The Canterbury Puzzles_ and Other Curious Problems Henry Ernest Dudeney 2500 downloads.pdf

The Project Gutenberg eBook of The Canterbury Puzzles, and
Other Curious Problems
This eBook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms of
the Project Gutenberg License included with this eBook or online at
www.gutenberg.org. If you are not located in the United States, you will
have to check the laws of the country where you are located before using
this eBook.

Title: The Canterbury Puzzles, and Other Curious Problems

Author: Henry Ernest Dudeney

Release date: December 27, 2008 [eBook #27635]

Language: English

Other information and formats: www.gutenberg.org/ebooks/27635

Credits: E-text prepared by Malcolm Farmer, Christine D., and the Project
Gutenberg Online Distributed Proofreading Team

*** START OF THE PROJECT GUTENBERG EBOOK THE
CANTERBURY PUZZLES, AND OTHER CURIOUS PROBLEMS ***

E-text prepared by Malcolm Farmer, Christine D.,
and the Project Gutenberg Online Distributed Proofreading Team
(http://www.pgdp.net)

PRESS OPINIONS ON "THE CANTERBURY
PUZZLES."
"It is a book of remarkable ingenuity and interest."—Educational Times.
"The most ingenious brain in England ... a fascinating new book."—
Evening News.
"A capital book of posers."—Daily News.
"The Puzzles ... reach the limit of ingenuity and intricacy; and it is well for
the sanity of his readers that the author gives a list of solutions at the end of
the book."—Observer.
"A book that will provide much entertainment for Christmas gatherings ...
ingenious puzzles and problems invented by 'Sphinx,' the Puzzle King."—
The Captain.
"Mr. Dudeney, whose reputation is world-wide as the puzzle and problem
maker of the age ... sure to find a wide circulation ... as attractive in
appearance as its contents are fascinating."—English Mechanic and World
of Science.
"An exceedingly ingenious constructor and solver of fascinating puzzles,
mathematical and otherwise."—School Guardian.
"A book which ought to be highly popular ... it is all mighty ingenious, and
very intelligently put before the reader."—Sheffield Telegraph.
"It is matter for delight that Mr. Henry E. Dudeney has collected into a
volume those mysterious puzzles of his which have appeared in many
journals ... contains quite a number of ingenious new mental problems ... a
valuable introduction."—The Lady.
"For the long winter evenings Mr. Dudeney's book of puzzledom is to be
recommended. Mr. Dudeney has made a study of every kind of puzzle there

is ... he supplies you with every kind of brain-twister."—The Daily
Chronicle.
"Took up more of the reviewer's time than he could well afford to give it; he
wanted to solve some of the curious problems that it contains, and for
ingenious persons who want employment on a wet day, he promises from it
abundant scope."—Yorkshire Post.
"A well-known master puzzler ... provides an abundance of seasonable
occupation for the ingenious, with an introduction on the general question
of puzzles, which is one of the most interesting parts of the book. He is a
skilful inventor."—Nottingham Guardian.
"Will enjoy the entertainment provided ... ingenious and witty."—The
Guardian.
"Extremely ingenious book, which abounds in problems that will keep the
reader busy for hours—until in despair he turns to the answers at the
end."—Manchester Guardian.
"The setting of these perplexities is novel ... a dramatic background being
thus provided which prevents too great aridity.... The book should be much
in request."—The Morning Leader.

THE CANTERBURY PUZZLES

By the same Author
"AMUSEMENTS IN MATHEMATICS"

3s. 6d.

First Edition, 1907

THE
CANTERBURY PUZZLES

AND OTHER CURIOUS PROBLEMS

BY

HENRY ERNEST DUDENEY
AUTHOR OF
"AMUSEMENTS IN MATHEMATICS," ETC.

Second Edition
(With Some Fuller Solutions and Additional Notes)

THOMAS NELSON AND SONS, LTD.
LONDON, EDINBURGH, AND NEW YORK
1919

CONTENTS
Preface 9
Introduction 11
The Canterbury Puzzles 23
Puzzling Times at Solvamhall Castle 58
The Merry Monks of Riddlewell 68
The Strange Escape of the King's Jester 78
The Squire's Christmas Puzzle Party 86
Adventures of the Puzzle Club 94
The Professor's Puzzles 110
Miscellaneous Puzzles 118
Solutions 163
Index 251

PREFACE
When preparing this new edition for the press, my first inclination was to
withdraw a few puzzles that appeared to be of inferior interest, and to
substitute others for them. But, on second thoughts, I decided to let the
book stand in its original form and add extended solutions and some short
notes to certain problems that have in the past involved me in
correspondence with interested readers who desired additional information.
I have also provided—what was clearly needed for reference—an index.
The very nature and form of the book prevented any separation of the
puzzles into classes, but a certain amount of classification will be found in
the index. Thus, for example, if the reader has a predilection for problems
with Moving Counters, or for Magic Squares, or for Combination and
Group Puzzles, he will find that in the index these are brought together for
his convenience.
Though the problems are quite different, with the exception of just one or
two little variations or extensions, from those in my book Amusements in
Mathematics, each work being complete in itself, I have thought it would
help the reader who happens to have both books before him if I made
occasional references that would direct him to solutions and analyses in the
later book calculated to elucidate matter in these pages. This course has also
obviated the necessity of my repeating myself. For the sake of brevity,
Amusements in Mathematics is throughout referred to as A. in M.
HENRY E. DUDENEY.
The Authors' Club,
July 2, 1919.

INTRODUCTION
Readers of The Mill on the Floss will remember that whenever Mr. Tulliver
found himself confronted by any little difficulty he was accustomed to
make the trite remark, "It's a puzzling world." There can be no denying the
fact that we are surrounded on every hand by posers, some of which the
intellect of man has mastered, and many of which may be said to be
impossible of solution. Solomon himself, who may be supposed to have
been as sharp as most men at solving a puzzle, had to admit "there be three
things which are too wonderful for me; yea, four which I know not: the way
of an eagle in the air; the way of a serpent upon a rock; the way of a ship in
the midst of the sea; and the way of a man with a maid."
Probing into the secrets of Nature is a passion with all men; only we select
different lines of research. Men have spent long lives in such attempts as to
turn the baser metals into gold, to discover perpetual motion, to find a cure
for certain malignant diseases, and to navigate the air.
From morning to night we are being perpetually brought face to face with
puzzles. But there are puzzles and puzzles. Those that are usually devised
for recreation and pastime may be roughly divided into two classes: Puzzles
that are built up on some interesting or informing little principle; and
puzzles that conceal no principle whatever—such as a picture cut at random
into little bits to be put together again, or the juvenile imbecility known as
the "rebus," or "picture puzzle." The former species may be said to be
adapted to the amusement of the sane man or woman; the latter can be
confidently recommended to the feeble-minded.
The curious propensity for propounding puzzles is not peculiar to any race
or to any period of history. It is simply innate in every intelligent man,
woman, and child that has ever lived, though it is always showing itself in
different forms; whether the individual be a Sphinx of Egypt, a Samson of
Hebrew lore, an Indian fakir, a Chinese philosopher, a mahatma of Tibet, or
a European mathematician makes little difference.

Theologian, scientist, and artisan are perpetually engaged in attempting to
solve puzzles, while every game, sport, and pastime is built up of problems
of greater or less difficulty. The spontaneous question asked by the child of
his parent, by one cyclist of another while taking a brief rest on a stile, by a
cricketer during the luncheon hour, or by a yachtsman lazily scanning the
horizon, is frequently a problem of considerable difficulty. In short, we are
all propounding puzzles to one another every day of our lives—without
always knowing it.
A good puzzle should demand the exercise of our best wit and ingenuity,
and although a knowledge of mathematics and a certain familiarity with the
methods of logic are often of great service in the solution of these things,
yet it sometimes happens that a kind of natural cunning and sagacity is of
considerable value. For many of the best problems cannot be solved by any
familiar scholastic methods, but must be attacked on entirely original lines.
This is why, after a long and wide experience, one finds that particular
puzzles will sometimes be solved more readily by persons possessing only
naturally alert faculties than by the better educated. The best players of such
puzzle games as chess and draughts are not mathematicians, though it is just
possible that often they may have undeveloped mathematical minds.
It is extraordinary what fascination a good puzzle has for a great many
people. We know the thing to be of trivial importance, yet we are impelled
to master it; and when we have succeeded there is a pleasure and a sense of
satisfaction that are a quite sufficient reward for our trouble, even when
there is no prize to be won. What is this mysterious charm that many find
irresistible? Why do we like to be puzzled? The curious thing is that
directly the enigma is solved the interest generally vanishes. We have done
it, and that is enough. But why did we ever attempt to do it?
The answer is simply that it gave us pleasure to seek the solution—that the
pleasure was all in the seeking and finding for their own sakes. A good
puzzle, like virtue, is its own reward. Man loves to be confronted by a
mystery, and he is not entirely happy until he has solved it. We never like to
feel our mental inferiority to those around us. The spirit of rivalry is innate
in man; it stimulates the smallest child, in play or education, to keep level
with his fellows, and in later life it turns men into great discoverers,

inventors, orators, heroes, artists, and (if they have more material aims)
perhaps millionaires.
In starting on a tour through the wide realm of Puzzledom we do well to
remember that we shall meet with points of interest of a very varied
character. I shall take advantage of this variety. People often make the
mistake of confining themselves to one little corner of the realm, and
thereby miss opportunities of new pleasures that lie within their reach
around them. One person will keep to acrostics and other word puzzles,
another to mathematical brain-rackers, another to chess problems (which
are merely puzzles on the chess-board, and have little practical relation to
the game of chess), and so on. This is a mistake, because it restricts one's
pleasures, and neglects that variety which is so good for the brain.
And there is really a practical utility in puzzle-solving. Regular exercise is
supposed to be as necessary for the brain as for the body, and in both cases
it is not so much what we do as the doing of it from which we derive
benefit. The daily walk recommended by the doctor for the good of the
body, or the daily exercise for the brain, may in itself appear to be so much
waste of time; but it is the truest economy in the end. Albert Smith, in one
of his amusing novels, describes a woman who was convinced that she
suffered from "cobwigs on the brain." This may be a very rare complaint,
but in a more metaphorical sense many of us are very apt to suffer from
mental cobwebs, and there is nothing equal to the solving of puzzles and
problems for sweeping them away. They keep the brain alert, stimulate the
imagination, and develop the reasoning faculties. And not only are they
useful in this indirect way, but they often directly help us by teaching us
some little tricks and "wrinkles" that can be applied in the affairs of life at
the most unexpected times and in the most unexpected ways.
There is an interesting passage in praise of puzzles in the quaint letters of
Fitzosborne. Here is an extract: "The ingenious study of making and solving
puzzles is a science undoubtedly of most necessary acquirement, and
deserves to make a part in the meditation of both sexes. It is an art, indeed,
that I would recommend to the encouragement of both the Universities, as it
affords the easiest and shortest method of conveying some of the most
useful principles of logic. It was the maxim of a very wise prince that 'he
who knows not how to dissemble knows not how to reign'; and I desire you

to receive it as mine, that 'he who knows not how to riddle knows not how
to live.'"
How are good puzzles invented? I am not referring to acrostics, anagrams,
charades, and that sort of thing, but to puzzles that contain an original idea.
Well, you cannot invent a good puzzle to order, any more than you can
invent anything else in that manner. Notions for puzzles come at strange
times and in strange ways. They are suggested by something we see or hear,
and are led up to by other puzzles that come under our notice. It is useless
to say, "I will sit down and invent an original puzzle," because there is no
way of creating an idea; you can only make use of it when it comes. You
may think this is wrong, because an expert in these things will make scores
of puzzles while another person, equally clever, cannot invent one "to save
his life," as we say. The explanation is very simple. The expert knows an
idea when he sees one, and is able by long experience to judge of its value.
Fertility, like facility, comes by practice.
Sometimes a new and most interesting idea is suggested by the blunder of
somebody over another puzzle. A boy was given a puzzle to solve by a
friend, but he misunderstood what he had to do, and set about attempting
what most likely everybody would have told him was impossible. But he
was a boy with a will, and he stuck at it for six months, off and on, until he
actually succeeded. When his friend saw the solution, he said, "This is not
the puzzle I intended—you misunderstood me—but you have found out
something much greater!" And the puzzle which that boy accidentally
discovered is now in all the old puzzle books.
Puzzles can be made out of almost anything, in the hands of the ingenious
person with an idea. Coins, matches, cards, counters, bits of wire or string,
all come in useful. An immense number of puzzles have been made out of
the letters of the alphabet, and from those nine little digits and cipher, 1, 2,
3, 4, 5, 6, 7, 8, 9, and 0.
It should always be remembered that a very simple person may propound a
problem that can only be solved by clever heads—if at all. A child asked,
"Can God do everything?" On receiving an affirmative reply, she at once
said: "Then can He make a stone so heavy that He can't lift it?" Many wide-
awake grown-up people do not at once see a satisfactory answer. Yet the

difficulty lies merely in the absurd, though cunning, form of the question,
which really amounts to asking, "Can the Almighty destroy His own
omnipotence?" It is somewhat similar to the other question, "What would
happen if an irresistible moving body came in contact with an immovable
body?" Here we have simply a contradiction in terms, for if there existed
such a thing as an immovable body, there could not at the same time exist a
moving body that nothing could resist.
Professor Tyndall used to invite children to ask him puzzling questions, and
some of them were very hard nuts to crack. One child asked him why that
part of a towel that was dipped in water was of a darker colour than the dry
part. How many readers could give the correct reply? Many people are
satisfied with the most ridiculous answers to puzzling questions. If you ask,
"Why can we see through glass?" nine people out of ten will reply,
"Because it is transparent;" which is, of course, simply another way of
saying, "Because we can see through it."
Puzzles have such an infinite variety that it is sometimes very difficult to
divide them into distinct classes. They often so merge in character that the
best we can do is to sort them into a few broad types. Let us take three or
four examples in illustration of what I mean.
First there is the ancient Riddle, that draws upon the imagination and play
of fancy. Readers will remember the riddle of the Sphinx, the monster of
Bœotia who propounded enigmas to the inhabitants and devoured them if
they failed to solve them. It was said that the Sphinx would destroy herself
if one of her riddles was ever correctly answered. It was this: "What animal
walks on four legs in the morning, two at noon, and three in the evening?"
It was explained by Œdipus, who pointed out that man walked on his hands
and feet in the morning of life, at the noon of life he walked erect, and in
the evening of his days he supported his infirmities with a stick. When the
Sphinx heard this explanation, she dashed her head against a rock and
immediately expired. This shows that puzzle solvers may be really useful
on occasion.
Then there is the riddle propounded by Samson. It is perhaps the first prize
competition in this line on record, the prize being thirty sheets and thirty
changes of garments for a correct solution. The riddle was this: "Out of the

eater came forth meat, and out of the strong came forth sweetness." The
answer was, "A honey-comb in the body of a dead lion." To-day this sort of
riddle survives in such a form as, "Why does a chicken cross the road?" to
which most people give the answer, "To get to the other side;" though the
correct reply is, "To worry the chauffeur." It has degenerated into the
conundrum, which is usually based on a mere pun. For example, we have
been asked from our infancy, "When is a door not a door?" and here again
the answer usually furnished ("When it is a-jar") is not the correct one. It
should be, "When it is a negress (an egress)."
There is the large class of Letter Puzzles, which are based on the little
peculiarities of the language in which they are written—such as anagrams,
acrostics, word-squares, and charades. In this class we also find
palindromes, or words and sentences that read backwards and forwards
alike. These must be very ancient indeed, if it be true that Adam introduced
himself to Eve (in the English language, be it noted) with the palindromic
words, "Madam, I'm Adam," to which his consort replied with the modest
palindrome "Eve."
Then we have Arithmetical Puzzles, an immense class, full of diversity.
These range from the puzzle that the algebraist finds to be nothing but a
"simple equation," quite easy of direct solution, up to the profoundest
problems in the elegant domain of the theory of numbers.
Next we have the Geometrical Puzzle, a favourite and very ancient branch
of which is the puzzle in dissection, requiring some plane figure to be cut
into a certain number of pieces that will fit together and form another
figure. Most of the wire puzzles sold in the streets and toy-shops are
concerned with the geometry of position.
But these classes do not nearly embrace all kinds of puzzles even when we
allow for those that belong at once to several of the classes. There are many
ingenious mechanical puzzles that you cannot classify, as they stand quite
alone: there are puzzles in logic, in chess, in draughts, in cards, and in
dominoes, while every conjuring trick is nothing but a puzzle, the solution
to which the performer tries to keep to himself.
There are puzzles that look easy and are easy, puzzles that look easy and are
difficult, puzzles that look difficult and are difficult, and puzzles that look

difficult and are easy, and in each class we may of course have degrees of
easiness and difficulty. But it does not follow that a puzzle that has
conditions that are easily understood by the merest child is in itself easy.
Such a puzzle might, however, look simple to the uninformed, and only
prove to be a very hard nut to him after he had actually tackled it.
For example, if we write down nineteen ones to form the number
1,111,111,111,111,111,111, and then ask for a number (other than 1 or itself)
that will divide it without remainder, the conditions are perfectly simple,
but the task is terribly difficult. Nobody in the world knows yet whether that
number has a divisor or not. If you can find one, you will have succeeded in
doing something that nobody else has ever done.[A]
The number composed of seventeen ones, 11,111,111,111,111,111, has only
these two divisors, 2,071,723 and 5,363,222,357, and their discovery is an
exceedingly heavy task. The only number composed only of ones that we
know with certainty to have no divisor is 11. Such a number is, of course,
called a prime number.
The maxim that there are always a right way and a wrong way of doing
anything applies in a very marked degree to the solving of puzzles. Here the
wrong way consists in making aimless trials without method, hoping to hit
on the answer by accident—a process that generally results in our getting
hopelessly entangled in the trap that has been artfully laid for us.
Occasionally, however, a problem is of such a character that, though it may
be solved immediately by trial, it is very difficult to do by a process of pure
reason. But in most cases the latter method is the only one that gives any
real pleasure.
When we sit down to solve a puzzle, the first thing to do is to make sure, as
far as we can, that we understand the conditions. For if we do not
understand what it is we have to do, we are not very likely to succeed in
doing it. We all know the story of the man who was asked the question, "If a
herring and a half cost three-halfpence, how much will a dozen herrings
cost?" After several unsuccessful attempts he gave it up, when the
propounder explained to him that a dozen herrings would cost a shilling.
"Herrings!" exclaimed the other apologetically; "I was working it out in
haddocks!"

[A] See footnote on page 198.

It sometimes requires more care than the reader might suppose so to word
the conditions of a new puzzle that they are at once clear and exact and not
so prolix as to destroy all interest in the thing. I remember once
propounding a problem that required something to be done in the "fewest
possible straight lines," and a person who was either very clever or very
foolish (I have never quite determined which) claimed to have solved it in
only one straight line, because, as she said, "I have taken care to make all
the others crooked!" Who could have anticipated such a quibble?
Then if you give a "crossing the river" puzzle, in which people have to be
got over in a boat that will only hold a certain number or combination of
persons, directly the would-be solver fails to master the difficulty he boldly
introduces a rope to pull the boat across. You say that a rope is forbidden;
and he then falls back on the use of a current in the stream. I once thought I
had carefully excluded all such tricks in a particular puzzle of this class. But
a sapient reader made all the people swim across without using the boat at
all! Of course, some few puzzles are intended to be solved by some trick of
this kind; and if there happens to be no solution without the trick it is
perfectly legitimate. We have to use our best judgment as to whether a
puzzle contains a catch or not; but we should never hastily assume it. To
quibble over the conditions is the last resort of the defeated would-be
solver.
Sometimes people will attempt to bewilder you by curious little twists in
the meaning of words. A man recently propounded to me the old familiar
problem, "A boy walks round a pole on which is a monkey, but as the boy
walks the monkey turns on the pole so as to be always facing him on the
opposite side. Does the boy go around the monkey?" I replied that if he
would first give me his definition of "to go around" I would supply him
with the answer. Of course, he demurred, so that he might catch me either
way. I therefore said that, taking the words in their ordinary and correct
meaning, most certainly the boy went around the monkey. As was expected,
he retorted that it was not so, because he understood by "going around" a
thing that you went in such a way as to see all sides of it. To this I made the
obvious reply that consequently a blind man could not go around anything.

He then amended his definition by saying that the actual seeing all sides
was not essential, but you went in such a way that, given sight, you could
see all sides. Upon which it was suggested that consequently you could not
walk around a man who had been shut up in a box! And so on. The whole
thing is amusingly stupid, and if at the start you, very properly, decline to
admit any but a simple and correct definition of "to go around," there is no
puzzle left, and you prevent an idle, and often heated, argument.
When you have grasped your conditions, always see if you cannot simplify
them, for a lot of confusion is got rid of in this way. Many people are
puzzled over the old question of the man who, while pointing at a portrait,
says, "Brothers and sisters have I none, but that man's father is my father's
son." What relation did the man in the picture bear to the speaker? Here you
simplify by saying that "my father's son" must be either "myself" or "my
brother." But, since the speaker has no brother, it is clearly "myself." The
statement simplified is thus nothing more than, "That man's father is
myself," and it was obviously his son's portrait. Yet people fight over this
question by the hour!
There are mysteries that have never been solved in many branches of
Puzzledom. Let us consider a few in the world of numbers—little things the
conditions of which a child can understand, though the greatest minds
cannot master. Everybody has heard the remark, "It is as hard as squaring a
circle," though many people have a very hazy notion of what it means. If
you have a circle of given diameter and wish to find the side of a square
that shall contain exactly the same area, you are confronted with the
problem of squaring the circle. Well, it cannot be done with exactitude
(though we can get an answer near enough for all practical purposes),
because it is not possible to say in exact numbers what is the ratio of the
diameter to the circumference. But it is only in recent times that it has been
proved to be impossible, for it is one thing not to be able to perform a
certain feat, but quite another to prove that it cannot be done. Only
uninstructed cranks now waste their time in trying to square the circle.
Again, we can never measure exactly in numbers the diagonal of a square.
If you have a window pane exactly a foot on every side, there is the
distance from corner to corner staring you in the face, yet you can never say
in exact numbers what is the length of that diagonal. The simple person will

at once suggest that we might take our diagonal first, say an exact foot, and
then construct our square. Yes, you can do this, but then you can never say
exactly what is the length of the side. You can have it which way you like,
but you cannot have it both ways.
All my readers know what a magic square is. The numbers 1 to 9 can be
arranged in a square of nine cells, so that all the columns and rows and each
of the diagonals will add up 15. It is quite easy; and there is only one way
of doing it, for we do not count as different the arrangements obtained by
merely turning round the square and reflecting it in a mirror. Now if we
wish to make a magic square of the 16 numbers, 1 to 16, there are just 880
different ways of doing it, again not counting reversals and reflections. This
has been finally proved of recent years. But how many magic squares may
be formed with the 25 numbers, 1 to 25, nobody knows, and we shall have
to extend our knowledge in certain directions before we can hope to solve
the puzzle. But it is surprising to find that exactly 174,240 such squares
may be formed of one particular restricted kind only—the bordered square,
in which the inner square of nine cells is itself magic. And I have shown
how this number may be at once doubled by merely converting every
bordered square—by a simple rule—into a non-bordered one.
Then vain attempts have been made to construct a magic square by what is
called a "knight's tour" over the chess-board, numbering each square that
the knight visits in succession, 1, 2, 3, 4, etc.; and it has been done, with the
exception of the two diagonals, which so far have baffled all efforts. But it
is not certain that it cannot be done.
Though the contents of the present volume are in the main entirely original,
some very few old friends will be found; but these will not, I trust, prove
unwelcome in the new dress that they have received. The puzzles are of
every degree of difficulty, and so varied in character that perhaps it is not
too much to hope that every true puzzle lover will find ample material to
interest—and possibly instruct. In some cases I have dealt with the methods
of solution at considerable length, but at other times I have reluctantly felt
obliged to restrict myself to giving the bare answers. Had the full solutions
and proofs been given in the case of every puzzle, either half the problems
would have had to be omitted, or the size of the book greatly increased. And
the plan that I have adopted has its advantages, for it leaves scope for the

mathematical enthusiast to work out his own analysis. Even in those cases
where I have given a general formula for the solution of a puzzle, he will
find great interest in verifying it for himself.

THE CANTERBURY PUZZLES

A Chance-gathered company of pilgrims, on their way to the shrine of Saint
Thomas à Becket at Canterbury, met at the old Tabard Inn, later called the
Talbot, in Southwark, and the host proposed that they should beguile the
ride by each telling a tale to his fellow-pilgrims. This we all know was the
origin of the immortal Canterbury Tales of our great fourteenth-century
poet, Geoffrey Chaucer. Unfortunately, the tales were never completed, and
perhaps that is why the quaint and curious "Canterbury Puzzles," devised
and propounded by the same body of pilgrims, were not also recorded by
the poet's pen. This is greatly to be regretted, since Chaucer, who, as Leland
tells us, was an "ingenious mathematician" and the author of a learned
treatise on the astrolabe, was peculiarly fitted for the propounding of
problems. In presenting for the first time some of these old-world posers, I
will not stop to explain the singular manner in which they came into my
possession, but proceed at once, without unnecessary preamble, to give my
readers an opportunity of solving them and testing their quality. There are
certainly far more difficult puzzles extant, but difficulty and interest are two
qualities of puzzledom that do not necessarily go together.

1.—The Reve's Puzzle.

The Reve was a wily man and something of a scholar. As Chaucer tells us,
"There was no auditor could of him win," and "there could no man bring
him in arrear." The poet also noticed that "ever he rode the hindermost of
the route." This he did that he might the better, without interruption, work
out the fanciful problems and ideas that passed through his active brain.
When the pilgrims were stopping at a wayside tavern, a number of cheeses
of varying sizes caught his alert eye; and calling for four stools, he told the
company that he would show them a puzzle of his own that would keep
them amused during their rest. He then placed eight cheeses of graduating
sizes on one of the end stools, the smallest cheese being at the top, as
clearly shown in the illustration. "This is a riddle," quoth he, "that I did
once set before my fellow townsmen at Baldeswell, that is in Norfolk, and,
by Saint Joce, there was no man among them that could rede it aright. And
yet it is withal full easy, for all that I do desire is that, by the moving of one

cheese at a time from one stool unto another, ye shall remove all the cheeses
to the stool at the other end without ever putting any cheese on one that is
smaller than itself. To him that will perform this feat in the least number of
moves that be possible will I give a draught of the best that our good host
can provide." To solve this puzzle in the fewest possible moves, first with 8,
then with 10, and afterwards with 21 cheeses, is an interesting recreation.

2.—The Pardoner's Puzzle.

The gentle Pardoner, "that straight was come from the court of Rome,"
begged to be excused; but the company would not spare him. "Friends and
fellow-pilgrims," said he, "of a truth the riddle that I have made is but a
poor thing, but it is the best that I have been able to devise. Blame my lack
of knowledge of such matters if it be not to your liking." But his invention
was very well received. He produced the accompanying plan, and said that
it represented sixty-four towns through which he had to pass during some of
his pilgrimages, and the lines connecting them were roads. He explained
that the puzzle was to start from the large black town and visit all the other
towns once, and once only, in fifteen straight pilgrimages. Try to trace the
route in fifteen straight lines with your pencil. You may end where you like,
but note that the omission of a little road at the bottom is intentional, as it
seems that it was impossible to go that way.

3.—The Miller's Puzzle.

The Miller next took the company aside and showed them nine sacks of
flour that were standing as depicted in the sketch. "Now, hearken, all and
some," said he, "while that I do set ye the riddle of the nine sacks of flour.
And mark ye, my lords and masters, that there be single sacks on the
outside, pairs next unto them, and three together in the middle thereof. By
Saint Benedict, it doth so happen that if we do but multiply the pair, 28, by
the single one, 7, the answer is 196, which is of a truth the number shown
by the sacks in the middle. Yet it be not true that the other pair, 34, when so
multiplied by its neighbour, 5, will also make 196. Wherefore I do beg you,
gentle sirs, so to place anew the nine sacks with as little trouble as possible
that each pair when thus multiplied by its single neighbour shall make the
number in the middle." As the Miller has stipulated in effect that as few
bags as possible shall be moved, there is only one answer to this puzzle,
which everybody should be able to solve.

4.—The Knight's Puzzle.
This worthy man was, as Chaucer tells us, "a very perfect, gentle knight,"
and "In many a noble army had he been: At mortal battles had he been
fifteen." His shield, as he is seen showing it to the company at the "Tabard"
in the illustration, was, in the peculiar language of the heralds, "argent,
semée of roses, gules," which means that on a white ground red roses were
scattered or strewn, as seed is sown by the hand. When this knight was
called on to propound a puzzle, he said to the company, "This riddle a wight
did ask of me when that I fought with the lord of Palatine against the
heathen in Turkey. In thy hand take a piece of chalk and learn how many
perfect squares thou canst make with one of the eighty-seven roses at each
corner thereof." The reader may find it an interesting problem to count the
number of squares that may be formed on the shield by uniting four roses.

5.—The Wife of Bath's Riddles.
The frolicsome Wife of Bath, when called upon to favour the company,
protested that she had no aptitude for such things, but that her fourth
husband had had a liking for them, and she remembered one of his riddles
that might be new to her fellow pilgrims: "Why is a bung that hath been
made fast in a barrel like unto another bung that is just falling out of a
barrel?" As the company promptly answered this easy conundrum, the lady
went on to say that when she was one day seated sewing in her private
chamber her son entered. "Upon receiving," saith she, "the parental
command, 'Depart, my son, and do not disturb me!' he did reply, 'I am, of a
truth, thy son; but thou art not my mother, and until thou hast shown me
how this may be I shall not go forth.'" This perplexed the company a good
deal, but it is not likely to give the reader much difficulty.

6.—The Host's Puzzle.
Perhaps no puzzle of the whole collection caused more jollity or was found
more entertaining than that produced by the Host of the "Tabard," who
accompanied the party all the way. He called the pilgrims together and
spoke as follows: "My merry masters all, now that it be my turn to give
your brains a twist, I will show ye a little piece of craft that will try your
wits to their full bent. And yet methinks it is but a simple matter when the
doing of it is made clear. Here be a cask of fine London ale, and in my
hands do I hold two measures—one of five pints, and the other of three
pints. Pray show how it is possible for me to put a true pint into each of the
measures." Of course, no other vessel or article is to be used, and no
marking of the measures is allowed. It is a knotty little problem and a
fascinating one. A good many persons to-day will find it by no means an
easy task. Yet it can be done.

7.—The Clerk of Oxenford's Puzzle.

The silent and thoughtful Clerk of Oxenford, of whom it is recorded that
"Every farthing that his friends e'er lent, In books and learning was it
always spent," was prevailed upon to give his companions a puzzle. He
said, "Ofttimes of late have I given much thought to the study of those
strange talismans to ward off the plague and such evils that are yclept magic
squares, and the secret of such things is very deep and the number of such
squares truly great. But the small riddle that I did make yester eve for the
purpose of this company is not so hard that any may not find it out with a
little patience." He then produced the square shown in the illustration and
said that it was desired so to cut it into four pieces (by cuts along the lines)
that they would fit together again and form a perfect magic square, in which
the four columns, the four rows, and the two long diagonals should add up
34. It will be found that this is a just sufficiently easy puzzle for most
people's tastes.

8.—The Tapiser's Puzzle.

Then came forward the Tapiser, who was, of course, a maker of tapestry,
and must not be confounded with a tapster, who draws and sells ale.
He produced a beautiful piece of tapestry, worked in a simple chequered
pattern, as shown in the diagram. "This piece of tapestry, sirs," quoth he,
"hath one hundred and sixty-nine small squares, and I do desire you to tell
me the manner of cutting the tapestry into three pieces that shall fit together
and make one whole piece in shape of a perfect square.
"Moreover, since there be divers ways of so doing, I do wish to know that
way wherein two of the pieces shall together contain as much as possible of
the rich fabric." It is clear that the Tapiser intended the cuts to be made
along the lines dividing the squares only, and, as the material was not both
sides alike, no piece may be reversed, but care must be observed that the
chequered pattern matches properly.

9.—The Carpenter's Puzzle.
The Carpenter produced the carved wooden pillar that he is seen holding in
the illustration, wherein the knight is propounding his knotty problem to the
goodly company (No. 4), and spoke as follows: "There dwelleth in the city
of London a certain scholar that is learned in astrology and other strange
arts. Some few days gone he did bring unto me a piece of wood that had
three feet in length, one foot in breadth and one foot in depth, and did desire
that it be carved and made into the pillar that you do now behold. Also did
he promise certain payment for every cubic inch of wood cut away by the
carving thereof.
"Now I did at first weigh the block, and found it truly to contain thirty
pounds, whereas the pillar doth now weigh but twenty pounds. Of a truth I
have therefore cut away one cubic foot (which is to say one-third) of the
three cubic feet of the block; but this scholar withal doth hold that payment
may not thus be fairly made by weight, since the heart of the block may be
heavier, or perchance may be more light, than the outside. How then may I
with ease satisfy the scholar as to the quantity of wood that hath been cut
away?" This at first sight looks a difficult question, but it is so absurdly
simple that the method employed by the carpenter should be known to
everybody to-day, for it is a very useful little "wrinkle."

10.—The Puzzle of the Squire's Yeoman.
Chaucer says of the Squire's Yeoman, who formed one of his party of
pilgrims, "A forester was he truly as I guess," and tells us that "His arrows
drooped not with feathers low, And in his hand he bare a mighty bow."
When a halt was made one day at a wayside inn, bearing the old sign of the
"Chequers," this yeoman consented to give the company an exhibition of
his skill. Selecting nine good arrows, he said, "Mark ye, good sirs, how that
I shall shoot these nine arrows in such manner that each of them shall lodge
in the middle of one of the squares that be upon the sign of the 'Chequers,'
and yet of a truth shall no arrow be in line with any other arrow." The
diagram will show exactly how he did this, and no two arrows will be found
in line, horizontally, vertically, or diagonally. Then the Yeoman said: "Here
then is a riddle for ye. Remove three of the arrows each to one of its
neighbouring squares, so that the nine shall yet be so placed that none
thereof may be in line with another." By a "neighbouring square" is meant
one that adjoins, either laterally or diagonally.

