slip of paper, so that no one shall be helped by the answers of
others. What is the greatest number of three-penny-pieces that may be
laid flat on the surface of a half-crown, so that no piece lies on
another or overlaps the surface of the half-crown? It is amazing what
a variety of different answers one gets to this question. Very few
people will be found to give the correct number. Of course the answer
must be given without looking at the coins.
29.--THE BROKEN COINS.
A man had three coins--a sovereign, a shilling, and a penny--and he
found that exactly the same fraction of each coin had been broken
away. Now, assuming that the original intrinsic value of these coins
was the same as their nominal value--that is, that the sovereign was
worth a pound, the shilling worth a shilling, and the penny worth a
penny--what proportion of each coin has been lost if the value of the
three remaining fragments is exactly one pound?
30.--TWO QUESTIONS IN PROBABILITIES.
There is perhaps no class of puzzle over which people so frequently
blunder as that which involves what is called the theory of
probabilities. I will give two simple examples of the sort of puzzle I
mean. They are really quite easy, and yet many persons are tripped up
by them. A friend recently produced five pennies and said to me: "In
throwing these five pennies at the same time, what are the chances
that at least four of the coins will turn up either all heads or all
tails?" His own solution was quite wrong, but the correct answer ought
not to be hard to discover. Another person got a wrong answer to the
following little puzzle which I heard him propound: "A man placed
three sovereigns and one shilling in a bag. How much should be paid
for permission to draw one coin from it?" It is, of course, understood
that you are as likely to draw any one of the four coins as another.
31.--DOMESTIC ECONOMY.
Young Mrs. Perkins, of Putney, writes to me as follows: "I should be
very glad if you could give me the answer to a little sum that has
been worrying me a good deal lately. Here it is: We have only been
married a short time, and now, at the end of two years from the time
when we set up housekeeping, my husband tells me that he finds we have
spent a third of his yearly income in rent, rates, and taxes, one-half
in domestic expenses, and one-ninth in other ways. He has a balance of
L190 remaining in the bank. I know this last, because he accidentally
left out his pass-book the other day, and I peeped into it. Don't you
think that a husband ought to give his wife his entire confidence in
his money matters? Well, I do; and--will you believe it?--he has never
told me what his income really is, and I want, very naturally, to find
out. Can you tell me what it is from the figures I have given you?"
Yes; the answer can certainly be given from the figures contained in
Mrs. Perkins's letter. And my readers, if not warned, will be
practically unanimous in declaring the income to be--something
absurdly in excess of the correct answer!
32.--THE EXCURSION TICKET PUZZLE.
When the big flaming placards were exhibited at the little provincial
railway station, announcing that the Great ---- Company would run
cheap excursion trains to London for the Christmas holidays, the
inhabitants of Mudley-cum-Turmits were in quite a flutter of
excitement. Half an hour before the train came in the little booking
office was crowded with country passengers, all bent on visiting their
friends in the great Metropolis. The booking clerk was unaccustomed to
dealing with crowds of such a dimension, and he told me afterwards,
while wiping his manly brow, that what caused him so much trouble was
the fact that these rustics paid their fares in such a lot of small
money.
He said that he had enough farthings to supply a West End draper with
change for a week, and a sufficient number of threepenny pieces for
the congregations of three parish churches. "That excursion fare,"
said he, "is nineteen shillings and ninepence, and I should like to
know in just how many different ways it is possible for such an amount
to be paid in the current coin of this realm."
Here, then, is a puzzle: In how many different ways may nineteen
shillings and ninepence be paid in our current coin? Remember that the
fourpenny-piece is not now current.
33.--PUZZLE IN REVERSALS.
Most people know that if you take any sum of money in pounds,
shillings, and pence, in which the number of pounds (less than L12)
exceeds that of the pence, reverse it (calling the pounds pence and
the pence pounds), find the difference, then reverse and add this
difference, the result is always L12, 18s. 11d. But if we omit the
condition, "less than L12," and allow nought to represent shillings or
pence--(1) What is the lowest amount to which the rule will not apply?
(2) What is the highest amount to which it will apply? Of course, when
reversing such a sum as L14, 15s. 3d. it may be written L3, 16s. 2d.,
which is the same as L3, 15s. 14d.
34.--THE GROCER AND DRAPER.
A country "grocer and draper" had two rival assistants, who prided
themselves on their rapidity in serving customers. The young man on
the grocery side could weigh up two one-pound parcels of sugar per
minute, while the drapery assistant could cut three one-yard lengths
of cloth in the same time. Their employer, one slack day, set them a
race, giving the grocer a barrel of sugar and telling him to weigh up
forty-eight one-pound parcels of sugar While the draper divided a roll
of forty-eight yards of cloth into yard pieces. The two men were
interrupted together by customers for nine minutes, but the draper was
disturbed seventeen times as long as the grocer. What was the result
of the race?
35.--JUDKINS'S CATTLE.
Hiram B. Judkins, a cattle-dealer of Texas, had five droves of
animals, consisting of oxen, pigs, and sheep, with the same number of
animals in each drove. One morning he sold all that he had to eight
dealers. Each dealer bought the same number of animals, paying
seventeen dollars for each ox, four dollars for each pig, and two
dollars for each sheep; and Hiram received in all three hundred and
one dollars. What is the greatest number of animals he could have had?
And how many would there be of each kind?