11.—The Nun's Puzzle.
"I trow there be not one among ye," quoth the Nun, on a later occasion,
"that doth not know that many monks do oft pass the time in play at certain
games, albeit they be not lawful for them. These games, such as cards and
the game of chess, do they cunningly hide from the abbot's eye by putting
them away in holes that they have cut out of the very hearts of great books
that be upon their shelves. Shall the nun therefore be greatly blamed if she
do likewise? I will show a little riddle game that we do sometimes play
among ourselves when the good abbess doth hap to be away."

The Nun then produced the eighteen cards that are shown in the illustration.
She explained that the puzzle was so to arrange the cards in a pack, that by
placing the uppermost one on the table, placing the next one at the bottom
of the pack, the next one on the table, the next at the bottom of the pack,
and so on, until all are on the table, the eighteen cards shall then read
"CANTERBURY PILGRIMS." Of course each card must be placed on the
table to the immediate right of the one that preceded it. It is easy enough if
you work backwards, but the reader should try to arrive at the required
order without doing this, or using any actual cards.

12.—The Merchant's Puzzle.
Of the Merchant the poet writes, "Forsooth he was a worthy man withal."
He was thoughtful, full of schemes, and a good manipulator of figures. "His
reasons spake he eke full solemnly. Sounding away the increase of his
winning." One morning, when they were on the road, the Knight and the
Squire, who were riding beside him, reminded the Merchant that he had not
yet propounded the puzzle that he owed the company. He thereupon said,
"Be it so? Here then is a riddle in numbers that I will set before this merry
company when next we do make a halt. There be thirty of us in all riding
over the common this morn. Truly we may ride one and one, in what they
do call the single file, or two and two, or three and three, or five and five, or
six and six, or ten and ten, or fifteen and fifteen, or all thirty in a row. In no
other way may we ride so that there be no lack of equal numbers in the
rows. Now, a party of pilgrims were able thus to ride in as many as sixty-
four different ways. Prithee tell me how many there must perforce have
been in the company." The Merchant clearly required the smallest number
of persons that could so ride in the sixty-four ways.

13.—The Man of Law's Puzzle.
The Sergeant of the Law was "full rich of excellence. Discreet he was, and
of great reverence." He was a very busy man, but, like many of us to-day,
"he seemed busier than he was." He was talking one evening of prisons and
prisoners, and at length made the following remarks: "And that which I
have been saying doth forsooth call to my mind that this morn I bethought
me of a riddle that I will now put forth." He then produced a slip of vellum,
on which was drawn the curious plan that is now given. "Here," saith he,
"be nine dungeons, with a prisoner in every dungeon save one, which is
empty. These prisoners be numbered in order, 7, 5, 6, 8, 2, 1, 4, 3, and I
desire to know how they can, in as few moves as possible, put themselves
in the order 1, 2, 3, 4, 5, 6, 7, 8. One prisoner may move at a time along the
passage to the dungeon that doth happen to be empty, but never, on pain of
death, may two men be in any dungeon at the same time. How may it be
done?" If the reader makes a rough plan on a sheet of paper and uses
numbered counters, he will find it an interesting pastime to arrange the
prisoners in the fewest possible moves. As there is never more than one
vacant dungeon at a time to be moved into, the moves may be recorded in
this simple way: 3—2—1—6, and so on.

14.—The Weaver's Puzzle.
When the Weaver brought out a square piece of beautiful cloth, daintily
embroidered with lions and castles, as depicted in the illustration, the
pilgrims disputed among themselves as to the meaning of these ornaments.
The Knight, however, who was skilled in heraldry, explained that they were
probably derived from the lions and castles borne in the arms of Ferdinand
III., the King of Castile and Leon, whose daughter was the first wife of our
Edward I. In this he was undoubtedly correct. The puzzle that the Weaver
proposed was this. "Let us, for the nonce, see," saith he, "if there be any of
the company that can show how this piece of cloth may be cut into four
several pieces, each of the same size and shape, and each piece bearing a
lion and a castle." It is not recorded that anybody mastered this puzzle,
though it is quite possible of solution in a satisfactory manner. No cut may
pass through any part of a lion or a castle.

15.—The Cook's Puzzle.
We find that there was a cook among the company; and his services were no
doubt at times in great request, "For he could roast and seethe, and broil and
fry, And make a mortress and well bake a pie." One night when the pilgrims
were seated at a country hostelry, about to begin their repast, the cook
presented himself at the head of the table that was presided over by the
Franklin, and said, "Listen awhile, my masters, while that I do ask ye a
riddle, and by Saint Moden it is one that I cannot answer myself withal.
There be eleven pilgrims seated at this board on which is set a warden pie
and a venison pasty, each of which may truly be divided into four parts and
no more. Now, mark ye, five out of the eleven pilgrims can eat the pie, but
will not touch the pasty, while four will eat the pasty but turn away from the
pie. Moreover, the two that do remain be able and willing to eat of either.
By my halidame, is there any that can tell me in how many different ways
the good Franklin may choose whom he will serve?" I will just caution the
reader that if he is not careful he will find, when he sees the answer, that he
has made a mistake of forty, as all the company did, with the exception of
the Clerk of Oxenford, who got it right by accident, through putting down a
wrong figure.

Strange to say, while the company perplexed their wits about this riddle the
cook played upon them a merry jest. In the midst of their deep thinking and
hot dispute what should the cunning knave do but stealthily take away both
the pie and the pasty. Then, when hunger made them desire to go on with
the repast, finding there was nought upon the table, they called clamorously
for the cook.
"My masters," he explained, "seeing you were so deep set in the riddle, I
did take them to the next room, where others did eat them with relish ere
they had grown cold. There be excellent bread and cheese in the pantry."

16.—The Sompnour's Puzzle.
The Sompnour, or Summoner, who, according to Chaucer, joined the party
of pilgrims, was an officer whose duty was to summon delinquents to
appear in ecclesiastical courts. In later times he became known as the
apparitor. Our particular individual was a somewhat quaint though worthy
man. "He was a gentle hireling and a kind; A better fellow should a man not
find." In order that the reader may understand his appearance in the picture,
it must be explained that his peculiar headgear is duly recorded by the poet.
"A garland had he set upon his head, As great as if it were for an ale-stake."

One evening ten of the company stopped at a village inn and requested to
be put up for the night, but mine host could only accommodate five of

them. The Sompnour suggested that they should draw lots, and as he had
had experience in such matters in the summoning of juries and in other
ways, he arranged the company in a circle and proposed a "count out."
Being of a chivalrous nature, his little plot was so to arrange that the men
should all fall out and leave the ladies in possession. He therefore gave the
Wife of Bath a number and directed her to count round and round the circle,
in a clockwise direction, and the person on whom that number fell was
immediately to step out of the ring. The count then began afresh at the next
person. But the lady misunderstood her instructions, and selected in mistake
the number eleven and started the count at herself. As will be found, this
resulted in all the women falling out in turn instead of the men, for every
eleventh person withdrawn from the circle is a lady.
"Of a truth it was no fault of mine," said the Sompnour next day to the
company, "and herein is methinks a riddle. Can any tell me what number
the good Wife should have used withal, and at which pilgrim she should
have begun her count so that no other than the five men should have been
counted out?" Of course, the point is to find the smallest number that will
have the desired effect.

17.—The Monk's Puzzle.
The Monk that went with the party was a great lover of sport. "Greyhounds
he had as swift as fowl of flight: Of riding and of hunting for the hare Was
all his love, for no cost would he spare." One day he addressed the pilgrims
as follows:—
"There is a little matter that hath at times perplexed me greatly, though
certes it is of no great weight; yet may it serve to try the wits of some that
be cunning in such things. Nine kennels have I for the use of my dogs, and
they be put in the form of a square; though the one in the middle I do never
use, it not being of a useful nature. Now the riddle is to find in how many
different ways I may place my dogs in all or any of the outside kennels so
that the number of dogs on every side of the square may be just ten." The
small diagrams show four ways of doing it, and though the fourth way is
merely a reversal of the third, it counts as different. Any kennels may be left
empty. This puzzle was evidently a variation of the ancient one of the
Abbess and her Nuns.

18.—The Shipman's Puzzle.
Of this person we are told, "He knew well all the havens, as they were,
From Gothland to the Cape of Finisterre, And every creek in Brittany and
Spain: His barque yclepéd was the Magdalen." The strange puzzle in
navigation that he propounded was as follows.
"Here be a chart," quoth the Shipman, "of five islands, with the inhabitants
of which I do trade. In each year my good ship doth sail over every one of
the ten courses depicted thereon, but never may she pass along the same
course twice in any year. Is there any among the company who can tell me
in how many different ways I may direct the Magdalen's ten yearly
voyages, always setting out from the same island?"

CHART of ye MAGDALEN

19.—The Puzzle of the Prioress.
The Prioress, who went by the name of Eglantine, is best remembered on
account of Chaucer's remark, "And French she spake full fair and properly,
After the school of Stratford-atté-Bow, For French of Paris was to her
unknow." But our puzzle has to do less with her character and education
than with her dress. "And thereon hung a brooch of gold full sheen, On
which was written first a crownéd A." It is with the brooch that we are
concerned, for when asked to give a puzzle she showed this jewel to the
company and said: "A learned man from Normandy did once give me this
brooch as a charm, saying strange and mystic things anent it, how that it
hath an affinity for the square, and such other wise words that were too
subtle for me. But the good Abbot of Chertsey did once tell me that the
cross may be so cunningly cut into four pieces that they will join and make
a perfect square; though on my faith I know not the manner of doing it."
It is recorded that "the pilgrims did find no answer to the riddle, and the
Clerk of Oxenford thought that the Prioress had been deceived in the matter
thereof; whereupon the lady was sore vexed, though the gentle knight did
flout and gibe at the poor clerk because of his lack of understanding over
other of the riddles, which did fill him with shame and make merry the
company."

20.—The Puzzle of the Doctor of Physic.
This Doctor, learned though he was, for "In all this world to him there was
none like To speak of physic and of surgery," and "He knew the cause of
every malady," yet was he not indifferent to the more material side of life.
"Gold in physic is a cordial; Therefore he lovéd gold in special." The
problem that the Doctor propounded to the assembled pilgrims was this. He
produced two spherical phials, as shown in our illustration, and pointed out
that one phial was exactly a foot in circumference, and the other two feet in
circumference.
"I do wish," said the Doctor, addressing the company, "to have the exact
measures of two other phials, of a like shape but different in size, that may
together contain just as much liquid as is contained by these two." To find
exact dimensions in the smallest possible numbers is one of the toughest
nuts I have attempted. Of course the thickness of the glass, and the neck and
base, are to be ignored.

21.—The Ploughman's Puzzle.

The Ploughman—of whom Chaucer remarked, "A worker true and very
good was he, Living in perfect peace and charity"—protested that riddles
were not for simple minds like his, but he would show the good pilgrims, if
they willed it, one that he had frequently heard certain clever folk in his
own neighbourhood discuss. "The lord of the manor in the part of Sussex
whence I come hath a plantation of sixteen fair oak trees, and they be so set
out that they make twelve rows with four trees in every row. Once on a time
a man of deep learning, who happened to be travelling in those parts, did
say that the sixteen trees might have been so planted that they would make
so many as fifteen straight rows, with four trees in every row thereof. Can
ye show me how this might be? Many have doubted that 'twere possible to
be done." The illustration shows one of many ways of forming the twelve
rows. How can we make fifteen?

22.—The Franklin's Puzzle.

"A Franklin was in this company; White was his beard as is the daisy." We
are told by Chaucer that he was a great householder and an epicure.
"Without baked meat never was his house. Of fish and flesh, and that so
plenteous, It snowed in his house of meat and drink, Of every dainty that
men could bethink." He was a hospitable and generous man. "His table
dormant in his hall alway Stood ready covered all throughout the day." At
the repasts of the Pilgrims he usually presided at one of the tables, as we
found him doing on the occasion when the cook propounded his problem of
the two pies.
One day, at an inn just outside Canterbury, the company called on him to
produce the puzzle required of him; whereupon he placed on the table
sixteen bottles numbered 1, 2, 3, up to 15, with the last one marked 0.
"Now, my masters," quoth he, "it will be fresh in your memories how that
the good Clerk of Oxenford did show us a riddle touching what hath been
called the magic square. Of a truth will I set before ye another that may
seem to be somewhat of a like kind, albeit there be little in common betwixt
them. Here be set out sixteen bottles in form of a square, and I pray you so
place them afresh that they shall form a magic square, adding up to thirty in
all the ten straight ways. But mark well that ye may not remove more than

ten of the bottles from their present places, for therein layeth the subtlety of
the riddle." This is a little puzzle that may be conveniently tried with
sixteen numbered counters.

23.—The Squire's Puzzle.
The young Squire, twenty years of age, was the son of the Knight that
accompanied him on the historic pilgrimage. He was undoubtedly what in
later times we should call a dandy, for, "Embroideréd was he as is a mead,
All full of fresh flowers, white and red. Singing he was or fluting all the
day, He was as fresh as is the month of May." As will be seen in the
illustration to No. 26, while the Haberdasher was propounding his problem
of the triangle, this young Squire was standing in the background making a
drawing of some kind; for "He could songs make and well indite, Joust and
eke dance, and well portray and write."
The Knight turned to him after a while and said, "My son, what is it over
which thou dost take so great pains withal?" and the Squire answered, "I
have bethought me how I might portray in one only stroke a picture of our
late sovereign lord King Edward the Third, who hath been dead these ten
years. 'Tis a riddle to find where the stroke doth begin and where it doth
also end. To him who first shall show it unto me will I give the portraiture."

I am able to present a facsimile of the original drawing, which was won by
the Man of Law. It may be here remarked that the pilgrimage set out from
Southwark on 17th April 1387, and Edward the Third died in 1377.

24.—The Friar's Puzzle.
The Friar was a merry fellow, with a sweet tongue and twinkling eyes.
"Courteous he was and lowly of service. There was a man nowhere so
virtuous." Yet he was "the best beggar in all his house," and gave reasons
why "Therefore, instead of weeping and much prayer, Men must give silver
to the needy friar." He went by the name of Hubert. One day he produced
four money bags and spoke as follows: "If the needy friar doth receive in
alms five hundred silver pennies, prithee tell in how many different ways
they may be placed in the four bags." The good man explained that order
made no difference (so that the distribution 50, 100, 150, 200 would be the
same as 100, 50, 200, 150, or 200, 50, 100, 150), and one, two, or three
bags may at any time be empty.

25.—The Parson's Puzzle.
The Parson was a really devout and good man. "A better priest I trow there
nowhere is." His virtues and charity made him beloved by all his flock, to
whom he presented his teaching with patience and simplicity; "but first he
followed it himself." Now, Chaucer is careful to tell us that "Wide was his
parish, and houses far asunder, But he neglected nought for rain or
thunder;" and it is with his parochial visitations that the Parson's puzzle
actually dealt. He produced a plan of part of his parish, through which a
small river ran that joined the sea some hundreds of miles to the south. I
give a facsimile of the plan.

"Here, my worthy Pilgrims, is a strange riddle," quoth the Parson. "Behold
how at the branching of the river is an island. Upon this island doth stand
my own poor parsonage, and ye may all see the whereabouts of the village

church. Mark ye, also, that there be eight bridges and no more over the river
in my parish. On my way to church it is my wont to visit sundry of my
flock, and in the doing thereof I do pass over every one of the eight bridges
once and no more. Can any of ye find the path, after this manner, from the
house to the church, without going out of the parish? Nay, nay, my friends, I
do never cross the river in any boat, neither by swimming nor wading, nor
do I go underground like unto the mole, nor fly in the air as doth the eagle;
but only pass over by the bridges." There is a way in which the Parson
might have made this curious journey. Can the reader discover it? At first it
seems impossible, but the conditions offer a loophole.

26.—The Haberdasher's Puzzle.

Many attempts were made to induce the Haberdasher, who was of the party,
to propound a puzzle of some kind, but for a long time without success. At
last, at one of the Pilgrims' stopping-places, he said that he would show
them something that would "put their brains into a twist like unto a bell-
rope." As a matter of fact, he was really playing off a practical joke on the
company, for he was quite ignorant of any answer to the puzzle that he set
them. He produced a piece of cloth in the shape of a perfect equilateral
triangle, as shown in the illustration, and said, "Be there any among ye full
wise in the true cutting of cloth? I trow not. Every man to his trade, and the
scholar may learn from the varlet and the wise man from the fool. Show

me, then, if ye can, in what manner this piece of cloth may be cut into four
several pieces that may be put together to make a perfect square."
Now some of the more learned of the company found a way of doing it in
five pieces, but not in four. But when they pressed the Haberdasher for the
correct answer he was forced to admit, after much beating about the bush,
that he knew no way of doing it in any number of pieces. "By Saint
Francis," saith he, "any knave can make a riddle methinks, but it is for them
that may to rede it aright." For this he narrowly escaped a sound beating.
But the curious point of the puzzle is that I have found that the feat may
really be performed in so few as four pieces, and without turning over any
piece when placing them together. The method of doing this is subtle, but I
think the reader will find the problem a most interesting one.

27.—The Dyer's Puzzle.
One of the pilgrims was a Dyer, but Chaucer tells us nothing about him, the
Tales being incomplete. Time after time the company had pressed this
individual to produce a puzzle of some kind, but without effect. The poor
fellow tried his best to follow the examples of his friends the Tapiser, the
Weaver, and the Haberdasher; but the necessary idea would not come, rack
his brains as he would. All things, however, come to those who wait—and
persevere—and one morning he announced, in a state of considerable
excitement, that he had a poser to set before them. He brought out a square
piece of silk on which were embroidered a number of fleurs-de-lys in rows,
as shown in our illustration.
"Lordings," said the Dyer, "hearken anon unto my riddle. Since I was
awakened at dawn by the crowing of cocks—for which din may our host
never thrive—I have sought an answer thereto, but by St. Bernard I have
found it not. There be sixty-and-four flowers-de-luce, and the riddle is to
show how I may remove six of these so that there may yet be an even
number of the flowers in every row and every column."

The Dyer was abashed when every one of the company showed without any
difficulty whatever, and each in a different way, how this might be done.
But the good Clerk of Oxenford was seen to whisper something to the Dyer,
who added, "Hold, my masters! What I have said is not all. Ye must find in
how many different ways it may be done!" All agreed that this was quite
another matter. And only a few of the company got the right answer.

28.—The Great Dispute between the Friar and the
Sompnour.
Chaucer records the painful fact that the harmony of the pilgrimage was
broken on occasions by the quarrels between the Friar and the Sompnour.
At one stage the latter threatened that ere they reached Sittingbourne he
would make the Friar's "heart for to mourn;" but the worthy Host intervened
and patched up a temporary peace. Unfortunately trouble broke out again
over a very curious dispute in this way.

At one point of the journey the road lay along two sides of a square field,
and some of the pilgrims persisted, in spite of trespass, in cutting across
from corner to corner, as they are seen to be doing in the illustration. Now,
the Friar startled the company by stating that there was no need for the
trespass, since one way was exactly the same distance as the other! "On my

faith, then," exclaimed the Sompnour, "thou art a very fool!" "Nay," replied
the Friar, "if the company will but listen with patience, I shall presently
show how that thou art the fool, for thou hast not wit enough in thy poor
brain to prove that the diagonal of any square is less than two of the sides."
If the reader will refer to the diagrams that we have given, he will be able to
follow the Friar's argument. If we suppose the side of the field to be 100
yards, then the distance along the two sides, A to B, and B to C, is 200
yards. He undertook to prove that the diagonal distance direct from A to C
is also 200 yards. Now, if we take the diagonal path shown in Fig. 1, it is
evident that we go the same distance, for every one of the eight straight
portions of this path measures exactly 25 yards. Similarly in Fig. 2, the
zigzag contains ten straight portions, each 20 yards long: that path is also
the same length—200 yards. No matter how many steps we make in our
zigzag path, the result is most certainly always the same. Thus, in Fig. 3 the
steps are very small, yet the distance must be 200 yards; as is also the case
in Fig. 4, and would yet be if we needed a microscope to detect the steps. In
this way, the Friar argued, we may go on straightening out that zigzag path
until we ultimately reach a perfectly straight line, and it therefore follows
that the diagonal of a square is of exactly the same length as two of the
sides.

Now, in the face of it, this must be wrong; and it is in fact absurdly so, as
we can at once prove by actual measurement if we have any doubt. Yet the
Sompnour could not for the life of him point out the fallacy, and so upset
the Friar's reasoning. It was this that so exasperated him, and consequently,
like many of us to-day when we get entangled in an argument, he utterly
lost his temper and resorted to abuse. In fact, if some of the other pilgrims
had not interposed the two would have undoubtedly come to blows. The
reader will perhaps at once see the flaw in the Friar's argument.

29.—Chaucer's Puzzle.

Chaucer himself accompanied the pilgrims. Being a mathematician and a
man of a thoughtful habit, the Host made fun of him, he tells us, saying,
"Thou lookest as thou wouldst find a hare, For ever on the ground I see thee
stare." The poet replied to the request for a tale by launching into a long-
spun-out and ridiculous poem, intended to ridicule the popular romances of
the day, after twenty-two stanzas of which the company refused to hear any
more, and induced him to start another tale in prose. It is an interesting fact
that in the "Parson's Prologue" Chaucer actually introduces a little
astronomical problem. In modern English this reads somewhat as follows:
—
"The sun from the south line was descended so low that it was not to my
sight more than twenty-nine degrees. I calculate that it was four o'clock, for,
assuming my height to be six feet, my shadow was eleven feet, a little more
or less. At the same moment the moon's altitude (she being in mid-Libra)
was steadily increasing as we entered at the west end of the village." A

correspondent has taken the trouble to work this out, and finds that the local
time was 3.58 p.m., correct to a minute, and that the day of the year was the
22nd or 23rd of April, modern style. This speaks well for Chaucer's
accuracy, for the first line of the Tales tells us that the pilgrimage was in
April—they are supposed to have set out on 17th April 1387, as stated in
No. 23.
Though Chaucer made this little puzzle and recorded it for the interest of
his readers, he did not venture to propound it to his fellow-pilgrims. The
puzzle that he gave them was of a simpler kind altogether: it may be called
a geographical one. "When, in the year 1372, I did go into Italy as the
envoy of our sovereign lord King Edward the Third, and while there did
visit Francesco Petrarch, that learned poet did take me to the top of a certain
mountain in his country. Of a truth, as he did show me, a mug will hold less
liquor at the top of this mountain than in the valley beneath. Prythee tell me
what mountain this may be that has so strange a property withal." A very
elementary knowledge of geography will suffice for arriving at the correct
answer.

30.—The Puzzle of the Canon's Yeoman.
This person joined the party on the road. "'God save,' quoth he, 'this jolly
company! Fast have I ridden,' saith he, 'for your sake, Because I would I
might you overtake, To ride among this merry company.'" Of course, he was
asked to entertain the pilgrims with a puzzle, and the one he propounded
was the following. He showed them the diamond-shaped arrangement of
letters presented in the accompanying illustration, and said, "I do call it the
rat-catcher's riddle. In how many different ways canst thou read the words,
'Was it a rat I saw?'" You may go in any direction backwards and forwards,
upwards or downwards, only the successive letters in any reading must
always adjoin one another.

31.—The Manciple's Puzzle.
The Manciple was an officer who had the care of buying victuals for an Inn
of Court—like the Temple. The particular individual who accompanied the
party was a wily man who had more than thirty masters, and made fools of
them all. Yet he was a man "whom purchasers might take as an example
How to be wise in buying of their victual."
It happened that at a certain stage of the journey the Miller and the Weaver
sat down to a light repast. The Miller produced five loaves and the Weaver
three. The Manciple coming upon the scene asked permission to eat with
them, to which they agreed. When the Manciple had fed he laid down eight
pieces of money and said with a sly smile, "Settle betwixt yourselves how
the money shall be fairly divided. 'Tis a riddle for thy wits."
A discussion followed, and many of the pilgrims joined in it. The Reve and
the Sompnour held that the Miller should receive five pieces and the
Weaver three, the simple Ploughman was ridiculed for suggesting that the
Miller should receive seven and the Weaver only one, while the Carpenter,
the Monk, and the Cook insisted that the money should be divided equally
between the two men. Various other opinions were urged with considerable
vigour, until it was finally decided that the Manciple, as an expert in such
matters, should himself settle the point. His decision was quite correct.
What was it? Of course, all three are supposed to have eaten equal shares of
the bread.

PUZZLING TIMES AT SOLVAMHALL
CASTLE

Everybody that has heard of Solvamhall Castle, and of the quaint customs
and ceremonies that obtained there in the olden times, is familiar with the
fact that Sir Hugh de Fortibus was a lover of all kinds of puzzles and
enigmas. Sir Robert de Riddlesdale himself declared on one occasion, "By
the bones of Saint Jingo, this Sir Hugh hath a sharp wit. Certes, I wot not
the riddle that he may not rede withal." It is, therefore, a source of particular
satisfaction that the recent discovery of some ancient rolls and documents
relating mainly to the family of De Fortibus enables me to place before my
readers a few of the posers that racked people's brains in the good old days.
The selection has been made to suit all tastes, and while the majority will be
found sufficiently easy to interest those who like a puzzle that is a puzzle,
but well within the scope of all, two that I have included may perhaps be
found worthy of engaging the attention of the more advanced student of
these things.

32.—The Game of Bandy-Ball.
Bandy-ball, cambuc, or goff (the game so well known to-day by the name
of golf), is of great antiquity, and was a special favourite at Solvamhall
Castle. Sir Hugh de Fortibus was himself a master of the game, and he once
proposed this question.
They had nine holes, 300, 250, 200, 325, 275, 350, 225, 375, and 400 yards
apart. If a man could always strike the ball in a perfectly straight line and
send it exactly one of two distances, so that it would either go towards the
hole, pass over it, or drop into it, what would the two distances be that
would carry him in the least number of strokes round the whole course?
"Beshrew me," Sir Hugh would say, "if I know any who could do it in this
perfect way; albeit, the point is a pretty one."
Two very good distances are 125 and 75, which carry you round in 28
strokes, but this is not the correct answer. Can the reader get round in fewer
strokes with two other distances?

33.—Tilting at the Ring.
Another favourite sport at the castle was tilting at the ring. A horizontal bar
was fixed in a post, and at the end of a hanging supporter was placed a
circular ring, as shown in the above illustrated title. By raising or lowering
the bar the ring could be adjusted to the proper height—generally about the
level of the left eyebrow of the horseman. The object was to ride swiftly
some eighty paces and run the lance through the ring, which was easily
detached, and remained on the lance as the property of the skilful winner. It
was a very difficult feat, and men were not unnaturally proud of the rings
they had succeeded in capturing.
At one tournament at the castle Henry de Gournay beat Stephen Malet by
six rings. Each had his rings made into a chain—De Gournay's chain being
exactly sixteen inches in length, and Malet's six inches. Now, as the rings
were all of the same size and made of metal half an inch thick, the little
puzzle proposed by Sir Hugh was to discover just how many rings each
man had won.

34.—The Noble Demoiselle.
Seated one night in the hall of the castle, Sir Hugh desired the company to
fill their cups and listen while he told the tale of his adventure as a youth in
rescuing from captivity a noble demoiselle who was languishing in the
dungeon of the castle belonging to his father's greatest enemy. The story
was a thrilling one, and when he related the final escape from all the
dangers and horrors of the great Death's-head Dungeon with the fair but
unconscious maiden in his arms, all exclaimed, "'Twas marvellous valiant!"
But Sir Hugh said, "I would never have turned from my purpose, not even
to save my body from the bernicles."

Sir Hugh then produced a plan of the thirty-five cells in the dungeon and
asked his companions to discover the particular cell that the demoiselle
occupied. He said that if you started at one of the outside cells and passed
through every doorway once, and once only, you were bound to end at the
cell that was sought. Can you find the cell? Unless you start at the correct
outside cell it is impossible to pass through all the doorways once and once
only. Try tracing out the route with your pencil.

35.—The Archery Butt.
The butt or target used in archery at Solvamhall was not marked out in
concentric rings as at the present day, but was prepared in fanciful designs.
In the illustration is shown a numbered target prepared by Sir Hugh himself.
It is something of a curiosity, because it will be found that he has so
cleverly arranged the numbers that every one of the twelve lines of three
adds up to exactly twenty-two.
One day, when the archers were a little tired of their sport, Sir Hugh de
Fortibus said, "What ho, merry archers! Of a truth it is said that a fool's bolt
is soon shot, but, by my faith, I know not any man among you who shall do
that which I will now put forth. Let these numbers that are upon the butt be
set down afresh, so that the twelve lines thereof shall make twenty and three
instead of twenty and two."

To rearrange the numbers one to nineteen so that all the twelve lines shall
add up to twenty-three will be found a fascinating puzzle. Half the lines are,
of course, on the sides, and the others radiate from the centre.

36.—The Donjon Keep Window.
On one occasion Sir Hugh greatly perplexed his chief builder. He took this
worthy man to the walls of the donjon keep and pointed to a window there.
"Methinks," said he, "yon window is square, and measures, on the inside,
one foot every way, and is divided by the narrow bars into four lights,
measuring half a foot on every side."
"Of a truth that is so, Sir Hugh."

"Then I desire that another window be made higher up whose four sides
shall also be each one foot, but it shall be divided by bars into eight lights,
whose sides shall be all equal."
"Truly, Sir Hugh," said the bewildered chief builder, "I know not how it
may be done."

"By my halidame!" exclaimed De Fortibus in pretended rage, "let it be done
forthwith. I trow thou art but a sorry craftsman if thou canst not, forsooth,
set such a window in a keep wall."
It will be noticed that Sir Hugh ignores the thickness of the bars.

37.—The Crescent and the Cross.
When Sir Hugh's kinsman, Sir John de Collingham, came back from the
Holy Land, he brought with him a flag bearing the sign of a crescent, as
shown in the illustration. It was noticed that De Fortibus spent much time in
examining this crescent and comparing it with the cross borne by the
Crusaders on their own banners. One day, in the presence of a goodly
company, he made the following striking announcement:—
"I have thought much of late, friends and masters, of the conversion of the
crescent to the cross, and this has led me to the finding of matters at which I
marvel greatly, for that which I shall now make known is mystical and
deep. Truly it was shown to me in a dream that this crescent of the enemy
may be exactly converted into the cross of our own banner. Herein is a sign
that bodes good for our wars in the Holy Land."

Sir Hugh de Fortibus then explained that the crescent in one banner might
be cut into pieces that would exactly form the perfect cross in the other. It is

certainly rather curious; and I show how the conversion from crescent to
cross may be made in ten pieces, using every part of the crescent. The flag
was alike on both sides, so pieces may be turned over where required.

38.—The Amulet.
A strange man was one day found loitering in the courtyard of the castle,
and the retainers, noticing that his speech had a foreign accent, suspected
him of being a spy. So the fellow was brought before Sir Hugh, who could
make nothing of him. He ordered the varlet to be removed and examined, in
order to discover whether any secret letters were concealed about him. All
they found was a piece of parchment securely suspended from the neck,
bearing this mysterious inscription:—

To-day we know that Abracadabra was the supreme deity of the Assyrians,
and this curious arrangement of the letters of the word was commonly worn
in Europe as an amulet or charm against diseases. But Sir Hugh had never
heard of it, and, regarding the document rather seriously, he sent for a
learned priest.
"I pray you, Sir Clerk," said he, "show me the true intent of this strange
writing."
"Sir Hugh," replied the holy man, after he had spoken in a foreign tongue
with the stranger, "it is but an amulet that this poor wight doth wear upon
his breast to ward off the ague, the toothache, and such other afflictions of
the body."
"Then give the varlet food and raiment and set him on his way," said Sir
Hugh. "Meanwhile, Sir Clerk, canst thou tell me in how many ways this

word 'Abracadabra' may be read on the amulet, always starting from the A
at the top thereof?"
Place your pencil on the A at the top and count in how many different ways
you can trace out the word downwards, always passing from a letter to an
adjoining one.

39.—The Snail on the Flagstaff.
It would often be interesting if we could trace back to their origin many of
the best known puzzles. Some of them would be found to have been first
propounded in very ancient times, and there can be very little doubt that
while a certain number may have improved with age, others will have
deteriorated and even lost their original point and bearing. It is curious to
find in the Solvamhall records our familiar friend the climbing snail puzzle,
and it will be seen that in its modern form it has lost its original subtlety.

On the occasion of some great rejoicings at the Castle, Sir Hugh was
superintending the flying of flags and banners, when somebody pointed out
that a wandering snail was climbing up the flagstaff. One wise old fellow
said:—
"They do say, Sir Knight, albeit I hold such stories as mere fables, that the
snail doth climb upwards three feet in the daytime, but slippeth back two
feet by night."

"Then," replied Sir Hugh, "tell us how many days it will take this snail to
get from the bottom to the top of the pole."
"By bread and water, I much marvel if the same can be done unless we take
down and measure the staff."
"Credit me," replied the knight, "there is no need to measure the staff."
Can the reader give the answer to this version of a puzzle that we all know
so well?

40.—Lady Isabel's Casket.

Sir Hugh's young kinswoman and ward, Lady Isabel de Fitzarnulph, was
known far and wide as "Isabel the Fair." Amongst her treasures was a
casket, the top of which was perfectly square in shape. It was inlaid with
pieces of wood, and a strip of gold ten inches long by a quarter of an inch
wide.
When young men sued for the hand of Lady Isabel, Sir Hugh promised his
consent to the one who would tell him the dimensions of the top of the box
from these facts alone: that there was a rectangular strip of gold, ten inches
by 1/4-inch; and the rest of the surface was exactly inlaid with pieces of
wood, each piece being a perfect square, and no two pieces of the same
size. Many young men failed, but one at length succeeded. The puzzle is not

an easy one, but the dimensions of that strip of gold, combined with those
other conditions, absolutely determine the size of the top of the casket.

THE MERRY MONKS OF RIDDLEWELL
Their Quaint Puzzles and Enigmas.

"Friar Andrew," quoth the Lord Abbot, as he lay a-dying, "methinks I could
now rede thee the riddle of riddles—an I had—the time—and—" The good
friar put his ear close to the holy Abbot's lips, but alas! they were silenced
for ever. Thus passed away the life of the jovial and greatly beloved Abbot
of the old monastery of Riddlewell.
The monks of Riddlewell Abbey were noted in their day for the quaint
enigmas and puzzles that they were in the habit of propounding. The Abbey
was built in the fourteenth century, near a sacred spring known as the Red-
hill Well. This became in the vernacular Reddlewell and Riddlewell, and
under the Lord Abbot David the monks evidently tried to justify the latter
form by the riddles they propounded so well. The solving of puzzles
became the favourite recreation, no matter whether they happened to be of a
metaphysical, philosophical, mathematical, or mechanical kind. It grew into
an absorbing passion with them, and as I have shown above, in the case of
the Abbot this passion was strong even in death.
It would seem that the words "puzzle," "problem," "enigma," etc., did not
occur in their vocabulary. They were accustomed to call every poser a

"riddle," no matter whether it took the form of "Where was Moses when the
light went out?" or the Squaring of the Circle. On one of the walls in the
refectory were inscribed the words of Samson, "I will now put forth a riddle
to you," to remind the brethren of what was expected of them, and the rule
was that each monk in turn should propose some riddle weekly to the
community, the others being always free to cap it with another if disposed
to do so. Abbot David was, undoubtedly, the puzzle genius of the
monastery, and everybody naturally bowed to his decision. Only a few of
the Abbey riddles have been preserved, and I propose to select those that
seem most interesting. I shall try to make the conditions of the puzzles
perfectly clear, so that the modern reader may fully understand them, and be
amused in trying to find some of the solutions.

41.—The Riddle of the Fish-pond.

At the bottom of the Abbey meads was a small fish-pond where the monks
used to spend many a contemplative hour with rod and line. One day, when
they had had very bad luck and only caught twelve fishes amongst them,
Brother Jonathan suddenly declared that as there was no sport that day he
would put forth a riddle for their entertainment. He thereupon took twelve
fish baskets and placed them at equal distances round the pond, as shown in
our illustration, with one fish in each basket.
"Now, gentle anglers," said he, "rede me this riddle of the Twelve Fishes.
Start at any basket you like, and, always going in one direction round the
pond, take up one fish, pass it over two other fishes, and place it in the next
basket. Go on again; take up another single fish, and, having passed that
also over two fishes, place it in a basket; and so continue your journey. Six
fishes only are to be removed, and when these have been placed, there
should be two fishes in each of six baskets, and six baskets empty. Which of

you merry wights will do this in such a manner that you shall go round the
pond as few times as possible?"
I will explain to the reader that it does not matter whether the two fishes
that are passed over are in one or two baskets, nor how many empty baskets
you pass. And, as Brother Jonathan said, you must always go in one
direction round the pond (without any doubling back) and end at the spot
from which you set out.

42.—The Riddle of the Pilgrims.
One day, when the monks were seated at their repast, the Abbot announced
that a messenger had that morning brought news that a number of pilgrims
were on the road and would require their hospitality.
"You will put them," he said, "in the square dormitory that has two floors
with eight rooms on each floor. There must be eleven persons sleeping on
each side of the building, and twice as many on the upper floor as on the
lower floor. Of course every room must be occupied, and you know my rule
that not more than three persons may occupy the same room."
I give a plan of the two floors, from which it will be seen that the sixteen
rooms are approached by a well staircase in the centre. After the monks had
solved this little problem and arranged for the accommodation, the pilgrims
arrived, when it was found that they were three more in number than was at
first stated. This necessitated a reconsideration of the question, but the wily
monks succeeded in getting over the new difficulty without breaking the
Abbot's rules. The curious point of this puzzle is to discover the total
number of pilgrims.
Plan of Dormitory.

43.—The Riddle of the Tiled Hearth.
It seems that it was Friar Andrew who first managed to "rede the riddle of
the Tiled Hearth." Yet it was a simple enough little puzzle. The square
hearth, where they burnt their Yule logs and round which they had such
merry carousings, was floored with sixteen large ornamental tiles. When
these became cracked and burnt with the heat of the great fire, it was
decided to put down new tiles, which had to be selected from four different
patterns (the Cross, the Fleur-de-lys, the Lion, and the Star); but plain tiles
were also available. The Abbot proposed that they should be laid as shown
in our sketch, without any plain tiles at all; but Brother Richard broke in,—
"I trow, my Lord Abbot, that a riddle is required of me this day. Listen,
then, to that which I shall put forth. Let these sixteen tiles be so placed that
no tile shall be in line with another of the same design"—(he meant, of
course, not in line horizontally, vertically, or diagonally)—"and in such
manner that as few plain tiles as possible be required." When the monks
handed in their plans it was found that only Friar Andrew had hit upon the
correct answer, even Friar Richard himself being wrong. All had used too
many plain tiles.

44.—The Riddle of the Sack Wine.
One evening, when seated at table, Brother Benjamin was called upon by
the Abbot to give the riddle that was that day demanded of him.
"Forsooth," said he, "I am no good at the making of riddles, as thou
knowest full well; but I have been teasing my poor brain over a matter that I
trust some among you will expound to me, for I cannot rede it myself. It is
this. Mark me take a glass of sack from this bottle that contains a pint of
wine and pour it into that jug which contains a pint of water. Now, I fill the
glass with the mixture from the jug and pour it back into the bottle holding
the sack. Pray tell me, have I taken more wine from the bottle than water
from the jug? Or have I taken more water from the jug than wine from the
bottle?"
I gather that the monks got nearer to a great quarrel over this little poser
than had ever happened before. One brother so far forgot himself as to tell
his neighbour that "more wine had got into his pate than wit came out of it,"
while another noisily insisted that it all depended on the shape of the glass
and the age of the wine. But the Lord Abbot intervened, showed them what
a simple question it really was, and restored good feeling all round.

45.—The Riddle of the Cellarer.
Then Abbot David looked grave, and said that this incident brought to his
mind the painful fact that John the Cellarer had been caught robbing the
cask of best Malvoisie that was reserved for special occasions. He ordered
him to be brought in.

"Now, varlet," said the Abbot, as the ruddy-faced Cellarer came before him,
"thou knowest that thou wast taken this morning in the act of stealing good
wine that was forbidden thee. What hast thou to say for thyself?"
"Prithee, my Lord Abbot, forgive me!" he cried, falling on his knees. "Of a
truth, the Evil One did come and tempt me, and the cask was so handy, and
the wine was so good withal, and—and I had drunk of it ofttimes without
being found out, and—"
"Rascal! that but maketh thy fault the worse! How much wine hast thou
taken?"

"Alack-a-day! There were a hundred pints in the cask at the start, and I have
taken me a pint every day this month of June—it being to-day the thirtieth
thereof—and if my Lord Abbot can tell me to a nicety how much good wine
I have taken in all, let him punish me as he will."
"Why, knave, that is thirty pints."
"Nay, nay; for each time I drew a pint out of the cask, I put in a pint of
water in its stead!"
It is a curious fact that this is the only riddle in the old record that is not
accompanied by its solution. Is it possible that it proved too hard a nut for
the monks? There is merely the note, "John suffered no punishment for his
sad fault."

46.—The Riddle of the Crusaders.
On another occasion a certain knight, Sir Ralph de Bohun, was a guest of
the monks at Riddlewell Abbey. Towards the close of a sumptuous repast he
spoke as follows:—
"My Lord Abbot, knowing full well that riddles are greatly to thy liking, I
will, by your leave, put forth one that was told unto me in foreign lands. A
body of Crusaders went forth to fight the good cause, and such was their
number that they were able to form themselves into a square. But on the
way a stranger took up arms and joined them, and they were then able to
form exactly thirteen smaller squares. Pray tell me, merry monks, how
many men went forth to battle?"
Abbot David pushed aside his plate of warden pie, and made a few hasty
calculations.

"Sir Knight," said he at length, "the riddle is easy to rede. In the first place
there were 324 men, who would make a square 18 by 18, and afterwards
325 men would make 13 squares of 25 Crusaders each. But which of you

can tell me how many men there would have been if, instead of 13, they had
been able to form 113 squares under exactly the like conditions?"
The monks gave up this riddle, but the Abbot showed them the answer next
morning.

47.—The Riddle of St. Edmondsbury.
"It used to be told at St. Edmondsbury," said Father Peter on one occasion,
"that many years ago they were so overrun with mice that the good abbot
gave orders that all the cats from the country round should be obtained to
exterminate the vermin. A record was kept, and at the end of the year it was
found that every cat had killed an equal number of mice, and the total was
exactly 1,111,111 mice. How many cats do you suppose there were?"
"Methinks one cat killed the lot," said Brother Benjamin.
"Out upon thee, brother! I said 'cats.'"
"Well, then," persisted Benjamin, "perchance 1,111,111 cats each killed one
mouse."
"No," replied Father Peter, after the monks' jovial laughter had ended, "I
said 'mice;' and all I need add is this—that each cat killed more mice than
there were cats. They told me it was merely a question of the division of
numbers, but I know not the answer to the riddle."
The correct answer is recorded, but it is not shown how they arrived at it.

48.—The Riddle of the Frogs' Ring.
One Christmas the Abbot offered a prize of a large black jack mounted in
silver, to be engraved with the name of the monk who should put forth the
best new riddle. This tournament of wit was won by Brother Benedict, who,
curiously enough, never before or after gave out anything that did not excite
the ridicule of his brethren. It was called the "Frogs' Ring."
A ring was made with chalk on the floor of the hall, and divided into
thirteen compartments, in which twelve discs of wood (called "frogs") were
placed in the order shown in our illustration, one place being left vacant.
The numbers 1 to 6 were painted white and the numbers 7 to 12 black. The
puzzle was to get all the white numbers where the black ones were, and vice
versa. The white frogs move round in one direction, and the black ones the
opposite way. They may move in any order one step at a time, or jumping
over one of the opposite colour to the place beyond, just as we play
draughts to-day. The only other condition is that when all the frogs have
changed sides, the 1 must be where the 12 now is and the 12 in the place
now occupied by 1. The puzzle was to perform the feat in as few moves as
possible. How many moves are necessary?
I will conclude in the words of the old writer: "These be some of the riddles
which the monks of Riddlewell did set forth and expound each to the others
in the merry days of the good Abbot David."

THE STRANGE ESCAPE OF THE KING'S
JESTER.
A PUZZLING ADVENTURE.
At one time I was greatly in favour with the king, and his Majesty never
seemed to weary of the companionship of the court fool. I had a gift for
making riddles and quaint puzzles which ofttimes caused great sport; for
albeit the king never found the right answer of one of these things in all his
life, yet would he make merry at the bewilderment of those about him.
But let every cobbler stick unto his last; for when I did set out to learn the
art of performing strange tricks in the magic, wherein the hand doth ever
deceive the eye, the king was affrighted, and did accuse me of being a
wizard, even commanding that I should be put to death. Luckily my wit did
save my life. I begged that I might be slain by the royal hand and not by
that of the executioner.
"By the saints," said his Majesty, "what difference can it make unto thee?
But since it is thy wish, thou shalt have thy choice whether I kill thee or the
executioner."
"Your Majesty," I answered, "I accept the choice that thou hast so
graciously offered to me: I prefer that your Majesty should kill the
executioner."
Yet is the life of a royal jester beset with great dangers, and the king having
once gotten it into his royal head that I was a wizard, it was not long before
I again fell into trouble, from which my wit did not a second time in a like
way save me. I was cast into the dungeon to await my death. How, by the
help of my gift in answering riddles and puzzles, I did escape from captivity
I will now set forth; and in case it doth perplex any to know how some of
the strange feats were performed, I will hereafter make the manner thereof
plain to all.

49.—The Mysterious Rope.
My dungeon did not lie beneath the moat, but was
in one of the most high parts of the castle. So stout
was the door, and so well locked and secured
withal, that escape that way was not to be found.
By hard work I did, after many days, remove one
of the bars from the narrow window, and was able
to crush my body through the opening; but the
distance to the courtyard below was so exceeding
great that it was certain death to drop thereto. Yet
by great good fortune did I find in the corner of the
cell a rope that had been there left and lay hid in
the great darkness. But this rope had not length
enough, and to drop in safety from the end was
nowise possible. Then did I remember how the
wise man from Ireland did lengthen the blanket
that was too short for him by cutting a yard off the
bottom of the same and joining it on to the top. So
I made haste to divide the rope in half and to tie
the two parts thereof together again. It was then
full long, and did reach the ground, and I went
down in safety. How could this have been?

50.—The Underground Maze.
The only way out of the yard that I now was in was to descend a few stairs
that led up into the centre (A) of an underground maze, through the winding
of which I must pass before I could take my leave by the door (B). But I
knew full well that in the great darkness of this dreadful place I might well
wander for hours and yet return to the place from which I set out. How was
I then to reach the door with certainty? With a plan of the maze it is but a
simple matter to trace out the route, but how was the way to be found in the
place itself in utter darkness?

51.—The Secret Lock.
When I did at last reach the door it was fast closed, and on sliding a panel
set before a grating the light that came in thereby showed unto me that my
passage was barred by the king's secret lock. Before the handle of the door
might be turned, it was needful to place the hands of three several dials in
their proper places. If you but knew the proper letter for each dial, the secret
was of a truth to your hand; but as ten letters were upon the face of every
dial, you might try nine hundred and ninety-nine times and only succeed on
the thousandth attempt withal. If I was indeed to escape I must waste not a
moment.
Now, once had I heard the learned monk who did invent the lock say that he
feared that the king's servants, having such bad memories, would mayhap
forget the right letters; so perchance, thought I, he had on this account
devised some way to aid their memories. And what more natural than to
make the letters form some word? I soon found a word that was English,
made of three letters—one letter being on each of the three dials. After that
I had pointed the hands properly to the letters the door opened and I passed
out. What was the secret word?

52.—Crossing the Moat.
I was now face to face with the castle moat, which was, indeed, very wide
and very deep. Alas! I could not swim, and my chance of escape seemed of
a truth hopeless, as, doubtless, it would have been had I not espied a boat
tied to the wall by a rope. But after I had got into it I did find that the oars
had been taken away, and that there was nothing that I could use to row me
across. When I had untied the rope and pushed off upon the water the boat
lay quite still, there being no stream or current to help me. How, then, did I
yet take the boat across the moat?

53.—The Royal Gardens.
It was now daylight, and still had I to pass through the royal gardens outside
of the castle walls. These gardens had once been laid out by an old king's
gardener, who had become bereft of his senses, but was allowed to amuse
himself therein. They were square, and divided into 16 parts by high walls,
as shown in the plan thereof, so that there were openings from one garden
to another, but only two different ways of entrance. Now, it was needful that
I enter at the gate A and leave by the other gate B; but as there were
gardeners going and coming about their work, I had to slip with agility from
one garden to another, so that I might not be seen, but escape unobserved. I
did succeed in so doing, but afterwards remembered that I had of a truth
entered every one of the 16 gardens once, and never more than once. This
was, indeed, a curious thing. How might it have been done?

54.—Bridging the Ditch.
I now did truly think that at last was I a free man, but I had quite forgot that
I must yet cross a deep ditch before I might get right away. This ditch was
10 feet wide, and I durst not attempt to jump it, as I had sprained an ankle in
leaving the garden. Looking around for something to help me over my
difficulty, I soon found eight narrow planks of wood lying together in a
heap. With these alone, and the planks were each no more than 9 feet long, I
did at last manage to make a bridge across the ditch. How was this done?

Being now free I did hasten to the house of a friend who provided me with
a horse and a disguise, with which I soon succeeded in placing myself out
of all fear of capture.
Through the goodly offices of divers persons at the king's court I did at
length obtain the royal pardon, though, indeed, I was never restored to that
full favour that was once my joy and pride.
Ofttimes have I been asked by many that do know me to set forth to them
the strange manner of my escape, which more than one hath deemed to be
of a truth wonderful, albeit the feat was nothing astonishing withal if we do
but remember that from my youth upwards I had trained my wit to the

making and answering of cunning enigmas. And I do hold that the study of
such crafty matters is good, not alone for the pleasure that is created
thereby, but because a man may never be sure that in some sudden and
untoward difficulty that may beset him in passing through this life of ours
such strange learning may not serve his ends greatly, and, mayhap, help him
out of many difficulties.
I am now an aged man, and have not quite lost all my taste for quaint
puzzles and conceits; but, of a truth, never have I found greater pleasure in
making out the answers to any of these things than I had in mastering them
that did enable me, as the king's jester in disgrace, to gain my freedom from
the castle dungeon and so save my life.

THE SQUIRE'S CHRISTMAS PUZZLE PARTY
A fine specimen of the old English country gentleman was Squire Davidge,
of Stoke Courcy Hall, in Somerset. When the last century was yet in its
youth, there were few men in the west country more widely known and
more generally respected and beloved than he. A born sportsman, his fame
extended to Exmoor itself, where his daring and splendid riding in pursuit
of the red deer had excited the admiration and envy of innumerable younger
huntsmen. But it was in his own parish, and particularly in his own home,
that his genial hospitality, generosity, and rare jovial humour made him the
idol of his friends—and even of his relations, which sometimes means a
good deal.
At Christmas it was always an open house at Stoke Courcy Hall, for if there
was one thing more than another upon which Squire Davidge had very
pronounced views, it was on the question of keeping up in a royal fashion
the great festival of Yule-tide. "Hark ye, my lads," he would say to his sons:
"our country will begin to fall on evil days if ever we grow indifferent to the
claims of those Christmas festivities that have helped to win us the proud
name of Merrie England." Therefore, when I say that Christmas at Stoke
Courcy was kept up in the good old happy, rollicking, festive style that our
grandfathers and great-grandfathers so dearly loved, it will be unnecessary
for me to attempt a description. We have a faithful picture of these merry
scenes in the Bracebridge Hall of Washington Irving. I must confine myself
in this sketch to one special feature in the Squire's round of jollification
during the season of peace and good will.
He took a curious and intelligent interest in puzzles of every kind, and there
was always one night devoted to what was known as "Squire Davidge's
Puzzle Party." Every guest was expected to come armed with some riddle or
puzzle for the bewilderment and possible delectation of the company. The
old gentleman always presented a new watch to the guest who was most
successful in his answers. It is a pity that all the puzzles were not preserved;
but I propose to present to my readers a few selected from a number that
have passed down to a surviving member of the family, who has kindly

allowed me to use them on this occasion. There are some very easy ones, a
few that are moderately difficult, and one hard brain-racker, so all should be
able to find something to their taste.
The little record is written in the neat angular hand of a young lady of that
day, and the puzzles, the conditions of which I think it best to give mainly
in my own words for the sake of greater clearness, appear to have been all
propounded on one occasion.

55.—The Three Teacups.

One young lady—of whom our fair historian records with delightful
inconsequence: "This Miss Charity Lockyer has since been married to a
curate from Taunton Vale"—placed three empty teacups on a table, and
challenged anybody to put ten lumps of sugar in them so that there would
be an odd number of lumps in every cup. "One young man, who has been to
Oxford University, and is studying the law, declared with some heat that,
beyond a doubt, there was no possible way of doing it, and he offered to
give proof of the fact to the company." It must have been interesting to see
his face when he was shown Miss Charity's correct answer.

56.—The Eleven Pennies.

A guest asked some one to favour him with eleven pennies, and he passed
the coins to the company, as depicted in our illustration. The writer says:
"He then requested us to remove five coins from the eleven, add four coins
and leave nine. We could not but think there must needs be ten pennies left.
We were a good deal amused at the answer hereof."

57.—The Christmas Geese.
Squire Hembrow, from Weston Zoyland—wherever that may be—proposed
the following little arithmetical puzzle, from which it is probable that
several somewhat similar modern ones have been derived: Farmer Rouse
sent his man to market with a flock of geese, telling him that he might sell
all or any of them, as he considered best, for he was sure the man knew how
to make a good bargain. This is the report that Jabez made, though I have
taken it out of the old Somerset dialect, which might puzzle some readers in
a way not desired. "Well, first of all I sold Mr. Jasper Tyler half of the flock
and half a goose over; then I sold Farmer Avent a third of what remained
and a third of a goose over; then I sold Widow Foster a quarter of what
remained and three-quarters of a goose over; and as I was coming home,
whom should I meet but Ned Collier: so we had a mug of cider together at
the Barley Mow, where I sold him exactly a fifth of what I had left, and
gave him a fifth of a goose over for the missus. These nineteen that I have
brought back I couldn't get rid of at any price." Now, how many geese did
Farmer Rouse send to market? My humane readers may be relieved to
know that no goose was divided or put to any inconvenience whatever by
the sales.

58.—The Chalked Numbers.

"We laughed greatly at a pretty jest on the part of Major Trenchard, a merry
friend of the Squire's. With a piece of chalk he marked a different number
on the backs of eight lads who were at the party." Then, it seems, he divided
them in two groups, as shown in the illustration, 1, 2, 3, 4 being on one
side, and 5, 7, 8, 9 on the other. It will be seen that the numbers of the left-
hand group add up to 10, while the numbers in the other group add up to 29.
The Major's puzzle was to rearrange the eight boys in two new groups, so
that the four numbers in each group should add up alike. The Squire's niece
asked if the 5 should not be a 6; but the Major explained that the numbers
were quite correct if properly regarded.

59.—Tasting the Plum Puddings.

"Everybody, as I suppose, knows well that the number of different
Christmas plum puddings that you taste will bring you the same number of
lucky days in the new year. One of the guests (and his name has escaped my
memory) brought with him a sheet of paper on which were drawn sixty-four
puddings, and he said the puzzle was an allegory of a sort, and he intended
to show how we might manage our pudding-tasting with as much dispatch
as possible." I fail to fully understand this fanciful and rather overstrained
view of the puzzle. But it would appear that the puddings were arranged
regularly, as I have shown them in the illustration, and that to strike out a
pudding was to indicate that it had been duly tasted. You have simply to put

the point of your pencil on the pudding in the top corner, bearing a sprig of
holly, and strike out all the sixty-four puddings through their centres in
twenty-one straight strokes. You can go up or down or horizontally, but not
diagonally or obliquely; and you must never strike out a pudding twice, as
that would imply a second and unnecessary tasting of those indigestible
dainties. But the peculiar part of the thing is that you are required to taste
the pudding that is seen steaming hot at the end of your tenth stroke, and to
taste the one decked with holly in the bottom row the very last of all.

60.—Under the Mistletoe Bough.
"At the party was a widower who has but lately come into these parts," says
the record; "and, to be sure, he was an exceedingly melancholy man, for he
did sit away from the company during the most part of the evening. We
afterwards heard that he had been keeping a secret account of all the kisses
that were given and received under the mistletoe bough. Truly, I would not
have suffered any one to kiss me in that manner had I known that so unfair
a watch was being kept. Other maids beside were in a like way shocked, as
Betty Marchant has since told me." But it seems that the melancholy
widower was merely collecting material for the following little osculatory
problem.
The company consisted of the Squire and his wife and six other married
couples, one widower and three widows, twelve bachelors and boys, and
ten maidens and little girls. Now, everybody was found to have kissed
everybody else, with the following exceptions and additions: No male, of
course, kissed a male. No married man kissed a married woman, except his
own wife. All the bachelors and boys kissed all the maidens and girls twice.
The widower did not kiss anybody, and the widows did not kiss each other.
The puzzle was to ascertain just how many kisses had been thus given
under the mistletoe bough, assuming, as it is charitable to do, that every kiss
was returned—the double act being counted as one kiss.

61.—The Silver Cubes.
The last extract that I will give is one that will, I think, interest those readers
who may find some of the above puzzles too easy. It is a hard nut, and
should only be attempted by those who flatter themselves that they possess
strong intellectual teeth.
"Master Herbert Spearing, the son of a widow lady in our parish, proposed
a puzzle in arithmetic that looks simple, but nobody present was able to
solve it. Of a truth I did not venture to attempt it myself, after the young
lawyer from Oxford, who they say is very learned in the mathematics and a
great scholar, failed to show us the answer. He did assure us that he
believed it could not be done, but I have since been told that it is possible,
though, of a certainty, I may not vouch for it. Master Herbert brought with
him two cubes of solid silver that belonged to his mother. He showed that,
as they measured two inches every way, each contained eight cubic inches
of silver, and therefore the two contained together sixteen cubic inches.
That which he wanted to know was—'Could anybody give him exact
dimensions for two cubes that should together contain just seventeen cubic
inches of silver?'" Of course the cubes may be of different sizes.

The idea of a Christmas Puzzle Party, as devised by the old Squire, seems to
have been excellent, and it might well be revived at the present day by
people who are fond of puzzles and who have grown tired of Book Teas and
similar recent introductions for the amusement of evening parties. Prizes

could be awarded to the best solvers of the puzzles propounded by the
guests.

Adventures of the Puzzle Club

When it recently became known that the bewildering mystery of the Prince
and the Lost Balloon was really solved by the members of the Puzzle Club,
the general public was quite unaware that any such club existed. The fact is
that the members always deprecated publicity; but since they have been
dragged into the light in connection with this celebrated case, so many
absurd and untrue stories have become current respecting their doings that I
have been permitted to publish a correct account of some of their more
interesting achievements. It was, however, decided that the real names of
the members should not be given.
The club was started a few years ago to bring together those interested in
the solution of puzzles of all kinds, and it contains some of the profoundest
mathematicians and some of the most subtle thinkers resident in London.
These have done some excellent work of a high and dry kind. But the main
body soon took to investigating the problems of real life that are perpetually
cropping up.
It is only right to say that they take no interest in crimes as such, but only
investigate a case when it possesses features of a distinctly puzzling
character. They seek perplexity for its own sake—something to unravel. As
often as not the circumstances are of no importance to anybody, but they
just form a little puzzle in real life, and that is sufficient.

62.—The Ambiguous Photograph.
A good example of the lighter kind of problem that occasionally comes
before them is that which is known amongst them by the name of "The
Ambiguous Photograph." Though it is perplexing to the inexperienced, it is
regarded in the club as quite a trivial thing. Yet it serves to show the close
observation of these sharp-witted fellows. The original photograph hangs on
the club wall, and has baffled every guest who has examined it. Yet any
child should be able to solve the mystery. I will give the reader an
opportunity of trying his wits at it.
Some of the members were one evening seated together in their clubhouse
in the Adelphi. Those present were: Henry Melville, a barrister not
overburdened with briefs, who was discussing a problem with Ernest
Russell, a bearded man of middle age, who held some easy post in Somerset
House, and was a Senior Wrangler and one of the most subtle thinkers of
the club; Fred Wilson, a journalist of very buoyant spirits, who had more
real capacity than one would at first suspect; John Macdonald, a Scotsman,
whose record was that he had never solved a puzzle himself since the club
was formed, though frequently he had put others on the track of a deep
solution; Tim Churton, a bank clerk, full of cranky, unorthodox ideas as to
perpetual motion; also Harold Tomkins, a prosperous accountant,
remarkably familiar with the elegant branch of mathematics—the theory of
numbers.
Suddenly Herbert Baynes entered the room, and everybody saw at once
from his face that he had something interesting to communicate. Baynes
was a man of private means, with no occupation.
"Here's a quaint little poser for you all," said Baynes. "I have received it to-
day from Dovey."
Dovey was proprietor of one of the many private detective agencies that
found it to their advantage to keep in touch with the club.

"Is it another of those easy cryptograms?" asked Wilson. "If so, I would
suggest sending it upstairs to the billiard-marker."
"Don't be sarcastic, Wilson," said Melville. "Remember, we are indebted to
Dovey for the great Railway Signal Problem that gave us all a week's
amusement in the solving."
"If you fellows want to hear," resumed Baynes, "just try to keep quiet while
I relate the amusing affair to you. You all know of the jealous little Yankee
who married Lord Marksford two years ago? Lady Marksford and her
husband have been in Paris for two or three months. Well, the poor creature
soon got under the influence of the green-eyed monster, and formed the
opinion that Lord Marksford was flirting with other ladies of his
acquaintance.
"Now, she has actually put one of Dovey's spies on to that excellent
husband of hers; and the myrmidon has been shadowing him about for a
fortnight with a pocket camera. A few days ago he came to Lady Marksford
in great glee. He had snapshotted his lordship while actually walking in the
public streets with a lady who was not his wife."
"'What is the use of this at all?' asked the jealous woman.
"'Well, it is evidence, your ladyship, that your husband was walking with
the lady. I know where she is staying, and in a few days shall have found
out all about her.'
"'But, you stupid man,' cried her ladyship, in tones of great contempt, 'how
can any one swear that this is his lordship, when the greater part of him,
including his head and shoulders, is hidden from sight? And—and'—she
scrutinized the photo carefully—'why, I guess it is impossible from this
photograph to say whether the gentleman is walking with the lady or going
in the opposite direction!'
"Thereupon she dismissed the detective in high dudgeon. Dovey has
himself just returned from Paris, and got this account of the incident from
her ladyship. He wants to justify his man, if possible, by showing that the
photo does disclose which way the man is going. Here it is. See what you
fellows can make of it."

Our illustration is a faithful drawing made from the original photograph. It
will be seen that a slight but sudden summer shower is the real cause of the
difficulty.
All agreed that Lady Marksford was right—that it is impossible to
determine whether the man is walking with the lady or not.
"Her ladyship is wrong," said Baynes, after everybody had made a close
scrutiny. "I find there is important evidence in the picture. Look at it
carefully."

"Of course," said Melville, "we can tell nothing from the frock-coat. It may
be the front or the tails. Blessed if I can say! Then he has his overcoat over
his arm, but which way his arm goes it is impossible to see."
"How about the bend of the legs?" asked Churton.

"Bend! why, there isn't any bend," put in Wilson, as he glanced over the
other's shoulder. "From the picture you might suspect that his lordship has
no knees. The fellow took his snapshot just when the legs happened to be
perfectly straight."

"I'm thinking that perhaps——" began Macdonald, adjusting his eye-
glasses.
"Don't think, Mac," advised Wilson. "It might hurt you. Besides, it is no use
you thinking that if the dog would kindly pass on things would be easy. He
won't."
"The man's general pose seems to me to imply movement to the left,"
Tomkins thought.
"On the contrary," Melville declared, "it appears to me clearly to suggest
movement to the right."
"Now, look here, you men," said Russell, whose opinions always carried
respect in the club. "It strikes me that what we have to do is to consider the
attitude of the lady rather than that of the man. Does her attention seem to
be directed to somebody by her side?"

Everybody agreed that it was impossible to say.
"I've got it!" shouted Wilson. "Extraordinary that none of you have seen it.
It is as clear as possible. It all came to me in a flash!"
"Well, what is it?" asked Baynes.
"Why, it is perfectly obvious. You see which way the dog is going—to the
left. Very well. Now, Baynes, to whom does the dog belong?"
"To the detective!"
The laughter against Wilson that followed this announcement was simply
boisterous, and so prolonged that Russell, who had at the time possession of
the photo, seized the opportunity for making a most minute examination of
it. In a few moments he held up his hands to invoke silence.
"Baynes is right," he said. "There is important evidence there which settles
the matter with certainty. Assuming that the gentleman is really Lord
Marksford—and the figure, so far as it is visible, is his—I have no
hesitation myself in saying that—"
"Stop!" all the members shouted at once.
"Don't break the rules of the club, Russell, though Wilson did," said
Melville. "Recollect that 'no member shall openly disclose his solution to a
puzzle unless all present consent.'"
"You need not have been alarmed," explained Russell. "I was simply going
to say that I have no hesitation in declaring that Lord Marksford is walking
in one particular direction. In which direction I will tell you when you have
all 'given it up.'"

63.—The Cornish Cliff Mystery.
Though the incident known in the Club as "The Cornish Cliff Mystery" has
never been published, every one remembers the case with which it was
connected—an embezzlement at Todd's Bank in Cornhill a few years ago.
Lamson and Marsh, two of the firm's clerks, suddenly disappeared; and it
was found that they had absconded with a very large sum of money. There
was an exciting hunt for them by the police, who were so prompt in their
action that it was impossible for the thieves to get out of the country. They
were traced as far as Truro, and were known to be in hiding in Cornwall.
Just at this time it happened that Henry Melville and Fred Wilson were
away together on a walking tour round the Cornish coast. Like most people,
they were interested in the case; and one morning, while at breakfast at a
little inn, they learnt that the absconding men had been tracked to that very
neighbourhood, and that a strong cordon of police had been drawn round
the district, making an escape very improbable. In fact, an inspector and a
constable came into the inn to make some inquiries, and exchanged
civilities with the two members of the Puzzle Club. A few references to
some of the leading London detectives, and the production of a confidential
letter Melville happened to have in his pocket from one of them, soon
established complete confidence, and the inspector opened out.
He said that he had just been to examine a very important clue a quarter of a
mile from there, and expressed the opinion that Messrs. Lamson and Marsh
would never again be found alive. At the suggestion of Melville the four
men walked along the road together.
"There is our stile in the distance," said the inspector. "This constable found
beside it the pocket-book that I have shown you, containing the name of
Marsh and some memoranda in his handwriting. It had evidently been
dropped by accident. On looking over the stone stile he noticed the
footprints of two men—which I have already proved from particulars
previously supplied to the police to be those of the men we want—and I am
sure you will agree that they point to only one possible conclusion."

Arrived at the spot, they left the hard road and got over the stile. The
footprints of the two men were here very clearly impressed in the thin but
soft soil, and they all took care not to trample on the tracks. They followed
the prints closely, and found that they led straight to the edge of a cliff
forming a sheer precipice, almost perpendicular, at the foot of which the
sea, some two hundred feet below, was breaking among the boulders.

"Here, gentlemen, you see," said the inspector, "that the footprints lead
straight to the edge of the cliff, where there is a good deal of trampling
about, and there end. The soil has nowhere been disturbed for yards around,
except by the footprints that you see. The conclusion is obvious."
"That, knowing they were unable to escape capture, they decided not to be
taken alive, and threw themselves over the cliff?" asked Wilson.
"Exactly. Look to the right and the left, and you will find no footprints or
other marks anywhere. Go round there to the left, and you will be satisfied
that the most experienced mountaineer that ever lived could not make a
descent, or even anywhere get over the edge of the cliff. There is no ledge
or foothold within fifty feet."

"Utterly impossible," said Melville, after an inspection. "What do you
propose to do?"
"I am going straight back to communicate the discovery to headquarters.
We shall withdraw the cordon and search the coast for the dead bodies."
"Then you will make a fatal mistake," said Melville. "The men are alive and
in hiding in the district. Just examine the prints again. Whose is the large
foot?"
"That is Lamson's, and the small print is Marsh's. Lamson was a tall man,
just over six feet, and Marsh was a little fellow."
"I thought as much," said Melville. "And yet you will find that Lamson
takes a shorter stride than Marsh. Notice, also, the peculiarity that Marsh
walks heavily on his heels, while Lamson treads more on his toes. Nothing
remarkable in that? Perhaps not; but has it occurred to you that Lamson
walked behind Marsh? Because you will find that he sometimes treads over
Marsh's footsteps, though you will never find Marsh treading in the steps of
the other."
"Do you suppose that the men walked backwards in their own footprints?"
asked the inspector.
"No; that is impossible. No two men could walk backwards some two
hundred yards in that way with such exactitude. You will not find a single
place where they have missed the print by even an eighth of an inch. Quite
impossible. Nor do I suppose that two men, hunted as they were, could have
provided themselves with flying-machines, balloons, or even parachutes.
They did not drop over the cliff."
Melville then explained how the men had got away. His account proved to
be quite correct, for it will be remembered that they were caught, hiding
under some straw in a barn, within two miles of the spot. How did they get
away from the edge of the cliff?

64.—The Runaway Motor-Car.
The little affair of the "Runaway Motor-car" is a good illustration of how a
knowledge of some branch of puzzledom may be put to unexpected use. A
member of the Club, whose name I have at the moment of writing
forgotten, came in one night and said that a friend of his was bicycling in
Surrey on the previous day, when a motor-car came from behind, round a
corner, at a terrific speed, caught one of his wheels, and sent him flying in
the road. He was badly knocked about, and fractured his left arm, while his
machine was wrecked. The motor-car was not stopped, and he had been
unable to trace it.
There were two witnesses to the accident, which was beyond question the
fault of the driver of the car. An old woman, a Mrs. Wadey, saw the whole
thing, and tried to take the number of the car. She was positive as to the
letters, which need not be given, and was certain also that the first figure
was a 1. The other figures she failed to read on account of the speed and
dust.
The other witness was the village simpleton, who just escapes being an
arithmetical genius, but is excessively stupid in everything else.
He is always working out sums in his head; and all he could say was that
there were five figures in the number, and that he found that when he
multiplied the first two figures by the last three they made the same figures,
only in different order—just as 24 multiplied by 651 makes 15,624 (the
same five figures), in which case the number of the car would have been
24,651; and he knew there was no 0 in the number.
"It will be easy enough to find that car," said Russell. "The known facts are
possibly sufficient to enable one to discover the exact number. You see,
there must be a limit to the five-figure numbers having the peculiarity
observed by the simpleton. And these are further limited by the fact that, as
Mrs. Wadey states, the number began with the figure 1. We have therefore
to find these numbers. It may conceivably happen that there is only one

such number, in which case the thing is solved. But even if there are several
cases, the owner of the actual car may easily be found.
"How will you manage that?" somebody asked.
"Surely," replied Russell, "the method is quite obvious. By the process of
elimination. Every owner except the one in fault will be able to prove an
alibi. Yet, merely guessing offhand, I think it quite probable that there is
only one number that fits the case. We shall see."

Russell was right, for that very night he sent the number by post, with the
result that the runaway car was at once traced, and its owner, who was
himself driving, had to pay the cost of the damages resulting from his
carelessness. What was the number of the car?

65.—The Mystery of Ravensdene Park.
The mystery of Ravensdene Park, which I will now present, was a tragic
affair, as it involved the assassination of Mr. Cyril Hastings at his country
house a short distance from London.
On February 17th, at 11 p.m., there was a heavy fall of snow, and though it
lasted only half an hour, the ground was covered to a depth of several
inches. Mr. Hastings had been spending the evening at the house of a
neighbour, and left at midnight to walk home, taking the short route that lay
through Ravensdene Park—that is, from D to A in the sketch-plan. But in
the early morning he was found dead, at the point indicated by the star in
our diagram, stabbed to the heart. All the seven gates were promptly closed,
and the footprints in the snow examined. These were fortunately very
distinct, and the police obtained the following facts:—

The footprints of Mr. Hastings were very clear, straight from D to the spot
where he was found. There were the footprints of the Ravensdene butler—

who retired to bed five minutes before midnight—from E to EE. There were
the footprints of the gamekeeper from A to his lodge at AA. Other footprints
showed that one individual had come in at gate B and left at gate BB, while
another had entered by gate C and left at gate CC.
Only these five persons had entered the park since the fall of snow. Now, it
was a very foggy night, and some of these pedestrians had consequently
taken circuitous routes, but it was particularly noticed that no track ever
crossed another track. Of this the police were absolutely certain, but they
stupidly omitted to make a sketch of the various routes before the snow had
melted and utterly effaced them.

The mystery was brought before the members of the Puzzle Club, who at
once set themselves the task of solving it. Was it possible to discover who
committed the crime? Was it the butler? Or the gamekeeper? Or the man
who came in at B and went out at BB? Or the man who went in at C and left
at CC? They provided themselves with diagrams—sketch-plans, like the
one we have reproduced, which simplified the real form of Ravensdene
Park without destroying the necessary conditions of the problem.
Our friends then proceeded to trace out the route of each person, in
accordance with the positive statements of the police that we have given. It
was soon evident that, as no path ever crossed another, some of the
pedestrians must have lost their way considerably in the fog. But when the
tracks were recorded in all possible ways, they had no difficulty in deciding

on the assassin's route; and as the police luckily knew whose footprints this
route represented, an arrest was made that led to the man's conviction.
Can our readers discover whether A, B, C, or E committed the deed? Just
trace out the route of each of the four persons, and the key to the mystery
will reveal itself.

66.—The Buried Treasure.
The problem of the Buried Treasure was of quite a different character. A
young fellow named Dawkins, just home from Australia, was introduced to
the club by one of the members, in order that he might relate an
extraordinary stroke of luck that he had experienced "down under," as the
circumstances involved the solution of a poser that could not fail to interest
all lovers of puzzle problems. After the club dinner, Dawkins was asked to
tell his story, which he did, to the following effect:—
"I have told you, gentlemen, that I was very much down on my luck. I had
gone out to Australia to try to retrieve my fortunes, but had met with no
success, and the future was looking very dark. I was, in fact, beginning to
feel desperate. One hot summer day I happened to be seated in a Melbourne
wineshop, when two fellows entered, and engaged in conversation. They
thought I was asleep, but I assure you I was very wide awake.
"'If only I could find the right field,' said one man, 'the treasure would be
mine; and as the original owner left no heir, I have as much right to it as
anybody else.'
"'How would you proceed?' asked the other.
"'Well, it is like this: The document that fell into my hands states clearly
that the field is square, and that the treasure is buried in it at a point exactly
two furlongs from one corner, three furlongs from the next corner, and four
furlongs from the next corner to that. You see, the worst of it is that nearly
all the fields in the district are square; and I doubt whether there are two of
exactly the same size. If only I knew the size of the field I could soon
discover it, and, by taking these simple measurements, quickly secure the
treasure.'
"'But you would not know which corner to start from, nor which direction
to go to the next corner.'

"'My dear chap, that only means eight spots at the most to dig over; and as
the paper says that the treasure is three feet deep, you bet that wouldn't take
me long.'

"Now, gentlemen," continued Dawkins, "I happen to be a bit of a
mathematician; and hearing the conversation, I saw at once that for a spot to
be exactly two, three, and four furlongs from successive corners of a square,
the square must be of a particular area. You can't get such measurements to
meet at one point in any square you choose. They can only happen in a field
of one size, and that is just what these men never suspected. I will leave you
the puzzle of working out just what that area is.
"Well, when I found the size of the field, I was not long in discovering the
field itself, for the man had let out the district in the conversation. And I did
not need to make the eight digs, for, as luck would have it, the third spot I
tried was the right one. The treasure was a substantial sum, for it has
brought me home and enabled me to start in a business that already shows
signs of being a particularly lucrative one. I often smile when I think of that

poor fellow going about for the rest of his life saying: 'If only I knew the
size of the field!' while he has placed the treasure safe in my own
possession. I tried to find the man, to make him some compensation
anonymously, but without success. Perhaps he stood in little need of the
money, while it has saved me from ruin."
Could the reader have discovered the required area of the field from those
details overheard in the wineshop? It is an elegant little puzzle, and
furnishes another example of the practical utility, on unexpected occasions,
of a knowledge of the art of problem-solving.

THE PROFESSOR'S PUZZLES
"Why, here is the Professor!" exclaimed Grigsby. "We'll make him show us
some new puzzles."
It was Christmas Eve, and the club was nearly deserted. Only Grigsby,
Hawkhurst, and myself, of all the members, seemed to be detained in town
over the season of mirth and mince-pies. The man, however, who had just
entered was a welcome addition to our number. "The Professor of Puzzles,"
as we had nicknamed him, was very popular at the club, and when, as on
the present occasion, things got a little slow, his arrival was a positive
blessing.
He was a man of middle age, cheery and kind-hearted, but inclined to be
cynical. He had all his life dabbled in puzzles, problems, and enigmas of
every kind, and what the Professor didn't know about these matters was
admittedly not worth knowing. His puzzles always had a charm of their
own, and this was mainly because he was so happy in dishing them up in
palatable form.
"You are the man of all others that we were hoping would drop in," said
Hawkhurst. "Have you got anything new?"
"I have always something new," was the reply, uttered with feigned conceit
—for the Professor was really a modest man—"I'm simply glutted with
ideas."
"Where do you get all your notions?" I asked.
"Everywhere, anywhere, during all my waking moments. Indeed, two or
three of my best puzzles have come to me in my dreams."
"Then all the good ideas are not used up?"
"Certainly not. And all the old puzzles are capable of improvement,
embellishment, and extension. Take, for example, magic squares. These
were constructed in India before the Christian era, and introduced into

Europe about the fourteenth century, when they were supposed to possess
certain magical properties that I am afraid they have since lost. Any child
can arrange the numbers one to nine in a square that will add up fifteen in
eight ways; but you will see it can be developed into quite a new problem if
you use coins instead of numbers."

67.—The Coinage Puzzle.
He made a rough diagram, and placed a crown and a florin in two of the
divisions, as indicated in the illustration.
"Now," he continued, "place the fewest possible current English coins in the
seven empty divisions, so that each of the three columns, three rows, and
two diagonals shall add up fifteen shillings. Of course, no division may be
without at least one coin, and no two divisions may contain the same
value."
"But how can the coins affect the question?" asked Grigsby.
"That you will find out when you approach the solution."
"I shall do it with numbers first," said Hawkhurst, "and then substitute
coins."
Five minutes later, however, he exclaimed, "Hang it all! I can't help getting
the 2 in a corner. May the florin be moved from its present position?"
"Certainly not."
"Then I give it up."
But Grigsby and I decided that we would work at it another time, so the
Professor showed Hawkhurst the solution privately, and then went on with
his chat.

68.—The Postage Stamps Puzzles.
"Now, instead of coins we'll substitute postage-stamps. Take ten current
English stamps, nine of them being all of different values, and the tenth a
duplicate. Stick two of them in one division and one in each of the others,
so that the square shall this time add up ninepence in the eight directions as
before."
"Here you are!" cried Grigsby, after he had been scribbling for a few
minutes on the back of an envelope.
The Professor smiled indulgently.
"Are you sure that there is a current English postage-stamp of the value of
threepence-halfpenny?"
"For the life of me, I don't know. Isn't there?"
"That's just like the Professor," put in Hawkhurst. "There never was such a
'tricky' man. You never know when you have got to the bottom of his
puzzles. Just when you make sure you have found a solution, he trips you
up over some little point you never thought of."
"When you have done that," said the Professor, "here is a much better one
for you. Stick English postage stamps so that every three divisions in a line
shall add up alike, using as many stamps as you choose, so long as they are
all of different values. It is a hard nut."

69.—The Frogs and Tumblers.

"What do you think of these?"
The Professor brought from his capacious pockets a number of frogs, snails,
lizards, and other creatures of Japanese manufacture—very grotesque in
form and brilliant in colour. While we were looking at them he asked the
waiter to place sixty-four tumblers on the club table. When these had been
brought and arranged in the form of a square, as shown in the illustration,
he placed eight of the little green frogs on the glasses as shown.

"Now," he said, "you see these tumblers form eight horizontal and eight
vertical lines, and if you look at them diagonally (both ways) there are
twenty-six other lines. If you run your eye along all these forty-two lines,
you will find no two frogs are anywhere in a line.
"The puzzle is this. Three of the frogs are supposed to jump from their
present position to three vacant glasses, so that in their new relative
positions still no two frogs shall be in a line. What are the jumps made?"
"I suppose——" began Hawkhurst.
"I know what you are going to ask," anticipated the Professor. "No; the
frogs do not exchange positions, but each of the three jumps to a glass that
was not previously occupied."
"But surely there must be scores of solutions?" I said.
"I shall be very glad if you can find them," replied the Professor with a dry
smile. "I only know of one—or rather two, counting a reversal, which
occurs in consequence of the position being symmetrical."

70.—Romeo and Juliet.
For some time we tried to make these little reptiles perform the feat allotted
to them, and failed. The Professor, however, would not give away his
solution, but said he would instead introduce to us a little thing that is
childishly simple when you have once seen it, but cannot be mastered by
everybody at the very first attempt.
"Waiter!" he called again. "Just take away these glasses, please, and bring
the chessboards."
"I hope to goodness," exclaimed Grigsby, "you are not going to show us
some of those awful chess problems of yours. 'White to mate Black in 427
moves without moving his pieces.' 'The bishop rooks the king, and pawns
his Giuoco Piano in half a jiff.'"
"No, it is not chess. You see these two snails. They are Romeo and Juliet.
Juliet is on her balcony, waiting the arrival of her love; but Romeo has been
dining, and forgets, for the life of him, the number of her house. The
squares represent sixty-four houses, and the amorous swain visits every
house once and only once before reaching his beloved. Now, make him do
this with the fewest possible turnings. The snail can move up, down, and
across the board and through the diagonals. Mark his track with this piece
of chalk."

"Seems easy enough," said Grigsby, running the chalk along the squares.
"Look! that does it."
"Yes," said the Professor: "Romeo has got there, it is true, and visited every
square once, and only once; but you have made him turn nineteen times,
and that is not doing the trick in the fewest turns possible."
Hawkhurst, curiously enough, hit on the solution at once, and the Professor
remarked that this was just one of those puzzles that a person might solve at
a glance or not master in six months.

71.—Romeo's Second Journey.
"It was a sheer stroke of luck on your part, Hawkhurst," he added. "Here is
a much easier puzzle, because it is capable of more systematic analysis; yet
it may just happen that you will not do it in an hour. Put Romeo on a white
square and make him crawl into every other white square once with the
fewest possible turnings. This time a white square may be visited twice, but
the snail must never pass a second time through the same corner of a square
nor ever enter the black squares."
"May he leave the board for refreshments?" asked Grigsby.
"No; he is not allowed out until he has performed his feat."

72.—The Frogs who would a-wooing go.
While we were vainly attempting to solve this puzzle, the Professor
arranged on the table ten of the frogs in two rows, as they will be found in
the illustration.

"That seems entertaining," I said. "What is it?"
"It is a little puzzle I made a year ago, and a favourite with the few people
who have seen it. It is called 'The Frogs who would a-wooing go.' Four of
them are supposed to go a-wooing, and after the four have each made a
jump upon the table, they are in such a position that they form five straight
rows with four frogs in every row."
"What's that?" asked Hawkhurst. "I think I can do that." A few minutes later
he exclaimed, "How's this?"
"They form only four rows instead of five, and you have moved six of
them," explained the Professor.
"Hawkhurst," said Grigsby severely, "you are a duffer. I see the solution at a
glance. Here you are! These two jump on their comrades' backs."
"No, no," admonished the Professor; "that is not allowed. I distinctly said
that the jumps were to be made upon the table. Sometimes it passes the wit
of man so to word the conditions of a problem that the quibbler will not
persuade himself that he has found a flaw through which the solution may
be mastered by a child of five."

After we had been vainly puzzling with these batrachian lovers for some
time, the Professor revealed his secret.
The Professor gathered up his Japanese reptiles and wished us good-night
with the usual seasonable compliments. We three who remained had one
more pipe together, and then also left for our respective homes. Each
believes that the other two racked their brains over Christmas in the
determined attempt to master the Professor's puzzles; but when we next met
at the club we were all unanimous in declaring that those puzzles which we
had failed to solve "we really had not had time to look at," while those we
had mastered after an enormous amount of labour "we had seen at the first
glance directly we got home."

MISCELLANEOUS PUZZLES

73.—The Game of Kayles.
Nearly all of our most popular games are of very ancient origin, though in
many cases they have been considerably developed and improved. Kayles
—derived from the French word quilles—was a great favourite in the
fourteenth century, and was undoubtedly the parent of our modern game of
ninepins. Kayle-pins were not confined in those days to any particular
number, and they were generally made of a conical shape and set up in a
straight row.
At first they were knocked down by a club that was thrown at them from a
distance, which at once suggests the origin of the pastime of "shying for
cocoanuts" that is to-day so popular on Bank Holidays on Hampstead Heath
and elsewhere. Then the players introduced balls, as an improvement on the
club.
In the illustration we get a picture of some of our fourteenth-century
ancestors playing at kayle-pins in this manner.
Now, I will introduce to my readers a new game of parlour kayle-pins, that
can be played across the table without any preparation whatever. You
simply place in a straight row thirteen dominoes, chess-pawns,
draughtsmen, counters, coins, or beans—anything will do—all close
together, and then remove the second one as shown in the picture.
It is assumed that the ancient players had become so expert that they could
always knock down any single kayle-pin, or any two kayle-pins that stood
close together. They therefore altered the game, and it was agreed that the
player who knocked down the last pin was the winner.
Therefore, in playing our table-game, all you have to do is to knock down
with your fingers, or take away, any single kayle-pin or two adjoining
kayle-pins, playing alternately until one of the two players makes the last
capture, and so wins. I think it will be found a fascinating little game, and I
will show the secret of winning.

Remember that the second kayle-pin must be removed before you begin to
play, and that if you knock down two at once those two must be close
together, because in the real game the ball could not do more than this.

74.—The Broken Chessboard.
There is a story of Prince Henry, son of William the Conqueror, afterwards
Henry I., that is so frequently recorded in the old chronicles that it is
doubtless authentic. The following version of the incident is taken from
Hayward's Life of William the Conqueror, published in 1613—
"Towards the end of his reigne he appointed his two sonnes Robert and
Henry, with joynt authoritie, governours of Normandie; the one to
suppresse either the insolence or levitie of the other. These went together to
visit the French king lying at Constance: where, entertaining the time with
varietie of disports, Henry played with Louis, then Daulphine of France, at
chesse, and did win of him very much.

"Hereat Louis beganne to growe warme in words, and was therein little
respected by Henry. The great impatience of the one and the small
forbearance of the other did strike in the end such a heat between them that
Louis threw the chessmen at Henry's face.
"Henry again stroke Louis with the chessboard, drew blood with the blowe,
and had presently slain him upon the place had he not been stayed by his
brother Robert.
"Hereupon they presently went to horse, and their spurres claimed so good
haste as they recovered Pontoise, albeit they were sharply pursued by the
French."

Now, tradition—on this point not trustworthy—says that the chessboard
broke into the thirteen fragments shown in our illustration. It will be seen
that there are twelve pieces, all different in shape, each containing five
squares, and one little piece of four squares only.
We thus have all the sixty-four squares of the chess-board, and the puzzle is
simply to cut them out and fit them together, so as to make a perfect board
properly chequered. The pieces may be easily cut out of a sheet of
"squared" paper, and, if mounted on cardboard, they will form a source of
perpetual amusement in the home.
If you succeed in constructing the chessboard, but do not record the
arrangement, you will find it just as puzzling the next time you feel
disposed to attack it.
Prince Henry himself, with all his skill and learning, would have found it an
amusing pastime.

75.—The Spider and the Fly.

Inside a rectangular room, measuring 30 feet in length and 12 feet in width
and height, a spider is at a point on the middle of one of the end walls, 1
foot from the ceiling, as at A; and a fly is on the opposite wall, 1 foot from
the floor in the centre, as shown at B. What is the shortest distance that the
spider must crawl in order to reach the fly, which remains stationary? Of
course the spider never drops or uses its web, but crawls fairly.

76.—The Perplexed Cellarman.
Here is a little puzzle culled from the traditions of an old monastery in the
west of England. Abbot Francis, it seems, was a very worthy man; and his
methods of equity extended to those little acts of charity for which he was
noted for miles round.

The Abbot, moreover, had a fine taste in wines. On one occasion he sent for
the cellarman, and complained that a particular bottling was not to his
palate.
"Pray tell me, Brother John, how much of this wine thou didst bottle
withal."

"A fair dozen in large bottles, my lord abbot, and the like in the small,"
replied the cellarman, "whereof five of each have been drunk in the
refectory."
"So be it. There be three varlets waiting at the gate. Let the two dozen
bottles be given unto them, both full and empty; and see that the dole be
fairly made, so that no man receive more wine than another, nor any
difference in bottles."
Poor John returned to his cellar, taking the three men with him, and then his
task began to perplex him. Of full bottles he had seven large and seven
small, and of empty bottles five large and five small, as shown in the
illustration. How was he to make the required equitable division?
He divided the bottles into three groups in several ways that at first sight
seemed to be quite fair, since two small bottles held just the same quantity
of wine as one large one. But the large bottles themselves, when empty,
were not worth two small ones.
Hence the abbot's order that each man must take away the same number of
bottles of each size.
Finally, the cellarman had to consult one of the monks who was good at
puzzles of this kind, and who showed him how the thing was done. Can you
find out just how the distribution was made?

77.—Making a Flag.
A good dissection puzzle in so
few as two pieces is rather a
rarity, so perhaps the reader
will be interested in the
following. The diagram
represents a piece of bunting,
and it is required to cut it into
two pieces (without any waste)
that will fit together and form a
perfectly square flag, with the
four roses symmetrically
placed. This would be easy
enough if it were not for the
four roses, as we should
merely have to cut from A to
B, and insert the piece at the bottom of the flag. But we are not allowed to
cut through any of the roses, and therein lies the difficulty of the puzzle. Of
course we make no allowance for "turnings."

78.—Catching the Hogs.
In the illustration Hendrick and Katrün are seen engaged in the exhilarating
sport of attempting the capture of a couple of hogs.
Why did they fail?

Strange as it may seem, a complete answer is afforded in the little puzzle
game that I will now explain.
Copy the simple diagram on a conveniently large sheet of cardboard or
paper, and use four marked counters to represent the Dutchman, his wife,
and the two hogs.

At the beginning of the game these must be placed on the squares on which
they are shown. One player represents Hendrick and Katrün, and the other
the hogs. The first player moves the Dutchman and his wife one square each
in any direction (but not diagonally), and then the second player moves both
pigs one square each (not diagonally); and so on, in turns, until Hendrick
catches one hog and Katrün the other.
This you will find would be absurdly easy if the hogs moved first, but this
is just what Dutch pigs will not do.

79.—The Thirty-one Game.
This is a game that used to be (and may be to this day, for aught I know) a
favourite means of swindling employed by card-sharpers at racecourses and
in railway carriages.
As, on its own merits, however, the game is particularly interesting, I will
make no apology for presenting it to my readers.
The cardsharper lays down the twenty-four cards shown in the illustration,
and invites the innocent wayfarer to try his luck or skill by seeing which of
them can first score thirty-one, or drive his opponent beyond, in the
following manner:—
One player turns down a card, say a 2, and counts "two"; the second player
turns down a card, say a 5, and, adding this to the score, counts "seven"; the
first player turns down another card, say a 1, and counts "eight"; and so the
play proceeds alternately until one of them scores the "thirty-one," and so
wins.
Now, the question is, in order to win, should you turn down the first card, or
courteously request your opponent to do so? And how should you conduct
your play? The reader will perhaps say: "Oh, that is easy enough. You must
play first, and turn down a 3; then, whatever your opponent does, he cannot
stop your making ten, or stop your making seventeen, twenty-four, and the
winning thirty-one. You have only to secure these numbers to win."
But this is just that little knowledge which is such a dangerous thing, and it
places you in the hands of the sharper.
You play 3, and the sharper plays 4 and counts "seven"; you play 3 and
count "ten"; the sharper turns down 3 and scores "thirteen"; you play 4 and
count "seventeen"; the sharper plays a 4 and counts "twenty-one"; you play
3 and make your "twenty-four."

Now the sharper plays the last 4 and scores "twenty-eight." You look in vain
for another 3 with which to win, for they are all turned down! So you are
compelled either to let him make the "thirty-one" or to go yourself beyond,
and so lose the game.
You thus see that your method of certainly winning breaks down utterly, by
what may be called the "method of exhaustion." I will give the key to the
game, showing how you may always win; but I will not here say whether
you must play first or second: you may like to find it out for yourself.

80.—The Chinese Railways.

Our illustration shows the plan of a Chinese city protected by pentagonal
fortifications. Five European Powers were scheming and clamouring for a
concession to run a railway to the place; and at last one of the Emperor's
more brilliant advisers said, "Let every one of them have a concession!" So
the Celestial Government officials were kept busy arranging the details. The
letters in the diagram show the different nationalities, and indicate not only
just where each line must enter the city, but also where the station
belonging to that line must be located. As it was agreed that the line of one
company must never cross the line of another, the representatives of the
various countries concerned were engaged so many weeks in trying to find
a solution to the problem, that in the meantime a change in the Chinese
Government was brought about, and the whole scheme fell through. Take
your pencil and trace out the route for the line A to A, B to B, C to C, and so
on, without ever allowing one line to cross another or pass through another
company's station.

81.—The Eight Clowns.

This illustration represents a troupe of clowns I once saw on the Continent.
Each clown bore one of the numbers 1 to 9 on his body. After going through
the usual tumbling, juggling, and other antics, they generally concluded
with a few curious little numerical tricks, one of which was the rapid
formation of a number of magic squares. It occurred to me that if clown No.
1 failed to appear (as happens in the illustration), this last item of their
performance might not be so easy. The reader is asked to discover how
these eight clowns may arrange themselves in the form of a square (one
place being vacant), so that every one of the three columns, three rows, and
each of the two diagonals shall add up the same. The vacant place may be at
any part of the square, but it is No. 1 that must be absent.

82.—The Wizard's Arithmetic.
Once upon a time a knight went to consult a certain famous wizard. The
interview had to do with an affair of the heart; but after the man of magic
had foretold the most favourable issues, and concocted a love-potion that
was certain to help his visitor's cause, the conversation drifted on to occult
subjects generally.

"And art thou learned also in the magic of numbers?" asked the knight.
"Show me but one sample of thy wit in these matters."
The old wizard took five blocks bearing numbers, and placed them on a
shelf, apparently at random, so that they stood in the order 41096, as shown

in our illustration. He then took in his hands an 8 and a 3, and held them
together to form the number 83.
"Sir Knight, tell me," said the wizard, "canst thou multiply one number into
the other in thy mind?"
"Nay, of a truth," the good knight replied. "I should need to set out upon the
task with pen and scrip."
"Yet mark ye how right easy a thing it is to a man learned in the lore of far
Araby, who knoweth all the magic that is hid in the philosophy of
numbers!"
The wizard simply placed the 3 next to the 4 on the shelf, and the 8 at the
other end. It will be found that this gives the answer quite correctly—
3410968. Very curious, is it not? How many other two-figure multipliers
can you find that will produce the same effect? You may place just as many
blocks as you like on the shelf, bearing any figures you choose.

83.—The Ribbon Problem.

If we take the ribbon by the ends and pull it out straight, we have the
number 0588235294117647. This number has the peculiarity that, if we
multiply it by any one of the numbers, 2, 3, 4, 5, 6, 7, 8, or 9, we get exactly
the same number in the circle, starting from a different place. For example,
multiply by 4, and the product is 2352941176470588, which starts from the
dart in the circle. So, if we multiply by 3, we get the same result starting
from the star. Now, the puzzle is to place a different arrangement of figures
on the ribbon that will produce similar results when so multiplied; only the
0 and the 7 appearing at the ends of the ribbon must not be removed.

84.—The Japanese Ladies and the Carpet.

Three Japanese ladies possessed a square ancestral carpet of considerable
intrinsic value, but treasured also as an interesting heirloom in the family.
They decided to cut it up and make three square rugs of it, so that each
should possess a share in her own house.
One lady suggested that the simplest way would be for her to take a smaller
share than the other two, because then the carpet need not be cut into more
than four pieces.
There are three easy ways of doing this, which I will leave the reader for the
present the amusement of finding for himself, merely saying that if you
suppose the carpet to be nine square feet, then one lady may take a piece

two feet square whole, another a two feet square in two pieces, and the third
a square foot whole.
But this generous offer would not for a moment be entertained by the other
two sisters, who insisted that the square carpet should be so cut that each
should get a square mat of exactly the same size.
Now, according to the best Western authorities, they would have found it
necessary to cut the carpet into seven pieces; but a correspondent in Tokio
assures me that the legend is that they did it in as few as six pieces, and he
wants to know whether such a thing is possible.
Yes; it can be done.
Can you cut out the six pieces that will form three square mats of equal
size?

85.—Captain Longbow and the Bears.
That eminent and more or less veracious traveller Captain Longbow has a
great grievance with the public. He claims that during a recent expedition in
Arctic regions he actually reached the North Pole, but cannot induce
anybody to believe him. Of course, the difficulty in such cases is to produce
proof, but he avers that future travellers, when they succeed in
accomplishing the same feat, will find evidence on the spot. He says that
when he got there he saw a bear going round and round the top of the pole
(which he declares is a pole), evidently perplexed by the peculiar fact that
no matter in what direction he looked it was always due south. Captain
Longbow put an end to the bear's meditations by shooting him, and
afterwards impaling him, in the manner shown in the illustration, as the
evidence for future travellers to which I have alluded.

When the Captain got one hundred miles south on his return journey he had
a little experience that is somewhat puzzling. He was surprised one
morning, on looking down from an elevation, to see no fewer than eleven
bears in his immediate vicinity. But what astonished him more than
anything else was the curious fact that they had so placed themselves that
there were seven rows of bears, with four bears in every row. Whether or
not this was the result of pure accident he cannot say, but such a thing might
have happened. If the reader tries to make eleven dots on a sheet of paper so
that there shall be seven rows of dots with four dots in every row, he will
find some difficulty; but the captain's alleged grouping of the bears is quite
possible. Can you discover how they were arranged?

86.—The English Tour.

This puzzle has to do with railway routes, and in these days of much
travelling should prove useful. The map of England shows twenty-four
towns, connected by a system of railways. A resident at the town marked A
at the top of the map proposes to visit every one of the towns once and only
once, and to finish up his tour at Z. This would be easy enough if he were
able to cut across country by road, as well as by rail, but he is not. How
does he perform the feat? Take your pencil and, starting from A, pass from
town to town, making a dot in the towns you have visited, and see if you
can end at Z.

87.—The Chifu-Chemulpo Puzzle.
Here is a puzzle that was once on sale in the London shops. It represents a
military train—an engine and eight cars. The puzzle is to reverse the cars,
so that they shall be in the order 8, 7, 6, 5, 4, 3, 2, 1, instead of 1, 2, 3, 4, 5,
6, 7, 8, with the engine left, as at first, on the side track. Do this in the
fewest possible moves. Every time the engine or a car is moved from the
main to the side track, or vice versa, it counts a move for each car or engine
passed over one of the points. Moves along the main track are not counted.
With 8 at the extremity, as shown, there is just room to pass 7 on to the side
track, run 8 up to 6, and bring down 7 again; or you can put as many as five
cars, or four and the engine, on the siding at the same time. The cars move
without the aid of the engine. The purchaser is invited to "try to do it in 20
moves." How many do you require?

88.—The Eccentric Market-woman.
Mrs. Covey, who keeps a little poultry farm in Surrey, is one of the most
eccentric women I ever met. Her manner of doing business is always
original, and sometimes quite weird and wonderful. She was once found
explaining to a few of her choice friends how she had disposed of her day's
eggs. She had evidently got the idea from an old puzzle with which we are
all familiar; but as it is an improvement on it, I have no hesitation in
presenting it to my readers. She related that she had that day taken a certain
number of eggs to market. She sold half of them to one customer, and gave
him half an egg over. She next sold a third of what she had left, and gave a
third of an egg over. She then sold a fourth of the remainder, and gave a
fourth of an egg over. Finally, she disposed of a fifth of the remainder, and
gave a fifth of an egg over. Then what she had left she divided equally
among thirteen of her friends. And, strange to say, she had not throughout
all these transactions broken a single egg. Now, the puzzle is to find the
smallest possible number of eggs that Mrs. Covey could have taken to
market. Can you say how many?

89.—The Primrose Puzzle.

Select the name of any flower that you think suitable, and that contains
eight letters. Touch one of the primroses with your pencil and jump over
one of the adjoining flowers to another, on which you mark the first letter of
your word. Then touch another vacant flower, and again jump over one in
another direction, and write down the second letter. Continue this (taking
the letters in their proper order) until all the letters have been written down,
and the original word can be correctly read round the garland. You must
always touch an unoccupied flower, but the flower jumped over may be
occupied or not. The name of a tree may also be selected. Only English
words may be used.

90.—The Round Table.
Seven friends, named Adams, Brooks, Cater, Dobson, Edwards, Fry, and
Green, were spending fifteen days together at the seaside, and they had a
round breakfast table at the hotel all to themselves. It was agreed that no
man should ever sit down twice with the same two neighbours. As they can
be seated, under these conditions, in just fifteen ways, the plan was quite
practicable. But could the reader have prepared an arrangement for every
sitting? The hotel proprietor was asked to draw up a scheme, but he
miserably failed.

91.—The Five Tea Tins.
Sometimes people will speak of mere counting as one of the simplest
operations in the world; but on occasions, as I shall show, it is far from easy.
Sometimes the labour can be diminished by the use of little artifices;
sometimes it is practically impossible to make the required enumeration
without having a very clear head indeed. An ordinary child, buying twelve
postage stamps, will almost instinctively say, when he sees there are four
along one side and three along the other, "Four times three are twelve;"
while his tiny brother will count them all in rows, "1, 2, 3, 4," etc. If the
child's mother has occasion to add up the numbers 1, 2, 3, up to 50, she will
most probably make a long addition sum of the fifty numbers; while her
husband, more used to arithmetical operations, will see at a glance that by
joining the numbers at the extremes there are 25 pairs of 51; therefore,
25×51=1,275. But his smart son of twenty may go one better and say, "Why
multiply by 25? Just add two 0's to the 51 and divide by 4, and there you
are!"
A tea merchant has five tin tea boxes of cubical shape, which he keeps on
his counter in a row, as shown in our illustration. Every box has a picture on
each of its six sides, so there are thirty pictures in all; but one picture on No.
1 is repeated on No. 4, and two other pictures on No. 4 are repeated on No.
3. There are, therefore, only twenty-seven different pictures. The owner
always keeps No. 1 at one end of the row, and never allows Nos. 3 and 5 to
be put side by side.

The tradesman's customer, having obtained this information, thinks it a
good puzzle to work out in how many ways the boxes may be arranged on
the counter so that the order of the five pictures in front shall never be twice
alike. He found the making of the count a tough little nut. Can you work out
the answer without getting your brain into a tangle? Of course, two similar
pictures may be in a row, as it is all a question of their order.

92.—The Four Porkers.
The four pigs are so placed, each in a separate sty, that although every one
of the thirty-six sties is in a straight line (either horizontally, vertically, or
diagonally), with at least one of the pigs, yet no pig is in line with another.
In how many different ways may the four pigs be placed to fulfil these
conditions? If you turn this page round you get three more arrangements,
and if you turn it round in front of a mirror you get four more. These are not
to be counted as different arrangements.

93.—The Number Blocks.
The children in the illustration have found that a large number of very
interesting and instructive puzzles may be made out of number blocks; that
is, blocks bearing the ten digits or Arabic figures—1, 2, 3, 4, 5, 6, 7, 8, 9,
and 0. The particular puzzle that they have been amusing themselves with is
to divide the blocks into two groups of five, and then so arrange them in the
form of two multiplication sums that one product shall be the same as the
other. The number of possible solutions is very considerable, but they have
hit on that arrangement that gives the smallest possible product. Thus, 3,485
multiplied by 2 is 6,970, and 6,970 multiplied by 1 is the same. You will
find it quite impossible to get any smaller result.

Now, my puzzle is to find the largest possible result. Divide the blocks into
any two groups of five that you like, and arrange them to form two
multiplication sums that shall produce the same product and the largest

amount possible. That is all, and yet it is a nut that requires some cracking.
Of course, fractions are not allowed, nor any tricks whatever. The puzzle is
quite interesting enough in the simple form in which I have given it.
Perhaps it should be added that the multipliers may contain two figures.

94.—Foxes and Geese.
Here is a little puzzle of the moving counters class that my readers will
probably find entertaining. Make a diagram of any convenient size similar
to that shown in our illustration, and provide six counters—three marked to
represent foxes and three to represent geese. Place the geese on the discs 1,
2, and 3, and the foxes on the discs numbered 10, 11, and 12
Now the puzzle is this. By moving one at a time, fox and goose alternately,
along a straight line from one disc to the next one, try to get the foxes on 1,
2, and 3, and the geese on 10, 11, and 12—that is, make them exchange
places—in the fewest possible moves.

But you must be careful never to let a fox and goose get within reach of
each other, or there will be trouble. This rule, you will find, prevents you
moving the fox from 11 on the first move, as on either 4 or 6 he would be
within reach of a goose. It also prevents your moving a fox from 10 to 9, or

from 12 to 7. If you play 10 to 5, then your next move may be 2 to 9 with a
goose, which you could not have played if the fox had not previously gone
from 10. It is perhaps unnecessary to say that only one fox or one goose can
be on a disc at the same time. Now, what is the smallest number of moves
necessary to make the foxes and geese change places?

95.—Robinson Crusoe's Table.
Here is a curious extract from Robinson Crusoe's diary. It is not to be found
in the modern editions of the Adventures, and is omitted in the old. This has
always seemed to me to be a pity.
"The third day in the morning, the wind having abated during the night, I
went down to the shore hoping to find a typewriter and other useful things
washed up from the wreck of the ship; but all that fell in my way was a
piece of timber with many holes in it. My man Friday had many times said
that we stood sadly in need of a square table for our afternoon tea, and I
bethought me how this piece of wood might be used for that purpose. And
since during the long time that Friday had now been with me I was not
wanting to lay a foundation of useful knowledge in his mind, I told him that
it was my wish to make the table from the timber I had found, without there
being any holes in the top thereof.

"Friday was sadly put to it to say how this might be, more especially as I
said it should consist of no more than two pieces joined together; but I
taught him how it could be done in such a way that the table might be as
large as was possible, though, to be sure, I was amused when he said, 'My
nation do much better: they stop up holes, so pieces sugars not fall
through.'"
Now, the illustration gives the exact proportion of the piece of wood with
the positions of the fifteen holes. How did Robinson Crusoe make the
largest possible square table-top in two pieces, so that it should not have
any holes in it?

96.—The Fifteen Orchards.

In the county of Devon, where the cider comes from, fifteen of the
inhabitants of a village are imbued with an excellent spirit of friendly
rivalry, and a few years ago they decided to settle by actual experiment a
little difference of opinion as to the cultivation of apple trees. Some said
they want plenty of light and air, while others stoutly maintained that they
ought to be planted pretty closely, in order that they might get shade and
protection from cold winds. So they agreed to plant a lot of young trees, a
different number in each orchard, in order to compare results.
One man had a single tree in his field, another had two trees, another had
three trees, another had four trees, another five, and so on, the last man
having as many as fifteen trees in his little orchard. Last year a very curious
result was found to have come about. Each of the fifteen individuals
discovered that every tree in his own orchard bore exactly the same number

of apples. But, what was stranger still, on comparing notes they found that
the total gathered in every allotment was almost the same. In fact, if the
man with eleven trees had given one apple to the man who had seven trees,
and the man with fourteen trees had given three each to the men with nine
and thirteen trees, they would all have had exactly the same.
Now, the puzzle is to discover how many apples each would have had (the
same in every case) if that little distribution had been carried out. It is quite
easy if you set to work in the right way.

97.—The Perplexed Plumber.
When I paid a visit to Peckham recently I found everybody asking, "What
has happened to Sam Solders, the plumber?" He seemed to be in a bad way,
and his wife was seriously anxious about the state of his mind. As he had
fitted up a hot-water apparatus for me some years ago which did not lead to
an explosion for at least three months (and then only damaged the
complexion of one of the cook's followers), I had considerable regard for
him.
"There he is," said Mrs. Solders, when I called to inquire. "That's how he's
been for three weeks. He hardly eats anything, and takes no rest, whilst his
business is so neglected that I don't know what is going to happen to me
and the five children. All day long—and night too—there he is, figuring and
figuring, and tearing his hair like a mad thing. It's worrying me into an early
grave."
I persuaded Mrs. Solders to explain matters to me. It seems that he had
received an order from a customer to make two rectangular zinc cisterns,
one with a top and the other without a top. Each was to hold exactly 1,000
cubic feet of water when filled to the brim. The price was to be a certain
amount per cistern, including cost of labour. Now Mr. Solders is a thrifty
man, so he naturally desired to make the two cisterns of such dimensions
that the smallest possible quantity of metal should be required. This was the
little question that was so worrying him.

Can my ingenious readers find the dimensions of the most economical
cistern with a top, and also the exact proportions of such a cistern without a
top, each to hold 1,000 cubic feet of water? By "economical" is meant the
method that requires the smallest possible quantity of metal. No margin
need be allowed for what ladies would call "turnings." I shall show how I
helped Mr. Solders out of his dilemma. He says: "That little wrinkle you
gave me would be useful to others in my trade."

98.—The Nelson Column.
During a Nelson celebration I was standing in Trafalgar Square with a
friend of puzzling proclivities. He had for some time been gazing at the
column in an abstracted way, and seemed quite unconscious of the casual
remarks that I addressed to him.
"What are you dreaming about?" I said at last.

"Two feet——" he murmured.
"Somebody's Trilbys?" I inquired.
"Five times round——"

"Two feet, five times round! What on earth are you saying?"
"Wait a minute," he said, beginning to figure something out on the back of
an envelope. I now detected that he was in the throes of producing a new
problem of some sort, for I well knew his methods of working at these
things.
"Here you are!" he suddenly exclaimed. "That's it! A very interesting little
puzzle. The height of the shaft of the Nelson column being 200 feet and its
circumference 16 feet 8 inches, it is wreathed in a spiral garland which
passes round it exactly five times. What is the length of the garland? It
looks rather difficult, but is really remarkably easy."
He was right. The puzzle is quite easy if properly attacked. Of course the
height and circumference are not correct, but chosen for the purposes of the
puzzle. The artist has also intentionally drawn the cylindrical shaft of the
column of equal circumference throughout. If it were tapering, the puzzle
would be less easy.

99.—The Two Errand Boys.
A country baker sent off his boy with a message to the butcher in the next
village, and at the same time the butcher sent his boy to the baker. One ran
faster than the other, and they were seen to pass at a spot 720 yards from the
baker's shop. Each stopped ten minutes at his destination and then started on
the return journey, when it was found that they passed each other at a spot
400 yards from the butcher's. How far apart are the two tradesmen's shops?
Of course each boy went at a uniform pace throughout.

100.—On the Ramsgate Sands.
Thirteen youngsters were seen dancing in a ring on the Ramsgate sands.
Apparently they were playing "Round the Mulberry Bush." The puzzle is
this. How many rings may they form without any child ever taking twice
the hand of any other child—right hand or left? That is, no child may ever
have a second time the same neighbour.

101.—The Three Motor-Cars.

Pope has told us that all chance is but "direction which thou canst not see,"
and certainly we all occasionally come across remarkable coincidences—
little things against the probability of the occurrence of which the odds are
immense—that fill us with bewilderment. One of the three motor men in the
illustration has just happened on one of these queer coincidences. He is
pointing out to his two friends that the three numbers on their cars contain
all the figures 1 to 9 and 0, and, what is more remarkable, that if the
numbers on the first and second cars are multiplied together they will make
the number on the third car. That is, 78, 345, and 26,910 contain all the ten
figures, and 78 multiplied by 345 makes 26,910. Now, the reader will be
able to find many similar sets of numbers of two, three, and five figures
respectively that have the same peculiarity. But there is one set, and one
only, in which the numbers have this additional peculiarity—that the second

number is a multiple of the first. In other words, if 345 could be divided by
78 without a remainder, the numbers on the cars would themselves fulfil
this extra condition. What are the three numbers that we want? Remember
that they must have two, three, and five figures respectively.

102.—A Reversible Magic Square.
Can you construct a square of sixteen different numbers so that it shall be
magic (that is, adding up alike in the four rows, four columns, and two
diagonals), whether you turn the diagram upside down or not? You must not
use a 3, 4, or 5, as these figures will not reverse; but a 6 may become a 9
when reversed, a 9 a 6, a 7 a 2, and a 2 a 7. The 1, 8, and 0 will read the
same both ways. Remember that the constant must not be changed by the
reversal.

103.—The Tube Railway.

The above diagram is the plan of an underground railway. The fare is
uniform for any distance, so long as you do not go twice along any portion
of the line during the same journey. Now a certain passenger, with plenty of
time on his hands, goes daily from A to F. How many different routes are
there from which he may select? For example, he can take the short direct
route, A, B, C, D, E, F, in a straight line; or he can go one of the long routes,
such as A, B, D, C, B, C, E, D, E, F. It will be noted that he has optional
lines between certain stations, and his selections of these lead to variations
of the complete route. Many readers will find it a very perplexing little
problem, though its conditions are so simple.

104.—The Skipper and the Sea-Serpent.
Mr. Simon Softleigh had spent most of his life between Tooting Bec and
Fenchurch Street. His knowledge of the sea was therefore very limited. So,
as he was taking a holiday on the south coast, he thought this was a
splendid opportunity for picking up a little useful information. He therefore
proceeded to "draw" the natives.

"I suppose," said Mr. Softleigh one morning to a jovial, weather-beaten
skipper, "you have seen many wonderful sights on the rolling seas?"
"Bless you, sir, yes," said the skipper. "P'raps you've never seen a vanilla
iceberg, or a mermaid a-hanging out her things to dry on the equatorial line,
or the blue-winged shark what flies through the air in pursuit of his prey, or
the sea-sarpint——"

"Have you really seen a sea-serpent? I thought it was uncertain whether
they existed."
"Uncertin! You wouldn't say there was anything uncertin about a sea-sarpint
if once you'd seen one. The first as I seed was when I was skipper of the
Saucy Sally. We was a-coming round Cape Horn with a cargo of shrimps
from the Pacific Islands when I looks over the port side and sees a
tremenjus monster like a snake, with its 'ead out of the water and its eyes
flashing fire, a-bearing down on our ship. So I shouts to the bo'sun to let
down the boat, while I runs below and fetches my sword—the same what I
used when I killed King Chokee, the cannibal chief as eat our cabin-boy—
and we pulls straight into the track of that there sea-sarpint. Well, to make a
long story short, when we come alongside o' the beast I just let drive at him
with that sword o' mine, and before you could say 'Tom Bowling' I cut him
into three pieces, all of exactually the same length, and afterwards we
hauled 'em aboard the Saucy Sally. What did I do with 'em? Well, I sold 'em
to a feller in Rio Janeiro. And what do you suppose he done with 'em? He
used 'em to make tyres for his motor-car—takes a lot to puncture a sea-
sarpint's skin."
"What was the length of the creature?" asked Simon.
"Well, each piece was equal in length to three-quarters the length of a piece
added to three-quarters of a cable. There's a little puzzle for you to work
out, young gentleman. How many cables long must that there sea-sarpint
'ave been?"
Now, it is not at all to the discredit of Mr. Simon Softleigh that he never
succeeded in working out the correct answer to that little puzzle, for it may
confidently be said that out of a thousand readers who attempt the solution
not one will get it exactly right.

105.—The Dorcas Society.
At the close of four and a half months' hard work, the ladies of a certain
Dorcas Society were so delighted with the completion of a beautiful silk
patchwork quilt for the dear curate that everybody kissed everybody else,
except, of course, the bashful young man himself, who only kissed his
sisters, whom he had called for, to escort home. There were just a gross of
osculations altogether. How much longer would the ladies have taken over
their needlework task if the sisters of the curate referred to had played lawn
tennis instead of attending the meetings? Of course we must assume that the
ladies attended regularly, and I am sure that they all worked equally well. A
mutual kiss here counts as two osculations.

106.—The Adventurous Snail.

A simple version of the puzzle of the climbing snail is familiar to
everybody. We were all taught it in the nursery, and it was apparently
intended to inculcate the simple moral that we should never slip if we can
help it. This is the popular story. A snail crawls up a pole 12 feet high,
ascending 3 feet every day and slipping back 2 feet every night. How long
does it take to get to the top? Of course, we are expected to say the answer
is twelve days, because the creature makes an actual advance of 1 foot in
every twenty-four hours. But the modern infant in arms is not taken in in
this way. He says, correctly enough, that at the end of the ninth day the snail
is 3 feet from the top, and therefore reaches the summit of its ambition on
the tenth day, for it would cease to slip when it had got to the top.
Let us, however, consider the original story. Once upon a time two
philosophers were walking in their garden, when one of them espied a

highly respectable member of the Helix Aspersa family, a pioneer in
mountaineering, in the act of making the perilous ascent of a wall 20 feet
high. Judging by the trail, the gentleman calculated that the snail ascended 3
feet each day, sleeping and slipping back 2 feet every night.
"Pray tell me," said the philosopher to his friend, who was in the same line
of business, "how long will it take Sir Snail to climb to the top of the wall
and descend the other side? The top of the wall, as you know, has a sharp
edge, so that when he gets there he will instantly begin to descend, putting
precisely the same exertion into his daily climbing down as he did in his
climbing up, and sleeping and slipping at night as before."
This is the true version of the puzzle, and my readers will perhaps be
interested in working out the exact number of days. Of course, in a puzzle
of this kind the day is always supposed to be equally divided into twelve
hours' daytime and twelve hours' night.

107.—The Four Princes.
The dominions of a certain Eastern monarch formed a perfectly square tract
of country. It happened that the king one day discovered that his four sons
were not only plotting against each other, but were in secret rebellion
against himself. After consulting with his advisers he decided not to exile
the princes, but to confine them to the four corners of the country, where
each should be given a triangular territory of equal area, beyond the
boundaries of which they would pass at the cost of their lives. Now, the
royal surveyor found himself confronted by great natural difficulties, owing
to the wild character of the country. The result was that while each was
given exactly the same area, the four triangular districts were all of different
shapes, somewhat in the manner shown in the illustration. The puzzle is to
give the three measurements for each of the four districts in the smallest
possible numbers—all whole furlongs. In other words, it is required to find
(in the smallest possible numbers) four rational right-angled triangles of
equal area.

108.—Plato and the Nines.
Both in ancient and in modern times the number nine has been considered
to possess peculiarly mystic qualities. We know, for instance, that there
were nine Muses, nine rivers of Hades, and that Vulcan was nine days
falling down from heaven. Then it has been confidently held that nine
tailors make a man; while we know that there are nine planets, nine days'
wonders, and that a cat has nine lives—and sometimes nine tails.
Most people are acquainted with some of the curious properties of the
number nine in ordinary arithmetic. For example, write down a number
containing as many figures as you like, add these figures together, and
deduct the sum from the first number. Now, the sum of the figures in this
new number will always be a multiple of nine.
There was once a worthy man at Athens who was not only a cranky
arithmetician, but also a mystic. He was deeply convinced of the magic
properties of the number nine, and was perpetually strolling out to the
groves of Academia to bother poor old Plato with his nonsensical ideas
about what he called his "lucky number." But Plato devised a way of getting
rid of him. When the seer one day proposed to inflict on him a lengthy
disquisition on his favourite topic, the philosopher cut him short with the
remark, "Look here, old chappie" (that is the nearest translation of the
original Greek term of familiarity): "when you can bring me the solution of
this little mystery of the three nines I shall be happy to listen to your
treatise, and, in fact, record it on my phonograph for the benefit of
posterity."

Plato then showed, in the manner depicted in our illustration, that three
nines may be arranged so as to represent the number eleven, by putting
them into the form of a fraction. The puzzle he then propounded was so to
arrange the three nines that they will represent the number twenty.
It is recorded of the old crank that, after working hard at the problem for
nine years, he one day, at nine o'clock on the morning of the ninth day of
the ninth month, fell down nine steps, knocked out nine teeth, and expired
in nine minutes. It will be remembered that nine was his lucky number. It
was evidently also Plato's.
In solving the above little puzzle, only the most elementary arithmetical
signs are necessary. Though the answer is absurdly simple when you see it,
many readers will have no little difficulty in discovering it. Take your pencil
and see if you can arrange the three nines to represent twenty.

109.—Noughts and Crosses.
Every child knows how to play this game. You make a square of nine cells,
and each of the two players, playing alternately, puts his mark (a nought or
a cross, as the case may be) in a cell with the object of getting three in a
line. Whichever player first gets three in a line wins with the exulting cry:—

"Tit, tat, toe,
My last go;
Three jolly butcher boys
All in a row."

It is a very ancient game. But if the two players have a perfect knowledge
of it, one of three things must always happen. (1) The first player should
win; (2) the first player should lose; or (3) the game should always be
drawn. Which is correct?

110.—Ovid's Game.
Having examined "Noughts and Crosses," we will now consider an
extension of the game that is distinctly mentioned in the works of Ovid. It
is, in fact, the parent of "Nine Men's Morris," referred to by Shakespeare in
A Midsummer Night's Dream (Act ii., Scene 2). Each player has three
counters, which they play alternately on to the nine points shown in the
diagram, with the object of getting three in a line and so winning. But after
the six counters are played they then proceed to move (always to an
adjacent unoccupied point) with the same object. In the example below
White played first, and Black has just played on point 7. It is now White's
move, and he will undoubtedly play from 8 to 9, and then, whatever Black
may do, he will continue with 5 to 6, and so win. That is the simple game.
Now, if both players are equally perfect at the game what should happen?
Should the first player always win? Or should the second player win? Or
should every game be a draw? One only of these things should always
occur. Which is it?

111.—The Farmer's Oxen.
A child may propose a problem that a sage cannot answer. A farmer
propounded the following question: "That ten-acre meadow of mine will
feed twelve bullocks for sixteen weeks or eighteen bullocks for eight
weeks. How many bullocks could I feed on a forty-acre field for six weeks,
the grass growing regularly all the time?"
It will be seen that the sting lies in the tail. That steady growth of the grass
is such a reasonable point to be considered, and yet to some readers it will
cause considerable perplexity. The grass is, of course, assumed to be of
equal length and uniform thickness in every case when the cattle begin to
eat. The difficulty is not so great as it appears, if you properly attack the
question.

112.—The Great Grangemoor Mystery.
Mr. Stanton Mowbray was a very wealthy man, a reputed millionaire,
residing in that beautiful old mansion that has figured so much in English
history, Grangemoor Park. He was a bachelor, spent most of the year at
home, and lived quietly enough.
According to the evidence given, on the day preceding the night of the
crime he received by the second post a single letter, the contents of which
evidently gave him a shock. At ten o'clock at night he dismissed the
servants, saying that he had some important business matters to look into,
and would be sitting up late. He would require no attendance. It was
supposed that after all had gone to bed he had admitted some person to the
house, for one of the servants was positive that she had heard loud
conversation at a very late hour.
Next morning, at a quarter to seven o'clock, one of the man-servants, on
entering the room, found Mr. Mowbray lying on the floor, shot through the
head, and quite dead. Now we come to the curious circumstance of the case.
It was clear that after the bullet had passed out of the dead man's head it had
struck the tall clock in the room, right in the very centre of the face, and
actually welded together the three hands; for the clock had a seconds hand
that revolved round the same dial as the hour and minute hands. But
although the three hands had become welded together exactly as they stood
in relation to each other at the moment of impact, yet they were free to
revolve round the swivel in one piece, and had been stupidly spun round
several times by the servants before Mr. Wiley Slyman was called upon the
spot. But they would not move separately.

Now, inquiries by the police in the neighbourhood led to the arrest in
London of a stranger who was identified by several persons as having been
seen in the district the day before the murder, but it was ascertained beyond
doubt at what time on the fateful morning he went away by train. If the
crime took place after his departure, his innocence was established. For this
and other reasons it was of the first importance to fix the exact time of the
pistol shot, the sound of which nobody in the house had heard. The clock
face in the illustration shows exactly how the hands were found. Mr.
Slyman was asked to give the police the benefit of his sagacity and
experience, and directly he was shown the clock he smiled and said:
"The matter is supremely simple. You will notice that the three hands
appear to be at equal distances from one another. The hour hand, for
example, is exactly twenty minutes removed from the minute hand—that is,
the third of the circumference of the dial. You attach a lot of importance to

the fact that the servants have been revolving the welded hands, but their
act is of no consequence whatever; for although they were welded
instantaneously, as they are free on the swivel, they would swing round of
themselves into equilibrium. Give me a few moments, and I can tell you
beyond any doubt the exact time that the pistol was fired."
Mr. Wiley Slyman took from his pocket a notebook, and began to figure it
out. In a few minutes he handed the police inspector a slip of paper, on
which he had written the precise moment of the crime. The stranger was
proved to be an old enemy of Mr. Mowbray's, was convicted on other
evidence that was discovered; but before he paid the penalty for his wicked
act, he admitted that Mr. Slyman's statement of the time was perfectly
correct.
Can you also give the exact time?

113.—Cutting a Wood Block.
An economical carpenter had a block of wood measuring eight inches long
by four inches wide by three and three-quarter inches deep. How many
pieces, each measuring two and a half inches by one inch and a half by one
inch and a quarter, could he cut out of it? It is all a question of how you cut
them out. Most people would have more waste material left over than is
necessary. How many pieces could you get out of the block?

114.—The Tramps and the Biscuits.
Four merry tramps bought, borrowed, found, or in some other manner
obtained possession of a box of biscuits, which they agreed to divide
equally amongst themselves at breakfast next morning. In the night, while
the others were fast asleep under the greenwood tree, one man approached
the box, devoured exactly a quarter of the number of biscuits, except the
odd one left over, which he threw as a bribe to their dog. Later in the night a
second man awoke and hit on the same idea, taking a quarter of what
remained and giving the odd biscuit to the dog. The third and fourth men
did precisely the same in turn, taking a quarter of what they found and
giving the odd biscuit to the dog. In the morning they divided what
remained equally amongst them, and again gave the odd biscuit to the
animal. Every man noticed the reduction in the contents of the box, but,
believing himself to be alone responsible, made no comments. What is the
smallest possible number of biscuits that there could have been in the box
when they first acquired it?

SOLUTIONS
THE CANTERBURY PUZZLES

1.—The Reve's Puzzle.
The 8 cheeses can be removed in 33 moves, 10 cheeses in 49 moves, and 21
cheeses in 321 moves. I will give my general method of solution in the
cases of 3, 4, and 5 stools.
Write out the following table to any required length:—

Stools. Number of Cheeses.
312 3 4 5 6 7Natural Numbers.
4 1 3 6 10 15 21 28Triangular Numbers.
5 1 410 20 35 56 84Triangular Pyramids.
Number of Moves.
3 1 3 7 15 31 63 127
4 1 517 49 129 321 769
5 1 731 1113511023 2815

The first row contains the natural numbers. The second row is found by
adding the natural numbers together from the beginning. The numbers in
the third row are obtained by adding together the numbers in the second
row from the beginning. The fourth row contains the successive powers of
2, less 1. The next series is found by doubling in turn each number of that
series and adding the number that stands above the place where you write
the result. The last row is obtained in the same way. This table will at once
give solutions for any number of cheeses with three stools, for triangular
numbers with four stools, and for pyramidal numbers with five stools. In
these cases there is always only one method of solution—that is, of piling
the cheeses.
In the case of three stools, the first and fourth rows tell us that 4 cheeses
may be removed in 15 moves, 5 in 31, 7 in 127. The second and fifth rows
show that, with four stools, 10 may be removed in 49, and 21 in 321 moves.
Also, with five stools, we find from the third and sixth rows that 20 cheeses
require 111 moves, and 35 cheeses 351 moves. But we also learn from the
table the necessary method of piling. Thus, with four stools and 10 cheeses,

the previous column shows that we must make piles of 6 and 3, which will
take 17 and 7 moves respectively—that is, we first pile the six smallest
cheeses in 17 moves on one stool; then we pile the next 3 cheeses on
another stool in 7 moves; then remove the largest cheese in 1 move; then
replace the 3 in 7 moves; and finally replace the 6 in 17: making in all the
necessary 49 moves. Similarly we are told that with five stools 35 cheeses
must form piles of 20, 10, and 4, which will respectively take 111, 49, and
15 moves.
If the number of cheeses in the case of four stools is not triangular, and in
the case of five stools pyramidal, then there will be more than one way of
making the piles, and subsidiary tables will be required. This is the case
with the Reve's 8 cheeses. But I will leave the reader to work out for
himself the extension of the problem.

2.—The Pardoner's Puzzle.
The diagram on page 165 will show how the Pardoner started from the large
black town and visited all the other towns once, and once only, in fifteen
straight pilgrimages.
See No. 320, "The Rook's Tour," in A. in M.

3.—The Miller's Puzzle.
The way to arrange the sacks of flour is as follows:—2, 78, 156, 39, 4. Here
each pair when multiplied by its single neighbour makes the number in the
middle, and only five of the sacks need be moved. There are just three other
ways in which they might have been arranged (4, 39, 156, 78, 2; or 3, 58,
174, 29, 6; or 6, 29, 174, 58, 3), but they all require the moving of seven
sacks.

4.—The Knight's Puzzle.
The Knight declared that as many as 575 squares could be marked off on
his shield, with a rose at every corner. How this result is achieved may be
realized by reference to the accompanying diagram:—Join A, B, C, and D,
and there are 66 squares of this size to be formed; the size A, E, F, G gives
48; A, H, I, J, 32; B, K, L, M, 19; B, N, O, P, 10; B, Q, R, S, 4; E, T, F, C,
57; I, U, V, P, 33; H, W, X, J, 15; K, Y, Z, M, 3; E, a, b, D, 82; H, d, M, D,
56; H, e, f, G, 42; K, g, f, C, 32; N, h, z, F, 24; K, h, m, b, 14; K, O, S, D,
16; K, n, p, G, 10; K, q, r, J, 6; Q, t, p, C, 4; Q, u, r, i, 2. The total number is
thus 575. These groups have been treated as if each of them represented a
different sized square. This is correct, with the one exception that the
squares of the form B, N, O, P are exactly the same size as those of the form
K, h, m, b.

5.—The Wife of Bath's Riddles.
The good lady explained that a bung that is made fast in a barrel is like
another bung that is falling out of a barrel because one of them is in secure
and the other is also insecure. The little relationship poser is readily
understood when we are told that the parental command came from the
father (who was also in the room) and not from the mother.

6.—The Host's Puzzle.
The puzzle propounded by the jovial host of the "Tabard" Inn of Southwark
had proved more popular than any other of the whole collection. "I see, my
merry masters," he cried, "that I have sorely twisted thy brains by my little
piece of craft. Yet it is but a simple matter for me to put a true pint of fine
old ale in each of these two measures, albeit one is of five pints and the
other of three pints, without using any other measure whatever."
The host of the "Tabard" Inn thereupon proceeded to explain to the pilgrims
how this apparently impossible task could be done. He first filled the 5-pint
and 3-pint measures, and then, turning the tap, allowed the barrel to run to
waste—a proceeding against which the company protested; but the wily
man showed that he was aware that the cask did not contain much more
than eight pints of ale. The contents, however, do not affect the solution of
the puzzle. He then closed the tap and emptied the 3-pint into the barrel;
filled the 3-pint from the 5-pint; emptied the 3-pint into the barrel;
transferred the two pints from the 5-pint to the 3-pint; filled the 5-pint from
the barrel, leaving one pint now in the barrel; filled 3-pint from 5-pint;
allowed the company to drink the contents of the 3-pint; filled the 3-pint
from the 5-pint, leaving one pint now in the 5-pint; drank the contents of the
3-pint; and finally drew off one pint from the barrel into the 3-pint. He had
thus obtained the required one pint of ale in each measure, to the great
astonishment of the admiring crowd of pilgrims.

7.—Clerk of Oxenford's Puzzle.
The illustration shows how the square is to be cut into four pieces, and how
these pieces are to be put together again to make a magic square. It will be
found that the four columns, four rows, and two long diagonals now add up
to 34 in every case.

8.—The Tapiser's Puzzle.
The piece of tapestry had to be cut along the lines into three pieces so as to
fit together and form a perfect square, with the pattern properly matched. It
was also stipulated in effect that one of the three pieces must be as small as
possible. The illustration shows how to make the cuts and how to put the
pieces together, while one of the pieces contains only twelve of the little
squares.

9.—The Carpenter's Puzzle.
The carpenter said that he made a box whose internal dimensions were
exactly the same as the original block of wood—that is, 3 feet by 1 foot by
1 foot. He then placed the carved pillar in this box and filled up all the
vacant space with a fine, dry sand, which he carefully shook down until he
could get no more into the box. Then he removed the pillar, taking great
care not to lose any of the sand, which, on being shaken down alone in the
box, filled a space equal to one cubic foot. This was, therefore, the quantity
of wood that had been cut away.

10.—The Puzzle of the Squire's Yeoman.
The illustration will show how three of the arrows were removed each to a
neighbouring square on the signboard of the "Chequers" Inn, so that still no
arrow was in line with another. The black dots indicate the squares on
which the three arrows originally stood.

11.—The Nun's Puzzle.

As there are eighteen cards bearing the letters "CANTERBURY
PILGRIMS," write the numbers 1 to 18 in a circle, as shown in the diagram.
Then write the first letter C against 1, and each successive letter against the
second number that happens to be vacant. This has been done as far as the
second R. If the reader completes the process by placing Y against 2, P
against 6, I against 10, and so on, he will get the letters all placed in the
following order:—CYASNPTREIRMBLUIRG, which is the required
arrangement for the cards, C being at the top of the pack and G at the
bottom.

12.—The Merchant's Puzzle.
This puzzle amounts to finding the smallest possible number that has
exactly sixty-four divisors, counting 1 and the number itself as divisors. The
least number is 7,560. The pilgrims might, therefore, have ridden in single
file, two and two, three and three, four and four, and so on, in exactly sixty-
four different ways, the last manner being in a single row of 7,560.
The Merchant was careful to say that they were going over a common, and
not to mention its size, for it certainly would not be possible along an
ordinary road!
To find how many different numbers will divide a given number, N, let N =
ap bq cr ..., where a, b, c ... are prime numbers. Then the number of divisors
will be (p + 1) (q + 1) (r + 1) ..., which includes as divisors 1 and N itself.
Thus in the case of my puzzle—

7,560 = 23 × 33 × 5× 7
Powers = 3 3 1 1
Therefore 4 × 4 × 2× 2= 64 divisors.

To find the smallest number that has a given number of divisors we must
proceed by trial. But it is important sometimes to note whether or not the
condition is that there shall be a given number of divisors and no more. For
example, the smallest number that has seven divisors and no more is 64,
while 24 has eight divisors, and might equally fulfil the conditions. The
stipulation as to "no more" was not necessary in the case of my puzzle, for
no smaller number has more than sixty-four divisors.

13.—The Man of Law's Puzzle.
The fewest possible moves for getting the prisoners into their dungeons in
the required numerical order are twenty-six. The men move in the following
order:—1, 2, 3, 1, 2, 6, 5, 3, 1, 2, 6, 5, 3, 1, 2, 4, 8, 7, 1, 2, 4, 8, 7, 4, 5, 6. As
there are never more than one vacant dungeon to be moved into, there can
be no ambiguity in the notation.

The diagram may be simplified by my "buttons and string" method, fully
explained in A. in M., p. 230. It then takes one of the simple forms of A or
B, and the solution is much easier. In A we use counters; in B we can
employ rooks on a corner of a chessboard. In both cases we have to get the
order

in the fewest possible moves.
See also solution to No. 94.

14.—The Weaver's Puzzle.
The illustration shows clearly how the Weaver cut his square of beautiful
cloth into four pieces of exactly the same size and shape, so that each piece
contained an embroidered lion and castle unmutilated in any way.

15.—The Cook's Puzzle.
There were four portions of warden pie and four portions of venison pasty
to be distributed among eight out of eleven guests. But five out of the
eleven will only eat the pie, four will only eat the pasty, and two are willing
to eat of either. Any possible combination must fall into one of the
following groups. (i.) Where the warden pie is distributed entirely among
the five first mentioned; (ii.) where only one of the accommodating pair is
given pie; (iii.) where the other of the pair is given pie; (iv.) where both of
the pair are given pie. The numbers of combinations are: (i.) = 75, (ii.) = 50,
(iii.) = 10, (iv.) = 10—making in all 145 ways of selecting the eight
participants. A great many people will give the answer as 185, by
overlooking the fact that in forty cases in class (iii.) precisely the same eight
guests would be sharing the meal as in class (ii.), though the
accommodating pair would be eating differently of the two dishes. This is
the point that upset the calculations of the company.

16.—The Sompnour's Puzzle.
The number that the Sompnour confided to the Wife of Bath was twenty-
nine, and she was told to begin her count at the Doctor of Physic, who will
be seen in the illustration standing the second on her right. The first count
of twenty-nine falls on the Shipman, who steps out of the ring. The second
count falls on the Doctor, who next steps out. The remaining three counts
fall respectively on the Cook, the Sompnour, and the Miller. The ladies
would, therefore, have been left in possession had it not been for the
unfortunate error of the good Wife. Any multiple of 2,520 added to 29
would also have served the same purpose, beginning the count at the
Doctor.

17.—The Monk's Puzzle.
The Monk might have placed dogs in the kennels in two thousand nine
hundred and twenty-six different ways, so that there should be ten dogs on
every side. The number of dogs might vary from twenty to forty, and as
long as the Monk kept his animals within these limits the thing was always
possible.
The general solution to this puzzle is difficult. I find that for n dogs on
every side of the square, the number of different ways is (n4 + 10n3 + 38n2
+ 62n + 33) / 48, where n is odd, and ((n4 + 10n3 + 38n2 + 68n) / 48) + 1,
where n is even, if we count only those arrangements that are fundamentally
different. But if we count all reversals and reflections as different, as the
Monk himself did, then n dogs (odd or even) may be placed in ((n4 + 6n3 +
14n2 + 15n) / 6) + 1 ways. In order that there may be n dogs on every side,
the number must not be less than 2n nor greater than 4n, but it may be any
number within these limits.
An extension of the principle involved in this puzzle is given in No. 42,
"The Riddle of the Pilgrims." See also "The Eight Villas" and "A Dormitory
Puzzle" in A. in M.

18.—The Shipman's Puzzle.
There are just two hundred and sixty-four different ways in which the ship
Magdalen might have made her ten annual voyages without ever going over
the same course twice in a year. Every year she must necessarily end her
tenth voyage at the island from which she first set out.

19.—The Puzzle of the Prioress.
The Abbot of Chertsey was quite correct. The curiously-shaped cross may
be cut into four pieces that will fit together and form a perfect square. How
this is done is shown in the illustration.
See also p. 31 in A. in M.

20.—The Puzzle of the Doctor of Physic.
Here we have indeed a knotty problem. Our text-books tell us that all
spheres are similar, and that similar solids are as the cubes of corresponding
lengths. Therefore, as the circumferences of the two phials were one foot
and two feet respectively and the cubes of one and two added together make
nine, what we have to find is two other numbers whose cubes added
together make nine. These numbers clearly must be fractional. Now, this
little question has really engaged the attention of learned men for two
hundred and fifty years; but although Peter de Fermat showed in the
seventeenth century how an answer may be found in two fractions with a
denominator of no fewer than twenty-one figures, not only are all the
published answers, by his method, that I have seen inaccurate, but nobody
has ever published the much smaller result that I now print. The cubes of
(415280564497 / 348671682660) and (676702467503 / 348671682660)
added together make exactly nine, and therefore these fractions of a foot are
the measurements of the circumferences of the two phials that the Doctor
required to contain the same quantity of liquid as those produced. An
eminent actuary and another correspondent have taken the trouble to cube
out these numbers, and they both find my result quite correct.
If the phials were one foot and three feet in circumference respectively, then
an answer would be that the cubes of (63284705 / 21446828) and
(28340511 / 21446828) added together make exactly 28. See also No. 61,
"The Silver Cubes."
Given a known case for the expression of a number as the sum or difference
of two cubes, we can, by formula, derive from it an infinite number of other
cases alternately positive and negative. Thus Fermat, starting from the
known case 13 + 23 = 9 (which we will call a fundamental case), first
obtained a negative solution in bigger figures, and from this his positive
solution in bigger figures still. But there is an infinite number of
fundamentals, and I found by trial a negative fundamental solution in
smaller figures than his derived negative solution, from which I obtained
the result shown above. That is the simple explanation.

We can say of any number up to 100 whether it is possible or not to express
it as the sum of two cubes, except 66. Students should read the Introduction
to Lucas's Théorie des Nombres, p. xxx.
Some years ago I published a solution for the case of

of which Legendre gave at some length a "proof" of impossibility; but I
have since found that Lucas anticipated me in a communication to
Sylvester.

21.—The Ploughman's Puzzle.
The illustration shows how the sixteen trees might have been planted so as
to form as many as fifteen straight rows with four trees in every row. This is
in excess of what was for a long time believed to be the maximum number
of rows possible; and though with our present knowledge I cannot
rigorously demonstrate that fifteen rows cannot be beaten, I have a strong
"pious opinion" that it is the highest number of rows obtainable.

22.—The Franklin's Puzzle.
The answer to this puzzle is shown in the illustration, where the numbers on
the sixteen bottles all add up to 30 in the ten straight directions. The trick
consists in the fact that, although the six bottles (3, 5, 6, 9, 10, and 15) in
which the flowers have been placed are not removed, yet the sixteen need
not occupy exactly the same position on the table as before. The square is,
in fact, formed one step further to the left.

23.—The Squire's Puzzle.
The portrait may be drawn in a single line because it contains only two
points at which an odd number of lines meet, but it is absolutely necessary
to begin at one of these points and end at the other. One point is near the
outer extremity of the King's left eye; the other is below it on the left cheek.

24.—The Friar's Puzzle.
The five hundred silver pennies might have been placed in the four bags, in
accordance with the stated conditions, in exactly 894,348 different ways. If
there had been a thousand coins there would be 7,049,112 ways. It is a
difficult problem in the partition of numbers. I have a single formula for the
solution of any number of coins in the case of four bags, but it was
extremely hard to construct, and the best method is to find the twelve
separate formulas for the different congruences to the modulus 12.

25.—The Parson's Puzzle.

A very little examination of the original drawing will have shown the reader
that, as he will have at first read the conditions, the puzzle is quite
impossible of solution. We have therefore to look for some loophole in the
actual conditions as they were worded. If the Parson could get round the
source of the river, he could then cross every bridge once and once only on
his way to church, as shown in the annexed illustration. That this was not
prohibited we shall soon find. Though the plan showed all the bridges in his
parish, it only showed "part of" the parish itself. It is not stated that the river
did not take its rise in the parish, and since it leads to the only possible
solution, we must assume that it did. The answer would be, therefore, as

shown. It should be noted that we are clearly prevented from considering
the possibility of getting round the mouth of the river, because we are told it
"joined the sea some hundred miles to the south," while no parish ever
extended a hundred miles!

26.—The Haberdasher's Puzzle.

The illustration will show how the triangular piece of cloth may be cut into
four pieces that will fit together and form a perfect square. Bisect AB in D
and BC in E; produce the line AE to F making EF equal to EB; bisect AF in
G and describe the arc AHF; produce EB to H, and EH is the length of the
side of the required square; from E with distance EH, describe the arc HJ,
and make JK equal to BE; now, from the points D and K drop
perpendiculars on EJ at L and M. If you have done this accurately, you will
now have the required directions for the cuts.

I exhibited this problem before the Royal Society, at Burlington House, on
17th May 1905, and also at the Royal Institution in the following month, in
the more general form:—"A New Problem on Superposition: a
demonstration that an equilateral triangle can be cut into four pieces that
may be reassembled to form a square, with some examples of a general
method for transforming all rectilinear triangles into squares by dissection."
It was also issued as a challenge to the readers of the Daily Mail (see issues
of 1st and 8th February 1905), but though many hundreds of attempts were
sent in there was not a single solver. Credit, however, is due to Mr. C. W.
M'Elroy, who alone sent me the correct solution when I first published the
problem in the Weekly Dispatch in 1902.
I add an illustration showing the puzzle in a rather curious practical form, as
it was made in polished mahogany with brass hinges for use by certain
audiences. It will be seen that the four pieces form a sort of chain, and that
when they are closed up in one direction they form the triangle, and when
closed in the other direction they form the square.

27.—The Dyer's Puzzle.
The correct answer is 18,816 different ways. The general formula for six
fleurs-de-lys for all squares greater than 22 is simply this: Six times the
square of the number of combinations of n things, taken three at a time,
where n represents the number of fleurs-de-lys in the side of the square. Of
course where n is even the remainders in rows and columns will be even,
and where n is odd the remainders will be odd.
For further solution, see No. 358 in A. in M.

28.—The Great Dispute between the Friar and the
Sompnour.
In this little problem we attempted to show how, by sophistical reasoning, it
may apparently be proved that the diagonal of a square is of precisely the
same length as two of the sides. The puzzle was to discover the fallacy,
because it is a very obvious fallacy if we admit that the shortest distance
between two points is a straight line. But where does the error come in?
Well, it is perfectly true that so long as our zigzag path is formed of "steps"
parallel to the sides of the square that path must be of the same length as the
two sides. It does not matter if you have to use the most powerful
microscope obtainable; the rule is always true if the path is made up of
steps in that way. But the error lies in the assumption that such a zigzag
path can ever become a straight line. You may go on increasing the number
of steps infinitely—that is, there is no limit whatever theoretically to the
number of steps that can be made—but you can never reach a straight line
by such a method. In fact it is just as much a "jump" to a straight line if you
have a billion steps as it is at the very outset to pass from the two sides to
the diagonal. It would be just as absurd to say we might go on dropping
marbles into a basket until they become sovereigns as to say we can
increase the number of our steps until they become a straight line. There is
the whole thing in a nutshell.

29.—Chaucer's Puzzle.
The surface of water or other liquid is always spherical, and the greater any
sphere is the less is its convexity. Hence the top diameter of any vessel at
the summit of a mountain will form the base of the segment of a greater
sphere than it would at the bottom. This sphere, being greater, must (from
what has been already said) be less convex; or, in other words, the spherical
surface of the water must be less above the brim of the vessel, and
consequently it will hold less at the top of a mountain than at the bottom.
The reader is therefore free to select any mountain he likes in Italy—or
elsewhere!

30.—The Puzzle of the Canon's Yeoman.
The number of different ways is 63,504. The general formula for such
arrangements, when the number of letters in the sentence is 2n + 1, and it is
a palindrome without diagonal readings, is [4(2n - 1)]2.
I think it will be well to give here a formula for the general solution of each
of the four most common forms of the diamond-letter puzzle. By the word
"line" I mean the complete diagonal. Thus in A, B, C, and D, the lines
respectively contain 5, 5, 7, and 9 letters. A has a non-palindrome line (the
word being BOY), and the general solution for such cases, where the line
contains 2n + 1 letters, is 4(2n - 1). Where the line is a single palindrome,
with its middle letter in the centre, as in B, the general formula is [4(2n -
1)]2. This is the form of the Rat-catcher's Puzzle, and therefore the
expression that I have given above. In cases C and D we have double
palindromes, but these two represent very different types. In C, where the
line contains 4n-1 letters, the general expression is 4(22n-2). But D is by far
the most difficult case of all.
I had better here state that in the diamonds under consideration (i.) no
diagonal readings are allowed—these have to be dealt with specially in
cases where they are possible and admitted; (ii.) readings may start
anywhere; (iii.) readings may go backwards and forwards, using letters
more than once in a single reading, but not the same letter twice in
immediate succession. This last condition will be understood if the reader
glances at C, where it is impossible to go forwards and backwards in a
reading without repeating the first O touched—a proceeding which I have
said is not allowed. In the case D it is very different, and this is what
accounts for its greater difficulty. The formula for D is this:

where the number of letters in the line is 4n + 1. In the example given there
are therefore 400 readings for n = 2.

See also Nos. 256, 257, and 258 in A. in M.

31.—The Manciple's Puzzle.
The simple Ploughman, who was so ridiculed for his opinion, was perfectly
correct: the Miller should receive seven pieces of money, and the Weaver
only one. As all three ate equal shares of the bread, it should be evident that
each ate 8/3 of a loaf. Therefore, as the Miller provided 15/3 and ate 8/3, he
contributed 7/3 to the Manciple's meal; whereas the Weaver provided 9/3,
ate 8/3, and contributed only 1/3. Therefore, since they contributed to the
Manciple in the proportion of 7 to 1, they must divide the eight pieces of
money in the same proportion.

PUZZLING TIMES AT SOLVAMHALL
CASTLE
SIR HUGH EXPLAINS HIS PROBLEMS
The friends of Sir Hugh de Fortibus were so perplexed over many of his
strange puzzles that at a gathering of his kinsmen and retainers he
undertook to explain his posers.

"Of a truth," said he, "some of the riddles that I have put forth would greatly
tax the wit of the unlettered knave to rede; yet will I try to show the manner
thereof in such way that all may have understanding. For many there be
who cannot of themselves do all these things, but will yet study them to
their gain when they be given the answers, and will take pleasure therein."

32.—The Game of Bandy-Ball.
Sir Hugh explained, in answer to this puzzle, that as the nine holes were
300, 250, 200, 325, 275, 350, 225, 375, and 400 yards apart, if a man could
always strike the ball in a perfectly straight line and send it at will a
distance of either 125 yards or 100 yards, he might go round the whole
course in 26 strokes. This is clearly correct, for if we call the 125 stroke the
"drive" and the 100 stroke the "approach," he could play as follows:—The
first hole could be reached in 3 approaches, the second in 2 drives, the third
in 2 approaches, the fourth in 2 approaches and 1 drive, the fifth in 3 drives
and 1 backward approach, the sixth in 2 drives and 1 approach, the seventh
in 1 drive and 1 approach, the eighth in 3 drives, and the ninth hole in 4
approaches. There are thus 26 strokes in all, and the feat cannot be
performed in fewer.

33.—Tilting at the Ring.

"By my halidame!" exclaimed Sir Hugh, "if some of yon varlets had been
put in chains, which for their sins they do truly deserve, then would they
well know, mayhap, that the length of any chain having like rings is equal
to the inner width of a ring multiplied by the number of rings and added to
twice the thickness of the iron whereof it is made. It may be shown that the
inner width of the rings used in the tilting was one inch and two-thirds
thereof, and the number of rings Stephen Malet did win was three, and
those that fell to Henry de Gournay would be nine."
The knight was quite correct, for 1-2/3 in. × 3 + 1 in. = 6 in., and 1-2/3 in. x
9 + 1 in. = 16 in. Thus De Gournay beat Malet by six rings. The drawing
showing the rings may assist the reader in verifying the answer and help
him to see why the inner width of a link multiplied by the number of links
and added to twice the thickness of the iron gives the exact length. It will be
noticed that every link put on the chain loses a length equal to twice the
thickness of the iron.

34.—The Noble Demoiselle.

"Some here have asked me," continued Sir Hugh, "how they may find the
cell in the Dungeon of the Death's-head wherein the noble maiden was cast.
Beshrew me! but 'tis easy withal when you do but know how to do it. In
attempting to pass through every door once, and never more, you must take
heed that every cell hath two doors or four, which be even numbers, except
two cells, which have but three. Now, certes, you cannot go in and out of
any place, passing through all the doors once and no more, if the number of
doors be an odd number. But as there be but two such odd cells, yet may
we, by beginning at the one and ending at the other, so make our journey in
many ways with success. I pray you, albeit, to mark that only one of these
odd cells lieth on the outside of the dungeon, so we must perforce start
therefrom. Marry, then, my masters, the noble demoiselle must needs have
been wasting in the other."
The drawing will make this quite clear to the reader. The two "odd cells"
are indicated by the stars, and one of the many routes that will solve the
puzzle is shown by the dotted line. It is perfectly certain that you must start
at the lower star and end at the upper one; therefore the cell with the star
situated over the left eye must be the one sought.

35.—The Archery Butt.

"It hath been said that the proof of a pudding is ever in the eating thereof,
and by the teeth of Saint George I know no better way of showing how this
placing of the figures may be done than by the doing of it. Therefore have I
in suchwise written the numbers that they do add up to twenty and three in
all the twelve lines of three that are upon the butt."
I think it well here to supplement the solution of De Fortibus with a few
remarks of my own. The nineteen numbers may be so arranged that the
lines will add up to any number we may choose to select from 22 to 38
inclusive, excepting 30. In some cases there are several different solutions,
but in the case of 23 there are only two. I give one of these. To obtain the
second solution exchange respectively 7, 10, 5, 8, 9, in the illustration, with

13, 4, 17, 2, 15. Also exchange 18 with 12, and the other numbers may
remain unmoved. In every instance there must be an even number in the
central place, and any such number from 2 to 18 may occur. Every solution
has its complementary. Thus, if for every number in the accompanying
drawing we substitute the difference between it and 20, we get the solution
in the case of 37. Similarly, from the arrangement in the original drawing,
we may at once obtain a solution for the case of 38.

36.—The Donjon Keep Window.
In this case Sir Hugh had greatly perplexed his chief builder by demanding
that he should make a window measuring one foot on every side and
divided by bars into eight lights, having all their sides equal. The illustration
will show how this was to be done. It will be seen that if each side of the
window measures one foot, then each of the eight triangular lights is six
inches on every side.
"Of a truth, master builder," said De Fortibus slyly to the architect, "I did
not tell thee that the window must be square, as it is most certain it never
could be."

37.—The Crescent and the Cross.
"By the toes of St. Moden," exclaimed Sir Hugh de Fortibus when this
puzzle was brought up, "my poor wit hath never shaped a more cunning
artifice or any more bewitching to look upon. It came to me as in a vision,
and ofttimes have I marvelled at the thing, seeing its exceeding difficulty.
My masters and kinsmen, it is done in this wise."

The worthy knight then pointed out that the crescent was of a particular and
somewhat irregular form—the two distances a to b and c to d being straight
lines, and the arcs ac and bd being precisely similar. He showed that if the
cuts be made as in Figure 1, the four pieces will fit together and form a
perfect square, as shown in Figure 2, if we there only regard the three
curved lines. By now making the straight cuts also shown in Figure 2, we
get the ten pieces that fit together, as in Figure 3, and form a perfectly
symmetrical Greek cross. The proportions of the crescent and the cross in
the original illustration were correct, and the solution can be demonstrated
to be absolutely exact and not merely approximate.
I have a solution in considerably fewer pieces, but it is far more difficult to
understand than the above method, in which the problem is simplified by
introducing the intermediate square.

38.—The Amulet.
The puzzle was to place your pencil on the A at the top of the amulet and
count in how many different ways you could trace out the word
"Abracadabra" downwards, always passing from a letter to an adjoining
one.

"Now, mark ye, fine fellows," said Sir Hugh to some who had besought him
to explain, "that at the very first start there be two ways open: whichever B
ye select, there will be two several ways of proceeding (twice times two are
four); whichever R ye select, there be two ways of going on (twice times
four are eight); and so on until the end. Each letter in order from A
downwards may so be reached in 2, 4, 8, 16, 32, etc., ways. Therefore, as
there be ten lines or steps in all from A to the bottom, all ye need do is to
multiply ten 2's together, and truly the result, 1024, is the answer thou dost
seek."

39.—The Snail on the Flagstaff.
Though there was no need to take down and measure the staff, it is
undoubtedly necessary to find its height before the answer can be given. It
was well known among the friends and retainers of Sir Hugh de Fortibus
that he was exactly six feet in height. It will be seen in the original picture
that Sir Hugh's height is just twice the length of his shadow. Therefore we
all know that the flagstaff will, at the same place and time of day, be also
just twice as long as its shadow. The shadow of the staff is the same length
as Sir Hugh's height; therefore this shadow is six feet long, and the flagstaff
must be twelve feet high. Now, the snail, by climbing up three feet in the
daytime and slipping back two feet by night, really advances one foot in a
day of twenty-four hours. At the end of nine days it is three feet from the
top, so that it reaches its journey's end on the tenth day.
The reader will doubtless here exclaim, "This is all very well; but how were
we to know the height of Sir Hugh? It was never stated how tall he was!"
No, it was not stated in so many words, but it was none the less clearly
indicated to the reader who is sharp in these matters. In the original
illustration to the donjon keep window Sir Hugh is shown standing against
a wall, the window in which is stated to be one foot square on the inside.
Therefore, as his height will be found by measurement to be just six times
the inside height of the window, he evidently stands just six feet in his
boots!

40.—Lady Isabel's Casket.
The last puzzle was undoubtedly a hard nut, but perhaps difficulty does not
make a good puzzle any the less interesting when we are shown the
solution. The accompanying diagram indicates exactly how the top of Lady
Isabel de Fitzarnulph's casket was inlaid with square pieces of rare wood
(no two squares alike) and the strip of gold 10 inches by a quarter of an
inch. This is the only possible solution, and it is a singular fact (though I
cannot here show the subtle method of working) that the number, sizes, and
order of those squares are determined by the given dimensions of the strip
of gold, and the casket can have no other dimensions than 20 inches square.
The number in a square indicates the length in inches of the side of that
square, so the accuracy of the answer can be checked almost at a glance.
Sir Hugh de Fortibus made some general concluding remarks on the
occasion that are not altogether uninteresting to-day.

"Friends and retainers," he said, "if the strange offspring of my poor wit
about which we have held pleasant counsel to-night hath mayhap had some
small interest for ye, let these matters serve to call to mind the lesson that
our fleeting life is rounded and beset with enigmas. Whence we came and
whither we go be riddles, and albeit such as these we may never bring
within our understanding, yet there be many others with which we and they
that do come after us will ever strive for the answer. Whether success do
attend or do not attend our labour, it is well that we make the attempt; for
'tis truly good and honourable to train the mind, and the wit, and the fancy
of man, for out of such doth issue all manner of good in ways unforeseen
for them that do come after us."

THE MERRY MONKS OF RIDDLEWELL

41.—The Riddle of the Fish-pond.
Number the fish baskets in the illustration from 1 to 12 in the direction that
Brother Jonathan is seen to be going. Starting from 1, proceed as follows,
where "1 to 4" means, take the fish from basket No. 1 and transfer it to
basket No. 4:—
1 to 4, 5 to 8, 9 to 12, 3 to 6, 7 to 10, 11 to 2, and complete the last
revolution to 1, making three revolutions in all. Or you can proceed this
way:—
4 to 7, 8 to 11, 12 to 3, 2 to 5, 6 to 9, 10 to 1.
It is easy to solve in four revolutions, but the solutions in three are more
difficult to discover.

42.—The Riddle of the Pilgrims.
If it were not for the Abbot's conditions that the number of guests in any
room may not exceed three, and that every room must be occupied, it would
have been possible to accommodate either 24, 27, 30, 33, 36, 39, or 42
pilgrims. But to accommodate 24 pilgrims so that there shall be twice as
many sleeping on the upper floor as on the lower floor, and eleven persons
on each side of the building, it will be found necessary to leave some of the
rooms empty. If, on the other hand, we try to put up 33, 36, 39 or 42
pilgrims, we shall find that in every case we are obliged to place more than
three persons in some of the rooms. Thus we know that the number of
pilgrims originally announced (whom, it will be remembered, it was
possible to accommodate under the conditions of the Abbot) must have
been 27, and that, since three more than this number were actually provided
with beds, the total number of pilgrims was 30. The accompanying diagram
shows how they might be arranged, and if in each instance we regard the
upper floor as placed above the lower one, it will be seen that there are
eleven persons on each side of the building, and twice as many above as
below.

43.—The Riddle of the Tiled Hearth.
The correct answer is shown in the illustration on page 196. No tile is in
line (either horizontally, vertically, or diagonally) with another tile of the
same design, and only three plain tiles are used. If after placing the four
lions you fall into the error of placing four other tiles of another pattern,
instead of only three, you will be left with four places that must be occupied
by plain tiles. The secret consists in placing four of one kind and only three
of each of the others.

44.—The Riddle of the Sack of Wine.
The question was: Did Brother Benjamin take more wine from the bottle
than water from the jug? Or did he take more water from the jug than wine
from the bottle? He did neither. The same quantity of wine was transferred
from the bottle as water was taken from the jug. Let us assume that the
glass would hold a quarter of a pint. There was a pint of wine in the bottle
and a pint of water in the jug. After the first manipulation the bottle contains
three-quarters of a pint of wine, and the jug one pint of water mixed with a
quarter of a pint of wine. Now, the second transaction consists in taking
away a fifth of the contents of the jug—that is, one-fifth of a pint of water
mixed with one-fifth of a quarter of a pint of wine. We thus leave behind in
the jug four-fifths of a quarter of a pint of wine—that is, one-fifth of a pint
—while we transfer from the jug to the bottle an equal quantity (one-fifth of
a pint) of water.

45.—The Riddle of the Cellarer.
There were 100 pints of wine in the cask, and on thirty occasions John the
Cellarer had stolen a pint and replaced it with a pint of water. After the first
theft the wine left in the cask would be 99 pints; after the second theft the
wine in the cask would be 9801/100 pints (the square of 99 divided by 100);
after the third theft there would remain 970299/10000 (the cube of 99
divided by the square of 100); after the fourth theft there would remain the
fourth power of 99 divided by the cube of 100; and after the thirtieth theft
there would remain in the cask the thirtieth power of 99 divided by the
twenty-ninth power of 100. This by the ordinary method of calculation
gives us a number composed of 59 figures to be divided by a number
composed of 58 figures! But by the use of logarithms it may be quickly
ascertained that the required quantity is very nearly 73-97/100 pints of wine
left in the cask. Consequently the cellarer stole nearly 26.03 pints. The
monks doubtless omitted the answer for the reason that they had no tables
of logarithms, and did not care to face the task of making that long and
tedious calculation in order to get the quantity "to a nicety," as the wily
cellarer had stipulated.
By a simplified process of calculation, I have ascertained that the exact
quantity of wine stolen would be
26.0299626611719577269984907683285057747323737647323555652999
pints. A man who would involve the monastery in a fraction of fifty-eight
decimals deserved severe punishment.

46.—The Riddle of the Crusaders.
The correct answer is that there would have been 602,176 Crusaders, who
could form themselves into a square 776 by 776; and after the stranger
joined their ranks, they could form 113 squares of 5,329 men—that is, 73
by 73. Or 113 × 732 - 1 = 7762. This is a particular case of the so-called
"Pellian Equation," respecting which see A. in M., p. 164.

47.—The Riddle of St. Edmondsbury.
The reader is aware that there are prime numbers and composite whole
numbers. Now, 1,111,111 cannot be a prime number, because if it were the
only possible answers would be those proposed by Brother Benjamin and
rejected by Father Peter. Also it cannot have more than two factors, or the
answer would be indeterminate. As a matter of fact, 1,111,111 equals 239 x
4649 (both primes), and since each cat killed more mice than there were
cats, the answer must be 239 cats. See also the Introduction, p. 18.
Treated generally, this problem consists in finding the factors, if any, of
numbers of the form (10n - 1)/9.
Lucas, in his L'Arithmétique Amusante, gives a number of curious tables
which he obtained from an arithmetical treatise, called the Talkhys, by Ibn
Albanna, an Arabian mathematician and astronomer of the first half of the
thirteenth century. In the Paris National Library are several manuscripts
dealing with the Talkhys, and a commentary by Alkalaçadi, who died in
1486. Among the tables given by Lucas is one giving all the factors of
numbers of the above form up to n = 18. It seems almost inconceivable that
Arabians of that date could find the factors where n = 17, as given in my
Introduction. But I read Lucas as stating that they are given in Talkhys,
though an eminent mathematician reads him differently, and suggests to me
that they were discovered by Lucas himself. This can, of course, be settled
by an examination of Talkhys, but this has not been possible during the war.
The difficulty lies wholly with those cases where n is a prime number. If n
= 2, we get the prime 11. The factors when n = 3, 5, 11, and 13 are
respectively (3 . 37), (41 . 271), (21,649 . 513,239), and (53 . 79 .
265371653). I have given in these pages the factors where n = 7 and 17.
The factors when n= 19, 23, and 37 are unknown, if there are any.[B] When
n = 29, the factors are (3,191 . 16,763 . 43,037. 62,003 . 77,843,839,397);
when n = 31, one factor is 2,791; and when n = 41, two factors are (83 .
1,231).

[B] Mr. Oscar Hoppe, of New York, informs me that, after reading my statement
in the Introduction, he was led to investigate the case of n = 19, and after long
and tedious work he succeeded in proving the number to be a prime. He
submitted his proof to the London Mathematical Society, and a specially
appointed committee of that body accepted the proof as final and conclusive. He
refers me to the Proceedings of the Society for 14th February 1918.

As for the even values of n, the following curious series of factors will
doubtless interest the reader. The numbers in brackets are primes.
n = 2 = (11)
n = 6 = (11) × 111 × 91
n = 10 = (11) × 11,111 × (9,091)
n = 14 = (11) × 1,111,111 × (909,091)
n = 18 = (11) × 111,111,111 × 90,909,091
Or we may put the factors this way:—
n = 2 = (11)
n = 6 = 111 × 1,001
n = 10 = 11,111 × 100,001
n = 14 = 1,111,111 × 10,000,001
n = 18 = 111,111,111 × 1,000,000,001
In the above two tables n is of the form 4m + 2. When n is of the form 4m
the factors may be written down as follows:—
n= 4 = (11) × (101)
n = 8 = (11) × (101) × 10,001
n = 12 = (11) × (101) × 100,010,001
n = 16 = (11) × (101) × 1,000,100,010,001.
When n = 2, we have the prime number 11; when n = 3, the factors are 3 .
37; when n = 6, they are 11 . 3 . 37 . 7. 13; when n = 9, they are 32 . 37 .

333,667. Therefore we know that factors of n = 18 are 11. 32 . 37 . 7 . 13 .
333,667, while the remaining factor is composite and can be split into 19 .
52579. This will show how the working may be simplified when n is not
prime.

48.—The Riddle of the Frogs' Ring.
The fewest possible moves in which this puzzle can be solved are 118. I
will give the complete solution. The black figures on white discs move in
the directions of the hands of a clock, and the white figures on black discs
the other way. The following are the numbers in the order in which they
move. Whether you have to make a simple move or a leaping move will be
clear from the position, as you never can have an alternative. The moves
enclosed in brackets are to be played five times over: 6, 7, 8, 6, 5, 4, 7, 8, 9,
10, 6, 5, 4, 3, 2, 7, 8, 9, 10, 11 (6, 5, 4, 3, 2, 1), 6, 5, 4, 3, 2, 12, (7, 8, 9, 10,
11, 12), 7, 8, 9, 10, 11, 1, 6, 5, 4, 3, 2, 12, 7, 8, 9, 10, 11, 6, 5, 4, 3, 2, 8, 9,
10, 11, 4, 3, 2, 10, 11, 2. We thus have made 118 moves within the
conditions, the black frogs have changed places with the white ones, and 1
and 12 are side by side in the positions stipulated.

The general solution in the case of this puzzle is 3n2 + 2n - 2 moves, where
the number of frogs of each colour is n. The law governing the sequence of
moves is easily discovered by an examination of the simpler cases, where n
= 2, 3, and 4.
If, instead of 11 and 12 changing places, the 6 and 7 must interchange, the
expression is n2 + 4n + 2 moves. If we give n the value 6, as in the example
of the Frogs' Ring, the number of moves would be 62.
For a general solution of the case where frogs of one colour reverse their
order, leaving the blank space in the same position, and each frog is allowed
to be moved in either direction (leaping, of course, over his own colour),
see "The Grasshopper Puzzle" in A. in M., p. 193.

THE STRANGE ESCAPE OF THE KING'S
JESTER
Although the king's jester promised that he would "thereafter make the
manner thereof plain to all," there is no record of his having ever done so. I
will therefore submit to the reader my own views as to the probable
solutions to the mysteries involved.

49.—The Mysterious Rope.
When the jester "divided his rope in half," it does not follow that he cut it
into two parts, each half the original length of the rope. No doubt he simply
untwisted the strands, and so divided it into two ropes, each of the original
length, but one-half the thickness. He would thus be able to tie the two
together and make a rope nearly twice the original length, with which it is
quite conceivable that he made good his escape from the dungeon.

50.—The Underground Maze.
How did the jester find his way out of the maze in the dark? He had simply
to grope his way to a wall and then keep on walking without once removing
his left hand (or right hand) from the wall. Starting from A, the dotted line
will make the route clear when he goes to the left. If the reader tries the
route to the right in the same way he will be equally successful; in fact, the
two routes unite and cover every part of the walls of the maze except those
two detached parts on the left-hand side—one piece like a U, and the other
like a distorted E. This rule will apply to the majority of mazes and puzzle
gardens; but if the centre were enclosed by an isolated wall in the form of a
split ring, the jester would simply have gone round and round this ring.

See the article, "Mazes, and How to Thread Them," in A. in M.

51.—The Secret Lock.
This puzzle entailed the finding of an English word of three letters, each
letter being found on a different dial. Now, there is no English word
composed of consonants alone, and the only vowel appearing anywhere on
the dials is Y. No English word begins with Y and has the two other letters
consonants, and all the words of three letters ending in Y (with two
consonants) either begin with an S or have H, L, or R as their second letter.
But these four consonants do not appear. Therefore Y must occur in the
middle, and the only word that I can find is "PYX," and there can be little
doubt that this was the word. At any rate, it solves our puzzle.

52.—Crossing the Moat.
No doubt some of my readers will smile at the statement that a man in a
boat on smooth water can pull himself across with the tiller rope! But it is a
fact. If the jester had fastened the end of his rope to the stern of the boat and
then, while standing in the bows, had given a series of violent jerks, the
boat would have been propelled forward. This has often been put to a
practical test, and it is said that a speed of two or three miles an hour may
be attained. See W. W. Rouse Ball's Mathematical Recreations.

53.—The Royal Gardens.

This puzzle must have struck many readers as being absolutely impossible.
The jester said: "I had, of a truth, entered every one of the sixteen gardens
once, and never more than once." If we follow the route shown in the
accompanying diagram, we find that there is no difficulty in once entering
all the gardens but one before reaching the last garden containing the exit B.
The difficulty is to get into the garden with a star, because if we leave the B
garden we are compelled to enter it a second time before escaping, and no
garden may be entered twice. The trick consists in the fact that you may
enter that starred garden without necessarily leaving the other. If, when the
jester got to the gateway where the dotted line makes a sharp bend, his
intention had been to hide in the starred garden, but after he had put one
foot through the doorway, upon the star, he discovered it was a false alarm
and withdrew, he could truly say: "I entered the starred garden, because I
put my foot and part of my body in it; and I did not enter the other garden
twice, because, after once going in I never left it until I made my exit at B."
This is the only answer possible, and it was doubtless that which the jester
intended.
See "The Languishing Maiden," in A. in M.

54.—Bridging the Ditch.

The solution to this puzzle is best explained by the illustration. If he had
placed his eight planks, in the manner shown, across the angle of the ditch,
he would have been able to cross without much trouble. The king's jester
might thus have well overcome all his difficulties and got safely away, as he
has told us that he succeeded in doing.

THE SQUIRE'S CHRISTMAS PUZZLE PARTY
HOW THE VARIOUS TRICKS WERE DONE
The record of one of Squire Davidge's annual "Puzzle Parties," made by the
old gentleman's young lady relative, who had often spent a merry Christmas
at Stoke Courcy Hall, does not contain the solutions of the mysteries. So I
will give my own answers to the puzzles and try to make them as clear as
possible to those who may be more or less novices in such matters.

55.—The Three Teacups.

Miss Charity Lockyer clearly must have had a trick up her sleeve, and I
think it highly probable that it was conceived on the following lines. She
proposed that ten lumps of sugar should be placed in three teacups, so that
there should be an odd number of lumps in every cup. The illustration
perhaps shows Miss Charity's answer, and the figures on the cups indicate
the number of lumps that have been separately placed in them. By placing
the cup that holds one lump inside the one that holds two lumps, it can be
correctly stated that every cup contains an odd number of lumps. One cup
holds seven lumps, another holds one lump, while the third cup holds three
lumps. It is evident that if a cup contains another cup it also contains the
contents of that second cup.
There are in all fifteen different solutions to this puzzle. Here they are:—

1 0 91 4 59 0 1
3 0 77 0 37 2 1
1 2 75 2 35 4 1
5 0 53 4 33 6 1
3 2 51 6 31 8 1

The first two numbers in a triplet represent respectively the number of
lumps to be placed in the inner and outer of the two cups that are placed one
inside the other. It will be noted that the outer cup of the pair may itself be
empty.

56.—The Eleven Pennies.
It is rather evident that the trick in this puzzle was as follows:—From the
eleven coins take five; then add four (to those already taken away) and you
leave nine—in the second heap of those removed!

57.—The Christmas Geese.
Farmer Rouse sent exactly 101 geese to market. Jabez first sold Mr. Jasper
Tyler half of the flock and half a goose over (that is, 50-½ + ½, or 51 geese,
leaving 50); he then sold Farmer Avent a third of what remained and a third
of a goose over (that is, 16-2/3 + 1/3, or 17 geese, leaving 33); he then sold
Widow Foster a quarter of what remained and three-quarters of a goose
over (that is, 8-1/4 + 3/4 or 9 geese, leaving 24); he next sold Ned Collier a
fifth of what he had left and gave him a fifth of a goose "for the missus"
(that is, 4-4/5 + 1/5 or 5 geese, leaving 19). He then took these 19 back to
his master.

58.—The Chalked Numbers.
This little jest on the part of Major Trenchard is another trick puzzle, and
the face of the roguish boy on the extreme right, with the figure 9 on his
back, showed clearly that he was in the secret, whatever that secret might
be. I have no doubt (bearing in mind the Major's hint as to the numbers
being "properly regarded") that his answer was that depicted in the
illustration, where boy No. 9 stands on his head and so converts his number
into 6. This makes the total 36—an even number—and by making boys 3
and 4 change places with 7 and 8, we get 1278 and 5346, the figures of
which, in each case, add up to 18. There are just three other ways in which
the boys may be grouped: 1368—2457, 1467—2358, and 2367—1458.

59.—Tasting the Plum Puddings.
The diagram will show how this puzzle is to be solved. It is the only way
within the conditions laid down. Starting at the pudding with holly at the
top left-hand corner, we strike out all the puddings in twenty-one straight
strokes, taste the steaming hot pudding at the end of the tenth stroke, and
end at the second sprig of holly.
Here we have an example of a chess rook's path that is not re-entrant, but
between two squares that are at the greatest possible distance from one
another. For if it were desired to move, under the condition of visiting every
square once and once only, from one corner square to the other corner
square on the same diagonal, the feat is impossible.
There are a good many different routes for passing from one sprig of holly
to the other in the smallest possible number of moves—twenty-one—but I
have not counted them. I have recorded fourteen of these, and possibly
there are more. Any one of these would serve our purpose, except for the
condition that the tenth stroke shall end at the steaming hot pudding. This
was introduced to stop a plurality of solutions—called by the maker of
chess problems "cooks." I am not aware of more than one solution to this
puzzle; but as I may not have recorded all the tours, I cannot make a
positive statement on the point at the time of writing.

60.—Under the Mistletoe Bough.
Everybody was found to have kissed everybody else once under the
mistletoe, with the following additions and exceptions: No male kissed a
male; no man kissed a married woman except his own wife; all the
bachelors and boys kissed all the maidens and girls twice; the widower did
not kiss anybody, and the widows did not kiss each other. Every kiss was
returned, and the double performance was to count as one kiss. In making a
list of the company, we can leave out the widower altogether, because he
took no part in the osculatory exercise.

7Married couples 14
3Widows 3
12Bachelors and Boys 12
10Maidens and Girls 10
Total 39Persons

Now, if every one of these 39 persons kissed everybody else once, the
number of kisses would be 741; and if the 12 bachelors and boys each
kissed the 10 maidens and girls once again, we must add 120, making a
total of 861 kisses. But as no married man kissed a married woman other
than his own wife, we must deduct 42 kisses; as no male kissed another
male, we must deduct 171 kisses; and as no widow kissed another widow,
we must deduct 3 kisses. We have, therefore, to deduct 42+171+3=216
kisses from the above total of 861, and the result, 645, represents exactly
the number of kisses that were actually given under the mistletoe bough.

61.—The Silver Cubes.
There is no limit to the number of different dimensions that will give two
cubes whose sum shall be exactly seventeen cubic inches. Here is the
answer in the smallest possible numbers. One of the silver cubes must
measure 2-23278/40831 inches along each edge, and the other must
measure 11663/40831 inch. If the reader likes to undertake the task of
cubing each number (that is, multiply each number twice by itself), he will
find that when added together the contents exactly equal seventeen cubic
inches. See also No. 20, "The Puzzle of the Doctor of Physic."

THE ADVENTURES OF THE PUZZLE CLUB

62.—The Ambiguous Photograph.
One by one the members of the Club succeeded in discovering the key to
the mystery of the Ambiguous Photograph, except Churton, who was at
length persuaded to "give it up." Herbert Baynes then pointed out to him
that the coat that Lord Marksford was carrying over his arm was a lady's
coat, because the buttons are on the left side, whereas a man's coat always
has the buttons on the right-hand side. Lord Marksford would not be likely
to walk about the streets of Paris with a lady's coat over his arm unless he
was accompanying the owner. He was therefore walking with the lady.
As they were talking a waiter brought a telegram to Baynes.
"Here you are," he said, after reading the message. "A wire from Dovey:
'Don't bother about photo. Find lady was the gentleman's sister, passing
through Paris.' That settles it. You might notice that the lady was lightly
clad, and therefore the coat might well be hers. But it is clear that the rain
was only a sudden shower, and no doubt they were close to their
destination, and she did not think it worth while to put the coat on."

63.—The Cornish Cliff Mystery.
Melville's explanation of the Cornish Cliff Mystery was very simple when
he gave it. Yet it was an ingenious trick that the two criminals adopted, and
it would have completely succeeded had not our friends from the Puzzle
Club accidentally appeared on the scene. This is what happened: When
Lamson and Marsh reached the stile, Marsh alone walked to the top of the
cliff, with Lamson's larger boots in his hands. Arrived at the edge of the
cliff, he changed the boots and walked backwards to the stile, carrying his
own boots.
This little manœuvre accounts for the smaller footprints showing a deeper
impression at the heel, and the larger prints a deeper impression at the toe;
for a man will walk more heavily on his heels when going forward, but will
make a deeper impression with the toes in walking backwards. It will also
account for the fact that the large footprints were sometimes impressed over
the smaller ones, but never the reverse; also for the circumstance that the
larger footprints showed a shorter stride, for a man will necessarily take a
smaller stride when walking backwards. The pocket-book was intentionally
dropped, to lead the police to discover the footprints, and so be put on the
wrong scent.

64.—The Runaway Motor-Car.
Russell found that there are just twelve five-figure numbers that have the
peculiarity that the first two figures multiplied by the last three—all the
figures being different, and there being no 0—will produce a number with
exactly the same five figures, in a different order. But only one of these
twelve begins with a 1—namely, 14926. Now, if we multiply 14 by 926, the
result is 12964, which contains the same five figures. The number of the
motor-car was therefore 14926.
Here are the other eleven numbers:—24651, 42678, 51246, 57834, 75231,
78624, 87435, 72936, 65281, 65983, and 86251.
Compare with the problems in "Digital Puzzles," section of A. in M., and
with Nos. 93 and 101 in these pages.

65.—The Mystery of Ravensdene Park.
The diagrams show that there are two different ways in which the routes of
the various persons involved in the Ravensdene Mystery may be traced,
without any path ever crossing another. It depends whether the butler, E,
went to the north or the south of the gamekeeper's cottage, and the
gamekeeper, A, went to the south or the north of the hall. But it will be
found that the only persons who could have approached Mr. Cyril Hastings
without crossing a path were the butler, E, and the man, C. It was, however,
a fact that the butler retired to bed five minutes before midnight, whereas
Mr. Hastings did not leave his friend's house until midnight. Therefore the
criminal must have been the man who entered the park at C.

66.—The Buried Treasure.
The field must have contained between 179 and 180 acres—to be more
exact, 179.37254 acres. Had the measurements been 3, 2, and 4 furlongs
respectively from successive corners, then the field would have been
209.70537 acres in area.
One method of solving this problem is as follows. Find the area of triangle
APB in terms of x, the side of the square. Double the result=xy. Divide by x
and then square, and we have the value of y2 in terms of x. Similarly find
value of z2 in terms of x; then solve the equation y2+z2=32, which will come
out in the form x4-20x2=-37. Therefore x2=10+(sqrt{63})=17.937254 square
furlongs, very nearly, and as there are ten acres in one square furlong, this
equals 179.37254 acres. If we take the negative root of the equation, we get
the area of the field as 20.62746 acres, in which case the treasure would
have been buried outside the field, as in Diagram 2. But this solution is
excluded by the condition that the treasure was buried in the field. The
words were, "The document ... states clearly that the field is square, and that
the treasure is buried in it."

THE PROFESSOR'S PUZZLES

67.—The Coinage Puzzle.
The point of this puzzle turns on the fact that if the magic square were to be
composed of whole numbers adding up 15 in all ways, the two must be
placed in one of the corners. Otherwise fractions must be used, and these
are supplied in the puzzle by the employment of sixpences and half-crowns.
I give the arrangement requiring the fewest possible current English coins
—fifteen. It will be seen that the amount in each corner is a fractional one,
the sum required in the total being a whole number of shillings.

68.—The Postage Stamps Puzzles.
The first of these puzzles is based on a similar principle, though it is really
much easier, because the condition that nine of the stamps must be of
different values makes their selection a simple matter, though how they are
to be placed requires a little thought or trial until one knows the rule
respecting putting the fractions in the corners. I give the solution.

I also show the solution to the second stamp puzzle. All the columns, rows,
and diagonals add up 1s. 6d. There is no stamp on one square, and the
conditions did not forbid this omission. The stamps at present in circulation
are these:—½d., 1d., 1-½d., 2d., 2-½d., 3d., 4d., 5d., 6d., 9d., 10d., 1s., 2s.
6d., 5s., 10s., £1, and £5. In the first solution the numbers are in
arithmetical progression—1, 1-½, 2, 2-½, 3, 3-½, 4, 4-½, 5. But any nine
numbers will form a magic square if we can write them thus:—

1 2 3
7 8 9
1314 15

where the horizontal differences are all alike and the vertical differences all
alike, but not necessarily the same as the horizontal. This happens in the
case of the second solution, the numbers of which may be written:—

0 1 2
5 6 7
1011 12

Also in the case of the solution to No. 67, the Coinage Puzzle, the numbers
are, in shillings:—

2 2½ 3
4½ 5 5½
7 7½ 8

If there are to be nine different numbers, 0 may occur once (as in the
solution to No. 22). Yet one might construct squares with negative numbers,
as follows:—

-2 -1 0
5 6 7
1213 14

69.—The Frogs and Tumblers.
It is perfectly true, as the Professor said, that there is only one solution (not
counting a reversal) to this puzzle. The frogs that jump are George in the
third horizontal row; Chang, the artful-looking batrachian at the end of the
fourth row; and Wilhelmina, the fair creature in the seventh row. George
jumps downwards to the second tumbler in the seventh row; Chang, who
can only leap short distances in consequence of chronic rheumatism,
removes somewhat unwillingly to the glass just above him—the eighth in
the third row; while Wilhelmina, with all the sprightliness of her youth and
sex, performs the very creditable saltatory feat of leaping to the fourth
tumbler in the fourth row. In their new positions, as shown in the
accompanying diagram, it will be found that of the eight frogs no two are in
line vertically, horizontally, or diagonally.

70.—Romeo and Juliet.
This is rather a difficult puzzle, though, as the Professor remarked when
Hawkhurst hit on the solution, it is "just one of those puzzles that a person
might solve at a glance" by pure luck. Yet when the solution, with its pretty,
symmetrical arrangement, is seen, it looks ridiculously simple.
It will be found that Romeo reaches Juliet's balcony after visiting every
house once and only once, and making fourteen turnings, not counting the
turn he makes at starting. These are the fewest turnings possible, and the
problem can only be solved by the route shown or its reversal.

71.—Romeo's Second Journey.

In order to take his trip through all the white squares only with the fewest
possible turnings, Romeo would do well to adopt the route I have shown, by
means of which only sixteen turnings are required to perform the feat. The
Professor informs me that the Helix Aspersa, or common or garden snail,
has a peculiar aversion to making turnings—so much so that one specimen
with which he made experiments went off in a straight line one night and
has never come back since.

72.—The Frogs who would a-wooing go.
This is one of those puzzles in which a plurality of solutions is practically
unavoidable. There are two or three positions into which four frogs may
jump so as to form five rows with four in each row, but the case I have
given is the most satisfactory arrangement.

The frogs that have jumped have left their astral bodies behind, in order to
show the reader the positions which they originally occupied. Chang, the
frog in the middle of the upper row, suffering from rheumatism, as
explained above in the Frogs and Tumblers solution, makes the shortest
jump of all—a little distance between the two rows; George and Wilhelmina
leap from the ends of the lower row to some distance N. by N.W. and N. by
N.E. respectively; while the frog in the middle of the lower row, whose
name the Professor forgot to state, goes direct S.

73.—The Game of Kayles.
To win at this game you must, sooner or later, leave your opponent an even
number of similar groups. Then whatever he does in one group you repeat
in a similar group. Suppose, for example, that you leave him these groups:
o.o.ooo.ooo. Now, if he knocks down a single, you knock down a single; if
he knocks down two in one triplet, you knock down two in the other triplet;
if he knocks down the central kayle in a triplet, you knock down the central
one in the other triplet. In this way you must eventually win. As the game is
started with the arrangement o.ooooooooooo, the first player can always
win, but only by knocking down the sixth or tenth kayle (counting the one
already fallen as the second), and this leaves in either case o.ooo.ooooooo,
as the order of the groups is of no importance. Whatever the second player
now does, this can always be resolved into an even number of equal groups.
Let us suppose that he knocks down the single one; then we play to leave
him oo.ooooooo. Now, whatever he does we can afterwards leave him
either ooo.ooo or o.oo.ooo. We know why the former wins, and the latter
wins also; because, however he may play, we can always leave him either
o.o, or o.o.o.o, or oo.oo, as the case may be. The complete analysis I can
now leave for the amusement of the reader.

74.—The Broken Chessboard.
The illustration will show how the thirteen pieces can be put together so as
to construct the perfect board, and the reverse problem of cutting these
particular pieces out will be found equally entertaining.

Compare with Nos. 293 and 294 in A. in M.

75.—The Spider and the Fly.
Though this problem was much discussed in the Daily Mail from 18th
January to 7th February 1905, when it appeared to create great public
interest, it was actually first propounded by me in the Weekly Dispatch of
14th June 1903.
Imagine the room to be a cardboard box. Then the box may be cut in
various different ways, so that the cardboard may be laid flat on the table. I
show four of these ways, and indicate in every case the relative positions of
the spider and the fly, and the straightened course which the spider must
take without going off the cardboard. These are the four most favourable
cases, and it will be found that the shortest route is in No. 4, for it is only 40
feet in length (add the square of 32 to the square of 24 and extract the
square root). It will be seen that the spider actually passes along five of the
six sides of the room! Having marked the route, fold the box up (removing
the side the spider does not use), and the appearance of the shortest course
is rather surprising. If the spider had taken what most persons will consider
obviously the shortest route (that shown in No. 1), he would have gone 42
feet! Route No. 2 is 43.174 feet in length, and Route No. 3 is 40.718 feet.

I will leave the reader to discover which are the shortest routes when the
spider and the fly are 2, 3, 4, 5, and 6 feet from the ceiling and the floor
respectively.

76.—The Perplexed Cellarman.
Brother John gave the first man three large bottles and one small bottleful
of wine, and one large and three small empty bottles. To each of the other
two men he gave two large and three small bottles of wine, and two large
and one small empty bottle. Each of the three then receives the same
quantity of wine, and the same number of each size of bottle.

77.—Making a Flag.
The diagram shows how the piece of bunting is to be cut into two pieces.
Lower the piece on the right one "tooth," and they will form a perfect
square, with the roses symmetrically placed.

It will be found interesting to compare this with No. 154 in A. in M.

78.—Catching the Hogs.
A very short examination of this puzzle game should convince the reader
that Hendrick can never catch the black hog, and that the white hog can
never be caught by Katrün.
Each hog merely runs in and out of one of the nearest corners and can never
be captured. The fact is, curious as it must at first sight appear, a Dutchman
cannot catch a black hog, and a Dutchwoman can never capture a white
one! But each can, without difficulty, catch one of the other colour.
So if the first player just determines that he will send Hendrick after the
white porker and Katrün after the black one, he will have no difficulty
whatever in securing both in a very few moves.
It is, in fact, so easy that there is no necessity whatever to give the line of
play. We thus, by means of the game, solve the puzzle in real life, why the
Dutchman and his wife could not catch their pigs: in their simplicity and
ignorance of the peculiarities of Dutch hogs, each went after the wrong
animal.
The little principle involved in this puzzle is that known to chess-players as
"getting the opposition." The rule, in the case of my puzzle (where the
moves resemble rook moves in chess, with the added condition that the
rook may only move to an adjoining square), is simply this. Where the
number of squares on the same row, between the man or woman and the
hog, is odd, the hog can never be captured; where the number of squares is
even, a capture is possible. The number of squares between Hendrick and
the black hog, and between Katrün and the white hog, is 1 (an odd number),
therefore these individuals cannot catch the animals they are facing. But the
number between Hendrick and the white hog, and between Katrün and the
black one, is 6 (an even number), therefore they may easily capture those
behind them.

79.—The Thirty-one Game.
By leading with a 5 the first player can always win. If your opponent plays
another 5, you play a 2 and score 12. Then as often as he plays a 5 you play
a 2, and if at any stage he drops out of the series, 3, 10, 17, 24, 31, you step
in and win. If after your lead of 5 he plays anything but another 5, you
make 10 or 17 and win. The first player may also win by leading a 1 or a 2,
but the play is complicated. It is, however, well worth the reader's study.

80.—The Chinese Railways.
This puzzle was artfully devised by the yellow man. It is not a matter for
wonder that the representatives of the five countries interested were
bewildered. It would have puzzled the engineers a good deal to construct
those circuitous routes so that the various trains might run with safety.
Diagram 1 shows directions for the five systems of lines, so that no line
shall ever cross another, and this appears to be the method that would
require the shortest possible mileage.

The reader may wish to know how many different solutions there are to the
puzzle. To this I should answer that the number is indeterminate, and I will

explain why. If we simply consider the case of line A alone, then one route
would be Diagram 2, another 3, another 4, and another 5. If 3 is different
from 2, as it undoubtedly is, then we must regard 5 as different from 4. But
a glance at the four diagrams, 2, 3, 4, 5, in succession will show that we
may continue this "winding up" process for ever; and as there will always
be an unobstructed way (however long and circuitous) from stations B and
E to their respective main lines, it is evident that the number of routes for
line A alone is infinite. Therefore the number of complete solutions must
also be infinite, if railway lines, like other lines, have no breadth; and
indeterminate, unless we are told the greatest number of parallel lines that it
is possible to construct in certain places. If some clear condition, restricting
these "windings up," were given, there would be no great difficulty in
giving the number of solutions. With any reasonable limitation of the kind,
the number would, I calculate, be little short of two thousand, surprising
though it may appear.

81.—The Eight Clowns.
This is a little novelty in magic squares. These squares may be formed with
numbers that are in arithmetical progression, or that are not in such
progression. If a square be formed of the former class, one place may be left
vacant, but only under particular conditions. In the case of our puzzle there
would be no difficulty in making the magic square with 9 missing; but with
1 missing (that is, using 2, 3, 4, 5, 6, 7, 8, and 9) it is not possible. But a
glance at the original illustration will show that the numbers we have to
deal with are not actually those just mentioned. The clown that has a 9 on
his body is portrayed just at the moment when two balls which he is
juggling are in mid-air. The positions of these balls clearly convert his
figure into the recurring decimal .̍9. Now, since the recurring decimal .̍9 is
equal to 9/9, and therefore to 1, it is evident that, although the clown who
bears the figure 1 is absent, the man who bears the figure 9 by this simple
artifice has for the occasion given his figure the value of the number 1. The
troupe can consequently be grouped in the following manner:—

7 5
246
38.̍9

Every column, every row, and each of the two diagonals now add up to 12.
This is the correct solution to the puzzle.

82.—The Wizard's Arithmetic.
This puzzle is both easy and difficult, for it is a very simple matter to find
one of the multipliers, which is 86. If we multiply 8 by 86, all we need do is
to place the 6 in front and the 8 behind in order to get the correct answer,
688. But the second number is not to be found by mere trial. It is 71, and
the number to be multiplied is no less than
1639344262295081967213114754098360655737704918032787. If you
want to multiply this by 71, all you have to do is to place another 1 at the
beginning and another 7 at the end—a considerable saving of labour! These
two, and the example shown by the wizard, are the only two-figure
multipliers, but the number to be multiplied may always be increased. Thus,
if you prefix to 41096 the number 41095890, repeated any number of times,
the result may always be multiplied by 83 in the wizard's peculiar manner.
If we add the figures of any number together and then, if necessary, again
add, we at last get a single-figure number. This I call the "digital root."
Thus, the digital root of 521 is 8, and of 697 it is 4. This digital analysis is
extensively dealt with in A. in M. Now, it is evident that the digital roots of
the two numbers required by the puzzle must produce the same root in sum
and product. This can only happen when the roots of the two numbers are 2
and 2, or 9 and 9, or 3 and 6, or 5 and 8. Therefore the two-figure multiplier
must have a digital root of 2, 3, 5, 6, 8, or 9. There are ten such numbers in
each case. I write out all the sixty, then I strike out all those numbers where
the second figure is higher than the first, and where the two figures are alike
(thirty-six numbers in all); also all remaining numbers where the first figure
is odd and the second figure even (seven numbers); also all multiples of 5
(three more numbers). The numbers 21 and 62 I reject on inspection, for
reasons that I will not enter into. I then have left, out of the original sixty,
only the following twelve numbers: 83, 63, 81, 84, 93, 42, 51, 87, 41, 86,
53, and 71. These are the only possible multipliers that I have really to
examine.
My process is now as curious as it is simple in working. First trying 83, I
deduct 10 and call it 73. Adding 0's to the second figure, I say if 30000, etc.,

ever has a remainder 43 when divided by 73, the dividend will be the
required multiplier for 83. I get the 43 in this way. The only multiplier of 3
that produces an 8 in the digits place is 6. I therefore multiply 73 by 6 and
get 438, or 43 after rejecting the 8. Now, 300,000 divided by 73 leaves the
remainder 43, and the dividend is 4,109. To this 1 add the 6 mentioned
above and get 41,096 x 83, the example given on page 129.
In trying the even numbers there are two cases to be considered. Thus,
taking 86, we may say that if 60000, etc., when divided by 76 leaves either
22 or 60 (because 3×6 and 8×6 both produce 8), we get a solution. But I
reject the former on inspection, and see that 60 divided by 76 is 0, leaving a
remainder 60. Therefore 8 x 86 = 688, the other example. It will be found in
the case of 71 that 100000, etc., divided by 61 gives a remainder 42, (7 × 61
= 427) after producing the long dividend at the beginning of this article,
with the 7 added.
The other multipliers fail to produce a solution, so 83, 86, and 71 are the
only three possible multipliers. Those who are familiar with the principle of
recurring decimals (as somewhat explained in my next note on No. 83, "The
Ribbon Problem") will understand the conditions under which the
remainders repeat themselves after certain periods, and will only find it
necessary in two or three cases to make any lengthy divisions. It clearly
follows that there is an unlimited number of multiplicands for each
multiplier.

83.—The Ribbon Problem.
The solution is as follows: Place this rather lengthy number on the ribbon,
0212765957446808510638297872340425531914393617. It may be
multiplied by any number up to 46 inclusive to give the same order of
figures in the ring. The number previously given can be multiplied by any
number up to 16. I made the limit 9 in order to put readers off the scent. The
fact is these two numbers are simply the recurring decimals that equal 1/17
and 1/47 respectively. Multiply the one by seventeen and the other by forty-
seven, and you will get all nines in each case.
In transforming a vulgar fraction, say 1/17, to a decimal fraction, we
proceed as below, adding as many noughts to the dividend as we like until
there is no remainder, or until we get a recurring series of figures, or until
we have carried it as far as we require, since every additional figure in a
never-ending decimal carries us nearer and nearer to exactitude.

17) 100(.058823
85
——
150
136
——
140
136
——
40
34
——
60
51
——
9

Now, since all powers of 10 can only contain factors of the powers of 2 and
5, it clearly follows that your decimal never will come to an end if any other
factor than these occurs in the denominator of your vulgar fraction. Thus,
½, 1/4, and 1/8 give us the exact decimals, .5, .25, and .125; 1/5 and 1/25
give us .2 and .04; 1/10 and 1/20 give us .1 and .05: because the
denominators are all composed of 2 and 5 factors. But if you wish to
convert 1/3, 1/6, or 1/7, your division sum will never end, but you will get
these decimals, .33333, etc., .166666, etc., and .142857142857142857, etc.,
where, in the first case, the 3 keeps on repeating for ever and ever; in the
second case the 6 is the repeater, and in the last case we get the recurring
period of 142857. In the case of 1/17 (in "The Ribbon Problem") we find
the circulating period to be .0588235294117647.
Now, in the division sum above, the successive remainders are 1, 10, 15, 14,
4, 6, 9, etc., and these numbers I have inserted around the inner ring of the
diagram. It will be seen that every number from 1 to 16 occurs once, and
that if we multiply our ribbon number by any one of the numbers in the
inner ring its position indicates exactly the point at which the product will
begin. Thus, if we multiply by 4, the product will be 235, etc.; if we
multiply by 6, 352, etc. We can therefore multiply by any number from 1 to
16 and get the desired result.

The kernel of the puzzle is this: Any prime number, with the exception of 2
and 5, which are the factors of 10, will exactly divide without remainder a
number consisting of as many nines as the number itself, less one. Thus
999999 (six 9's) is divisible by 7, sixteen 9's are divisible by 17, eighteen
9's by 19, and so on. This is always the case, though frequently fewer 9's
will suffice; for one 9 is divisible by 3, two by 11, six by 13, when our
ribbon rule for consecutive multipliers breaks down and another law comes
in. Therefore, since the 0 and 7 at the ends of the ribbon may not be
removed, we must seek a fraction with a prime denominator ending in 7
that gives a full period circulator. We try 37, and find that it gives a short
period decimal, .027, because 37 exactly divides 999; it, therefore, will not
do. We next examine 47, and find that it gives us the full period circulator,
in 46 figures, at the beginning of this article.
If you cut any of these full period circulators in half and place one half
under the other, you will find that they will add up all 9's; so you need only
work out one half and then write down the complements. Thus, in the
ribbon above, if you add 05882352 to 94117647 the result is 99999999, and

so with our long solution number. Note also in the diagram above that not
only are the opposite numbers on the outer ring complementary, always
making 9 when added, but that opposite numbers in the inner ring, our
remainders, are also complementary, adding to 17 in every case. I ought
perhaps to point out that in limiting our multipliers to the first nine numbers
it seems just possible that a short period circulator might give a solution in
fewer figures, but there are reasons for thinking it improbable.

84.—The Japanese Ladies and the Carpet.
If the squares had not to be all the same size, the carpet could be cut in four
pieces in any one of the three manners shown. In each case the two pieces
marked A will fit together and form one of the three squares, the other two
squares being entire. But in order to have the squares exactly equal in size,
we shall require six pieces, as shown in the larger diagram. No. 1 is a
complete square, pieces 4 and 5 will form a second square, and pieces 2, 3,
and 6 will form the third—all of exactly the same size.

If with the three equal squares we form the
rectangle IDBA, then the mean proportional of
the two sides of the rectangle will be the side of
a square of equal area. Produce AB to C, making
BC equal to BD. Then place the point of the
compasses at E (midway between A and C) and
describe the arc AC. I am showing the quite
general method for converting rectangles to
squares, but in this particular case we may, of
course, at once place our compasses at E, which
requires no finding. Produce the line BD, cutting
the arc in F, and BF will be the required side of

the square. Now mark off AG and DH, each equal to BF, and make the cut
IG, and also the cut HK from H, perpendicular to ID. The six pieces
produced are numbered as in the diagram on last page.
It will be seen that I have here given the
reverse method first: to cut the three small
squares into six pieces to form a large square.
In the case of our puzzle we can proceed as
follows:—
Make LM equal to half the diagonal ON.
Draw the line NM and drop from L a
perpendicular on NM. Then LP will be the
side of all the three squares of combined area
equal to the large square QNLO. The reader
can now cut out without difficulty the six
pieces, as shown in the numbered square on
the last page.

85.—Captain Longbow and the Bears.

It might have struck the reader that the story of the bear impaled on the
North Pole had no connection with the problem that followed. As a matter
of fact it is essential to a solution. Eleven bears cannot possibly be arranged
to form of themselves seven rows of bears with four bears in every row. But
it is a different matter when Captain Longbow informs us that "they had so
placed themselves that there were" seven rows of four bears. For if they
were grouped as shown in the diagram, so that three of the bears, as
indicated, were in line with the North Pole, that impaled animal would
complete the seventh row of four, which cannot be obtained in any other
way. It obviously does not affect the problem whether this seventh row is a
hundred miles long or a hundred feet, so long as they were really in a
straight line—a point that might perhaps be settled by the captain's pocket
compass.

86.—The English Tour.
It was required to show how a resident at the town marked A might visit
every one of the towns once, and only once, and finish up his tour at Z. This
puzzle conceals a little trick. After the solver has demonstrated to his
satisfaction that it cannot be done in accordance with the conditions as he at
first understood them, he should carefully examine the wording in order to
find some flaw. It was said, "This would be easy enough if he were able to
cut across country by road, as well as by rail, but he is not."

Now, although he is prohibited from cutting across country by road, nothing
is said about his going by sea! If, therefore, we carefully look again at the

map, we shall find that two towns, and two only, lie on the sea coast. When
he reaches one of these towns he takes his departure on board a coasting
vessel and sails to the other port. The annexed illustration shows, by a dark
line, the complete route.
This problem should be compared with No. 250, "The Grand Tour," in A. in
M. It can be simplified in practically an identical manner, but as there is
here no choice on the first stage from A, the solutions are necessarily quite
different. See also solution to No. 94.

87.—The Chifu-Chemulpo Puzzle.
The solution is as follows. You may accept the invitation to "try to do it in
twenty moves," but you will never succeed in performing the feat. The
fewest possible moves are twenty-six. Play the cars so as to reach the
following positions:—

E5678
———— = 10 moves.
1234
E56
———— = 2 moves.
123 87 4
56
———— = 5 moves.
E312 87 4
E
———— = 9 moves.
87654321

Twenty-six moves in all.

88.—The Eccentric Market-woman.
The smallest possible number of eggs that Mrs. Covey could have taken to
market is 719. After selling half the number and giving half an egg over she
would have 359 left; after the second transaction she would have 239 left;
after the third deal, 179; and after the fourth, 143. This last number she
could divide equally among her thirteen friends, giving each 11, and she
would not have broken an egg.

89.—The Primrose Puzzle.
The two words that solve this puzzle are BLUEBELL and PEARTREE.
Place the letters as follows: B 3-1, L 6-8, U 5-3, E 4-6, B 7-5, E 2-4, L 9-7,
L 9-2. This means that you take B, jump from 3 to 1, and write it down on
1; and so on. The second word can be inserted in the same order. The
solution depends on finding those words in which the second and eighth
letters are the same, and also the fourth and sixth the same, because these
letters interchange without destroying the words. MARITIMA (or sea-pink)
would also solve the puzzle if it were an English word.
Compare with No. 226 in A. in M.

90.—The Round Table.
Here is the way of arranging the seven men:—

AB C DE F G
AC DB GE F
ADB C F GE
AGB F E C D
AF C E GDB
AE DGF B C
AC E B GF D
ADGC F E B
AB F DE GC
AE F DC GB
AGE B DF C
AF GC B E D
AE B F C DG
AGC E DB F
AF DGB C E

Of course, at a circular table, A will be next to the man at the end of the
line.
I first gave this problem for six persons on ten days, in the Daily Mail for
the 13th and 16th October 1905, and it has since been discussed in various
periodicals by mathematicians. Of course, it is easily seen that the
maximum number of sittings for n persons is (n - 1)(n - 2)/2 ways. The
comparatively easy method for solving all cases where n is a prime+1 was
first discovered by Ernest Bergholt. I then pointed out the form and
construction of a solution that I had obtained for 10 persons, from which E.
D. Bewley found a general method for all even numbers. The odd numbers,
however, are extremely difficult, and for a long time no progress could be
made with their solution, the only numbers that could be worked being 7
(given above) and 5, 9, 17, and 33, these last four being all powers of 2+1.

At last, however (though not without much difficulty), I discovered a subtle
method for solving all cases, and have written out schedules for every
number up to 25 inclusive. The case of 11 has been solved also by W. Nash.
Perhaps the reader will like to try his hand at 13. He will find it an
extraordinarily hard nut.
The solutions for all cases up to 12 inclusive are given in A. in M., pp. 205,
206.

91.—The Five Tea Tins.
There are twelve ways of arranging the boxes without considering the
pictures. If the thirty pictures were all different the answer would be 93,312.
But the necessary deductions for cases where changes of boxes may be
made without affecting the order of pictures amount to 1,728, and the boxes
may therefore be arranged, in accordance with the conditions, in 91,584
different ways. I will leave my readers to discover for themselves how the
figures are to be arrived at.

92.—The Four Porkers.
The number of ways in which the four pigs may be placed in the thirty-six
sties in accordance with the conditions is seventeen, including the example
that I gave, not counting the reversals and reflections of these arrangements
as different. Jaenisch, in his Analyse Mathématique au jeu des Échecs
(1862), quotes the statement that there are just twenty-one solutions to the
little problem on which this puzzle is based. As I had myself only recorded
seventeen, I examined the matter again, and found that he was in error, and,
doubtless, had mistaken reversals for different arrangements.
Here are the seventeen answers. The figures indicate the rows, and their
positions show the columns. Thus, 104603 means that we place a pig in the
first row of the first column, in no row of the second column, in the fourth
row of the third column, in the sixth row of the fourth column, in no row of
the fifth column, and in the third row of the sixth column. The arrangement
E is that which I gave in diagram form:—

A. 104603
B. 136002
C. 140502
D. 140520
E. 160025
F. 160304
G. 201405
H. 201605
I. 205104
J. 206104
K. 241005
L. 250014
M. 250630
N. 260015
O. 261005

P. 261040
Q. 306104

It will be found that forms N and Q are semi-symmetrical with regard to the
centre, and therefore give only two arrangements each by reversal and
reflection; that form H is quarter-symmetrical, and gives only four
arrangements; while all the fourteen others yield by reversal and reflection
eight arrangements each. Therefore the pigs may be placed in (2 × 2) + (4 ×
1) + (8 × 14) = 120 different ways by reversing and reflecting all the
seventeen forms.
Three pigs alone may be placed so that every sty is in line with a pig,
provided that the pigs are not forbidden to be in line with one another; but
there is only one way of doing it (if we do not count reversals as different),
as follows: 105030.

93.—The Number Blocks.
Arrange the blocks so as to form the two multiplication sums 915 × 64 and
732 × 80, and the product in both cases will be the same: 58,560.

94.—Foxes and Geese.
The smallest possible number of moves is twenty-two—that is, eleven for
the foxes and eleven for the geese. Here is one way of solving the puzzle:

10—5 11—6 12—75—12 6—1 7—6
—— —— —— —— —— ——
1—8 2—9 3—4 8—3 9—104—9
12—7 1—8 6—1 7—2 8—3
—— —— —— —— ——
3—4 10—5 9—10 4—11 5—12

Of course, the reader will play the first move in the top line, then the first
move in the second line, then the second move in the top line, and so on
alternately.

In A. in M., p. 230, I have explained fully my "buttons and string" method
of solving puzzles on chequered boards. In Diagram A is shown the puzzle
in the form in which it may be presented on a portion of the chessboard
with six knights. A comparison with the illustration on page 141 will show
that I have there dispensed with the necessity of explaining the knight's
move to the uninstructed reader by lines that indicate those moves. The two
puzzles are the same thing in different dress. Now compare page 141 with
Diagram B, and it will be seen that by disentangling the strings I have
obtained a simplified diagram without altering the essential relations
between the buttons or discs. The reader will now satisfy himself without
any difficulty that the puzzle requires eleven moves for the foxes and eleven
for the geese. He will see that a goose on 1 or 3 must go to 8, to avoid being
one move from a fox and to enable the fox on 11 to come on to the ring. If
we play 1—8, then it is clearly best to play 10—5 and not 12—5 for the
foxes. When they are all on the circle, then they simply promenade round it
in a clockwise direction, taking care to reserve 8—3 and 5—12 for the final
moves. It is thus rendered ridiculously easy by this method. See also notes
on solutions to Nos. 13 and 85.

95.—Robinson Crusoe's Table.
The diagram shows how the piece of wood should be cut in two pieces to
form the square table-top. A, B, C, D are the corners of the table. The way
in which the piece E fits into the piece F will be obvious to the eye of the
reader. The shaded part is the wood that is discarded.

96.—The Fifteen Orchards.
The number must be the least common multiple of 1, 2, 3, etc., up to 15,
that, when divided by 7, leaves the remainder 1, by 9 leaves 3, by 11 leaves
10, by 13 leaves 3, and by 14 leaves 8. Such a number is 120. The next
number is 360,480, but as we have no record of a tree—especially a very
young one—bearing anything like such a large number of apples, we may
take 120 to be the only answer that is acceptable.

97.—The Perplexed Plumber.
The rectangular closed cistern that shall hold a given quantity of water and
yet have the smallest possible surface of metal must be a perfect cube—that
is, a cistern every side of which is a square. For 1,000 cubic feet of water
the internal dimensions will be 10 ft. × 10 ft. × 10 ft., and the zinc required
will be 600 square feet. In the case of a cistern without a top the proportions
will be exactly half a cube. These are the "exact proportions" asked for in
the second case. The exact dimensions cannot be given, but 12.6 ft. × 12.6
ft. × 6.3 ft. is a close approximation. The cistern will hold a little too much
water, at which the buyer will not complain, and it will involve the plumber
in a trifling loss not worth considering.

98.—The Nelson Column.
If you take a sheet of paper and mark it with a diagonal line, as in Figure A,
you will find that when you roll it into cylindrical form, with the line
outside, it will appear as in Figure B.
It will be seen that the spiral (in one complete turn)
is merely the hypotenuse of a right-angled triangle,
of which the length and width of the paper are the
other two sides. In the puzzle given, the lengths of
the two sides of the triangle are 40 ft. (one-fifth of
200 ft.) and 16 ft. 8 in. Therefore the hypotenuse is
43 ft. 4 in. The length of the garland is therefore five
times as long—216 ft. 8 in. A curious feature of the
puzzle is the fact that with the dimensions given the
result is exactly the sum of the height and the circumference.

99.—The Two Errand Boys.
All that is necessary is to add the two distances at which they meet to twice
their difference. Thus 720 + 400 + 640 = 1760 yards, or one mile, which is
the distance required. Or, put another way, three times the first distance less
the second distance will always give the answer, only the first distance
should be more than two-thirds of the second.

100.—On the Ramsgate Sands.
Just six different rings may be formed without breaking the conditions.
Here is one way of effecting the arrangements.

AB C D E F G HI J KL M
AC E G I K MB DF HJ L
ADG J MC F I L B E HK
AE I MD H L C GKB F J
AF K C H ME J B GL DI
AGMF L E K DJ C I B H

Join the ends and you have the six rings.
Lucas devised a simple mechanical method for obtaining the n rings that
may be formed under the conditions by 2n+1 children.

101.—The Three Motor-Cars.
The only set of three numbers, of two, three, and five figures respectively,
that will fulfil the required conditions is 27 × 594 = 16,038. These three
numbers contain all the nine digits and 0, without repetition; the first two
numbers multiplied together make the third, and the second is exactly
twenty-two times the first. If the numbers might contain one, four, and five
figures respectively, there would be many correct answers, such as 3 ×
5,694 = 17,082; but it is a curious fact that there is only one answer to the
problem as propounded, though it is no easy matter to prove that this is the
case.

102.—A Reversible Magic Square.

It will be seen that in the arrangement given every number is different, and
all the columns, all the rows, and each of the two diagonals, add up 179,
whether you turn the page upside down or not. The reader will notice that I
have not used the figures 3, 4, 5, 8, or 0.

103.—The Tube Railway.
There are 640 different routes. A general formula for puzzles of this kind is
not practicable. We have obviously only to consider the variations of route
between B and E. Here there are nine sections or "lines," but it is impossible
for a train, under the conditions, to traverse more than seven of these lines
in any route. In the following table by "directions" is meant the order of
stations irrespective of "routes." Thus, the "direction" BCDE gives nine
"routes," because there are three ways of getting from B to C, and three
ways of getting from D to E. But the "direction" BDCE admits of no
variation; therefore yields only one route.

2two-line directions of 3routes — 6
1three-line " " 1 " — 1
1 " " " 9 " — 9
2four-line " " 6 " — 12
2 " " " 18 " — 36
6five-line " " 6 " — 36
2 " " " 18 " — 36
2six-line " " 36 " — 72
12seven-line " " 36 " — 432
——
Total 640

We thus see that there are just 640 different routes in all, which is the
correct answer to the puzzle.

104.—The Skipper and the Sea-Serpent.
Each of the three pieces was clearly three cables long. But Simon persisted
in assuming that the cuts were made transversely, or across, and that
therefore the complete length was nine cables. The skipper, however,
explained (and the point is quite as veracious as the rest of his yarn) that his
cuts were made longitudinally—straight from the tip of the nose to the tip
of the tail! The complete length was therefore only three cables, the same as
each piece. Simon was not asked the exact length of the serpent, but how
long it must have been. It must have been at least three cables long, though
it might have been (the skipper's statement apart) anything from that up to
nine cables, according to the direction of the cuts.

105.—The Dorcas Society.
If there were twelve ladies in all, there would be 132 kisses among the
ladies alone, leaving twelve more to be exchanged with the curate—six to
be given by him and six to be received. Therefore, of the twelve ladies, six
would be his sisters. Consequently, if twelve could do the work in four and
a half months, six ladies would do it in twice the time—four and a half
months longer—which is the correct answer.
At first sight there might appear to be some ambiguity about the words,
"Everybody kissed everybody else, except, of course, the bashful young
man himself." Might this not be held to imply that all the ladies immodestly
kissed the curate, although they were not (except the sisters) kissed by him
in return? No; because, in that case, it would be found that there must have
been twelve girls, not one of whom was a sister, which is contrary to the
conditions. If, again, it should be held that the sisters might not, according
to the wording, have kissed their brother, although he kissed them, I reply
that in that case there must have been twelve girls, all of whom must have
been his sisters. And the reference to the ladies who might have worked
exclusively of the sisters shuts out the possibility of this.

106.—The Adventurous Snail.
At the end of seventeen days the snail will have climbed 17 ft., and at the
end of its eighteenth day-time task it will be at the top. It instantly begins
slipping while sleeping, and will be 2 ft. down the other side at the end of
the eighteenth day of twenty-four hours. How long will it take over the
remaining 18 ft.? If it slips 2 ft. at night it clearly overcomes the tendency to
slip 2 ft. during the daytime, in climbing up. In rowing up a river we have
the stream against us, but in coming down it is with us and helps us. If the
snail can climb 3 ft. and overcome the tendency to slip 2 ft. in twelve hours'
ascent, it could with the same exertion crawl 5 ft. a day on the level.
Therefore, in going down, the same exertion carries it 7 ft. in twelve hours
—that is, 5 ft. by personal exertion and 2 ft. by slip. This, with the night
slip, gives it a descending progress of 9 ft. in the twenty-four hours. It can,
therefore, do the remaining 18 ft. in exactly two days, and the whole
journey, up and down, will take it exactly twenty days.

107.—The Four Princes.
When Montucla, in his edition of Ozanam's Recreations in Mathematics,
declared that "No more than three right-angled triangles, equal to each
other, can be found in whole numbers, but we may find as many as we
choose in fractions," he curiously overlooked the obvious fact that if you
give all your sides a common denominator and then cancel that
denominator you have the required answer in integers!
Every reader should know that if we take any two numbers, m and n, then
m2 + n2, m2 - n2, and 2mn will be the three sides of a rational right-angled
triangle. Here m and n are called generating numbers. To form three such
triangles of equal area, we use the following simple formula, where m is the
greater number:—

mn + m2 + n2 = a
m2 - n2 = b
2mn + n2 = c
Now, if we form three triangles from the following pairs of generators, a
and b, a and c, a and b + c, they will all be of equal area. This is the little
problem respecting which Lewis Carroll says in his diary (see his Life and
Letters by Collingwood, p. 343), "Sat up last night till 4 a.m., over a
tempting problem, sent me from New York, 'to find three equal rational-
sided right-angled triangles.' I found two ... but could not find three!"
The following is a subtle formula by means of which we may always find a
R.A.T. equal in area to any given R.A.T. Let z = hypotenuse, b = base, h =
height, a = area of the given triangle; then all we have to do is to form a
R.A.T. from the generators z2 and 4a, and give each side the denominator 2z
(b2 - h2), and we get the required answer in fractions. If we multiply all
three sides of the original triangle by the denominator, we shall get at once a
solution in whole numbers.
The answer to our puzzle in smallest possible numbers is as follows:—

First Prince 51813201418
Second Prince 28024422458
Third Prince 23129602969
Fourth Prince 111 61606161

The area in every case is 341,880 square furlongs. I must here refrain from
showing fully how I get these figures. I will explain, however, that the first
three triangles are obtained, in the manner shown, from the numbers 3 and
4, which give the generators 37, 7; 37, 33; 37, 40. These three pairs of
numbers solve the indeterminate equation, a3b - b3a = 341,880. If we can
find another pair of values, the thing is done. These values are 56, 55, which
generators give the last triangle. The next best answer that I have found is
derived from 5 and 6, which give the generators 91, 11; 91, 85; 91, 96. The
fourth pair of values is 63, 42.
The reader will understand from what I have written above that there is no
limit to the number of rational-sided R.A.T.'s of equal area that may be
found in whole numbers.

108.—Plato and the Nines.
The following is the simple solution of the three nines puzzle:—

9+9
——
.9

To divide 18 by .9 (or nine-tenths) we, of course, multiply by 10 and divide
by 9. The result is 20, as required.

109.—Noughts and Crosses.
The solution is as follows: Between two players who thoroughly understand
the play every game should be drawn. Neither player could ever win except
through the blundering of his opponent. If Nought (the first player) takes
the centre, Cross must take a corner, or Nought may beat him with certainty.
If Nought takes a corner on his first play, Cross must take the centre at
once, or again be beaten with certainty. If Nought leads with a side, both
players must be very careful to prevent a loss, as there are numerous
pitfalls. But Nought may safely lead anything and secure a draw, and he can
only win through Cross's blunders.

110.—Ovid's Game.
The solution here is: The first player can always win, provided he plays to
the centre on his first move. But a good variation of the game is to bar the
centre for the first move of the first player. In that case the second player
should take the centre at once. This should always end in a draw, but to
ensure it the first player must play to two adjoining corners (such as 1 and
3) on his first and second moves. The game then requires great care on both
sides.

111.—The Farmer's Oxen.
Sir Isaac Newton has shown us, in his Universal Arithmetic, that we may
divide the bullocks in each case in two parts—one part to eat the increase,
and the other the accumulated grass. The first will vary directly as the size
of the field, and will not depend on the time; the second part will also vary
directly as the size of the field, and in addition inversely with the time. We
find from the farmer's statements that 6 bullocks keep down the growth in a
10-acre field, and 6 bullocks eat the grass on 10 acres in 16 weeks.
Therefore, if 6 bullocks keep down the growth on 10 acres, 24 will keep
down the growth on 40 acres.
Again, we find that if 6 bullocks eat the accumulated grass on 10 acres in
16 weeks, then

12eat the grass on 10 acres in 8weeks,
48 " " 40 " 8 "
192 " " 40 " 2 "
64 " " 40 " 6 "

Add the two results together (24 + 64), and we find that 88 oxen may be fed
on a 40-acre meadow for 6 weeks, the grass growing regularly all the time.

112.—The Great Grangemoor Mystery.
We were told that the bullet that killed Mr. Stanton Mowbray struck the
very centre of the clock face and instantly welded together the hour, minute,
and second hands, so that all revolved in one piece. The puzzle was to tell
from the fixed relative positions of the three hands the exact time when the
pistol was fired.
We were also told, and the illustration of the clock face bore out the
statement, that the hour and minute hands were exactly twenty divisions
apart, "the third of the circumference of the dial." Now, there are eleven
times in twelve hours when the hour hand is exactly twenty divisions ahead
of the minute hand, and eleven times when the minute hand is exactly
twenty divisions ahead of the hour hand. The illustration showed that we
had only to consider the former case. If we start at four o'clock, and keep on
adding 1 h. 5 m. 27-3/11 sec., we shall get all these eleven times, the last
being 2 h. 54 min. 32-8/11 sec. Another addition brings us back to four
o'clock. If we now examine the clock face, we shall find that the seconds
hand is nearly twenty-two divisions behind the minute hand, and if we look
at all our eleven times we shall find that only in the last case given above is
the seconds hand at this distance. Therefore the shot must have been fired at
2 h. 54 min. 32-8/11 sec. exactly, or, put the other way, at 5 min. 27-3/11
sec. to three o'clock. This is the correct and only possible answer to the
puzzle.

113.—Cutting a Wood Block.
Though the cubic contents are sufficient for twenty-five pieces, only
twenty-four can actually be cut from the block. First reduce the length of
the block by half an inch. The smaller piece cut off constitutes the portion
that cannot be used. Cut the larger piece into three slabs, each one and a
quarter inch thick, and it will be found that eight blocks may easily be cut
out of each slab without any further waste.

114.—The Tramps and the Biscuits.
The smallest number of biscuits must have been 1021, from which it is
evident that they were of that miniature description that finds favour in the
nursery. The general solution is that for n men the number must be m (nn+1)
- (n - 1), where m is any integer. Each man will receive m (n - 1)n - 1
biscuits at the final division, though in the case of two men, when m = 1, the
final distribution only benefits the dog. Of course, in every case each man
steals an nth of the number of biscuits, after giving the odd one to the dog.

INDEX
"Abracadabra," 64.
Age and Kinship Puzzles, 20, 28.
Albanna, Ibn, 198.
Ale, Measuring the, 29.
Algebraical Puzzles. See Arithmetical Puzzles.
Alkalaçadi, 198.
Amulet, The, 64, 190.
Archery Butt, The, 60, 187.
Arithmetical Puzzles, 18, 26, 34, 36, 45, 46, 51, 56, 59, 61, 64, 65, 73,
74, 75, 88, 89, 91, 92, 103, 107, 122, 125, 128, 129, 130, 135, 137,
139, 143, 147, 148, 150, 151, 152, 153, 154, 157, 158, 161.
Arrows, The Nine, 32.
Astronomical Problem, 55.

Bags, Four Money, 46.
Ball, W. W. Rouse, 202.
Bandy-Ball, The Game of, 58, 185.
Bears, Capt. Longbow and the, 132, 233.
Bergholt, Ernest, 237.
Bewley, E. D., 237.
Biscuits, The Tramps and the, 160, 250.
Block, Cutting a Wood, 160, 250.
Blocks, The Number, 139, 238.
Bottles, Sharing the, 122.
Bottles, The Sixteen, 45.
Bridges, The Eight, 48.
Bridging the Ditch, 83, 204.
Brooch, Cutting the, 41.
Buried Treasure, 107, 212.
Buttons and String Method, 171, 239.

Canon's Yeoman, Puzzle of the, 55, 181.
Canterbury Pilgrim's Puzzle, 33.

Canterbury Puzzles, 23, 163.
Card Puzzle, 125.
Carpenter's Puzzle, The, 31, 168.
Carpet, Japanese Ladies and, 131, 231.
Carroll, Lewis, 246.
Casket, Lady Isabel's, 67, 191.
Cats and Mice, 75.
Cellarer, The Riddle of the, 73, 196.
Cellarman, The Perplexed, 122, 222.
Chalked Numbers, The, 89, 206.
Chaucer's Puzzle, 54, 181.
Cheeses on Stools, 24.
Chessboard Problems, 21, 25, 32, 51, 72, 82, 90, 113, 114, 116, 119,
124, 138, 141.
Chessboard, The Broken, 119, 220.
Chifu-Chemulpo Puzzle, 134, 235.
Chinese Railways, The, 127, 224.
Christmas Puzzle Party, The Squire's, 86, 205.
Cisterns, Making, 145.
Clerk of Oxenford's Puzzle, The, 29, 167.
Cliff Mystery, The Cornish, 99, 210.
Clock Puzzle, 158.
Cloth, Cutting the, 50.
Clowns, The Eight, 128, 226.
Club, Adventures of the Puzzle, 94, 210.
Coinage Puzzle, The, 111, 214.
Coin Magic Square, 111.
Column, The Nelson, 146, 241.

Combination and Group Problems, 38, 39, 61, 70, 122, 137, 147.
Cook's Puzzle, The, 36, 171.
Cornish Cliff Mystery, The, 99, 210.
Counter Problems, Moving, 24, 35, 69, 77, 124, 135, 136, 141.
Counting out Puzzle, 39.
Crescent and the Cross, The, 63, 189.
Crossing River Problems, 82, 83.

Crusaders, The Riddle of the, 74, 197.
Crusoe's Table, Robinson, 142, 240.
Cubes, Sums of Two, 174, 209.
Cubes, The Silver, 92, 209.

Daily Mail, 179, 221, 236.
Decimals, Recurring, 228, 229.
Demoiselle, The Noble, 59, 186.
Diamond Letter Puzzles, 181.
Digital Analysis, 228.
Digital Puzzles, 18, 26, 90, 103, 129, 139, 148, 154.
Dispute between Friar and Sompnour, 51, 180.
Dissection Puzzles, 29, 30, 35, 41, 49, 63, 123, 131, 142.
Ditch, Bridging the, 83, 204.
Divisors of Numbers, To Find, 170.
Doctor of Physic, The Puzzle of the, 42, 174.
Donjon Keep Window, The, 62, 188.
Dorcas Society, The, 151, 245.
Dormitory Puzzle, The, 70.
Dungeon, The Death's-head, 60.
Dungeons, The Nine, 35.
Dyer's Puzzle, The, 50, 180.

Edward, Portrait of King, 46.
Eggs, Selling the, 135.
Eleven Pennies, The, 88, 206.
Errand Boys, The Two, 147, 242.
Escape of King's Jester, The Strange, 78, 201.
Executioner, The, 78.

Fallacy of Square's Diagonal, 52.
Farmer's Oxen, The, 157, 248.
Fermat, P. de, 174.
Fish-pond, The Riddle of the, 69, 194.
Flag, Making a, 123, 223.
Fleurs-de-Lys, Sixty-four, 50.
Flour, The Nine Sacks of, 26.

Fly, The Spider and the, 121, 221.
Footprints Puzzles, 101, 105.
Four Princes, The, 153, 246.
Foxes and Geese, 140, 239.
Franklin's Puzzle, The, 44, 176.
Friar and Sompnour's Dispute, 51, 180.
Friar's Puzzle, The, 46, 177.
Frogs and Tumblers, The, 113, 216.
Frogs who would a-wooing go, 116, 219.
Frogs' Ring, The Riddle of the, 76, 199.

Games, Puzzle, 118, 125, 156, 157.
Gardens, The Royal, 82, 203.
Geese, The Christmas, 88, 206.
Geometrical Problems, 52, 62, 67, 121, 131, 144, 146.
Grangemoor Mystery, The, 158, 249.
Group Problems, Combination and. See Combination and Group
Problems.

Haberdasher's Puzzle, The, 49, 178.
Hogs, Catching the, 124, 223.
Hoppe, Oscar, 198.
Host's Puzzle, The, 28, 166.

Isabel's Casket, Lady, 67, 191.

Jaenisch, 237.
Japanese Ladies and the Carpet, 131, 231.
Jester, Strange Escape of the King's, 78, 201.

Kayles, The Game of, 118, 220.
Kennels, The Nine, 39.
King's Jester, Strange Escape of the, 78, 201.
Knight's Puzzle, The, 26, 165.

Lady Isabel's Casket, 67, 191.
L'Arithmétique Amusante, 198.
Legendre, 175.

Letter Puzzles, 16.
Lock, The Secret, 80, 202.
Locomotive and Speed Puzzle, 147.
Longbow and the Bears, Capt., 132, 233.
Lucas, Edouard, 175, 198, 242.

M'Elroy, C. W., 179.
Magdalen, Chart of the, 41.
Magic Square, A Reversible, 149, 243.
Magic Square Problems, 21, 29, 44, 111, 112, 128, 149.
Manciple's Puzzle, The, 56, 183.
Man of Law's Puzzle, The, 34, 170.
Market Woman, The Eccentric, 135, 235.
Marksford and the Lady, Lord, 96.
Maze, The Underground, 79, 201.
Measuring, Weighing, and Packing Puzzles, 29, 31, 55, 72, 73, 160.
Merchant's Puzzle, The, 33, 170.
Merry Monks of Riddlewell, 68, 194.
Miller's Puzzle, The, 26, 164.
Miscellaneous Puzzles, 118, 220.
Mistletoe Bough, Under the, 91, 208.
Moat, Crossing the, 81, 202.
Money, Dividing the, 57.
Monks of Riddlewell, The Merry, 68, 194.
Monk's Puzzle, The, 39, 172.
Montucla, 246.
Motor-Car, The Runaway, 103, 211.
Motor-Cars, The Three, 147, 242.
Moving Counter Problems. See Counter Problems, Moving.

Nash, W., 237.
Nelson Column, The, 146, 241.
Newton, Sir Isaac, 248.
Nines, Plato and the, 154, 247.
Noble Demoiselle, The, 59, 186.
Noughts and Crosses, 156, 248.
Number Blocks, The, 139, 238.

Numbers on Motor-Cars, 103, 148.
Numbers Partition of, 46.
Numbers The Chalked, 89, 206.
Nun's Puzzle, The, 32, 169.

Ones, Numbers composed only of, 18, 75, 198.
Opposition in Chess, 224.
Orchards, The Fifteen, 143, 241.
Ovid's Game, 156, 248.
Oxen, The Farmer's, 157, 248.
Ozanam's Recreations, 246.

Packing Puzzles, Measuring, Weighing, and. See Measuring.
Palindromes, 17.
Pardoner's Puzzle, The, 25, 164.
Parental Command, A, 28.
Park, Mystery of Ravensdene, 105, 211.
Parson's Puzzle, The, 47, 177.
Party, The Squire's Christmas Puzzle, 86, 205.
Pellian Equation, 197.
Pennies, The Eleven, 88, 206.
Phials, The Two, 42.
Photograph, The Ambiguous, 94, 210.
Pie and the Pasty, The, 36.
Pilgrimages, The Fifteen, 25.
Pilgrims' Manner of Riding, 34.
Pilgrims The Riddle of the, 70, 194.
Pillar, The Carved Wooden, 31.
Plato and the Nines, 154, 247.
Ploughman's Puzzle, The, 43, 175.
Plumber, The Perplexed, 144, 241.
Plum Puddings, Tasting the, 90, 207.
Points and Lines Problems, 43, 116, 133.
Porkers, The Four, 138, 237.
Postage Stamps Puzzle, The, 112, 214.
Primrose Puzzle, The, 136, 235.
Princes, The Four, 153, 246.

Prioress, The Puzzle of the, 41, 173.

Professor's Puzzles, The, 110, 214.
Puzzle Club, Adventures of the, 94, 210.
Puzzles, How to solve, 18.
Puzzles, How they are made, 14.
Puzzles, Sophistical, 15.
Puzzles, The exact conditions of, 18.
Puzzles, The mysterious charm of, 12.
Puzzles, The nature of, 11.
Puzzles, The utility of, 13.
Puzzles, The variety of, 13, 16.
Puzzles, Unsolved, 20.
Puzzling Times at Solvamhall Castle, 58, 184.
Pyramids, Triangular, 163.

Railway Puzzle, 134.
Railway The Tube, 149, 243.
Railways, The Chinese, 127, 224.
Ramsgate Sands, On the, 147, 242.
Rat-catcher's Riddle, The, 56.
Ravensdene Park, Mystery of, 105, 211.
Reve's Puzzle, The, 24, 163.
Ribbon Problem, The, 130, 228.
Riddles, old, 16.
Riddlewell, The Merry Monks of, 68, 194.
River Crossing Problems, 82, 83.
Robinson Crusoe's Table, 142, 240.
Romeo and Juliet, 114, 217.
Romeo's Second Journey, 116, 218.
Rook's Path, The, 207.
Rope, The Mysterious, 79, 201.
Round Table, The, 137, 236.
Route Problems, Unicursal and. See Unicursal.

Sack Wine, The Riddle of the, 72, 196.

St. Edmondsbury, The Riddle of, 75, 197.
Sands, On the Ramsgate, 147, 242.
Sea-Serpent, The Skipper and the, 150, 244.
Shield, Squares on a, 27.
Shipman's Puzzle, The, 40, 173.
Skipper and the Sea-Serpent, The, 150, 244.
Snail on the Flagstaff, The, 66, 190.
Snail The Adventurous, 152, 245.
Snails, The Two, 115, 217.
Solvamhall Castle, Puzzling Times at, 58, 184.
Sompnour's and Friar's Dispute, 51, 180.
Sompnour's Puzzle, The, 38, 172.
Spherical Surface of Water, 181.
Spider and the Fly, The, 121, 221.
Square and Triangle, The, 49.
Square Field, The, 107.
Squares, Problem of, 74.
Square, Three Squares from One, 131, 231.
Squire's Christmas Puzzle Party, The, 86, 205.
Squire's Puzzle, The, 45, 176.
Squire's Yeoman, The Puzzle of the, 31, 168.
Stamps, Counting Postage, 137.
Stamps, Magic Squares of, 112.
Stamps, Puzzle, The Postage, 112, 214.
Superposition, Problem on, 179.
Sylvester, 175.

Table, Robinson Crusoe's, 142, 240.
Table, The Round, 137, 236.
Talkhys, 198.
Tapestry, Cutting the, 30.
Tapiser's Puzzle, The, 30, 167.
Teacups, The Three, 87, 205.
Tea Tins, The Five, 137, 237.
Thirty-one Game, The, 125, 224.
Tiled Hearth, The Riddle of the, 71, 195.
Tilting at the Ring, 59, 185.

Tour, The English, 134, 233.
Towns, Visiting the, 134.
Tramps and the Biscuits, The, 160, 250.
Treasure, The Buried, 107, 212.
Trees, The Sixteen Oak, 44.
Triangle and Square, 49.

Triangles of Equal Area, 153, 246.
Triangular numbers, 163.
Tube Railway, The, 149, 243.

Unicursal and Route Problems, 40, 45, 48, 56, 60, 83, 90, 105, 127,
134, 149.

Weaver's Puzzle, The, 35, 171.
Weekly Dispatch, 179, 221.
Weighing, and Packing Puzzles, Measuring. See Measuring.
Wife of Bath's Riddles, The, 27, 166.
Window, The Donjon Keep, 62, 188.
Wine, Stealing the, 73.
Wizard's Arithmetic, The, 129, 226.
Wood Block, Cutting a, 160, 250.
Wreath on Column, 146, 241.

PRINTED IN GREAT BRITAIN AT
THE PRESS OF THE PUBLISHERS.

*** END OF THE PROJECT GUTENBERG EBOOK THE
CANTERBURY PUZZLES, AND OTHER CURIOUS PROBLEMS ***

Updated editions will replace the previous one—the old editions will be
renamed.

Creating the works from print editions not protected by U.S. copyright law
means that no one owns a United States copyright in these works, so the
Foundation (and you!) can copy and distribute it in the United States
without permission and without paying copyright royalties. Special rules,
set forth in the General Terms of Use part of this license, apply to copying
and distributing Project Gutenberg™ electronic works to protect the
PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a
registered trademark, and may not be used if you charge for an eBook,
except by following the terms of the trademark license, including paying
royalties for use of the Project Gutenberg trademark. If you do not charge
anything for copies of this eBook, complying with the trademark license is
very easy. You may use this eBook for nearly any purpose such as creation
of derivative works, reports, performances and research. Project Gutenberg
eBooks may be modified and printed and given away—you may do
practically ANYTHING in the United States with eBooks not protected by
U.S. copyright law. Redistribution is subject to the trademark license,
especially commercial redistribution.

START: FULL LICENSE

THE FULL PROJECT GUTENBERG™ LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free
distribution of electronic works, by using or distributing this work (or any
other work associated in any way with the phrase “Project Gutenberg”), you
agree to comply with all the terms of the Full Project Gutenberg License
available with this file or online at www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing
Project Gutenberg electronic works

1.A. By reading or using any part of this Project Gutenberg electronic work,
you indicate that you have read, understand, agree to and accept all the
terms of this license and intellectual property (trademark/copyright)
agreement. If you do not agree to abide by all the terms of this agreement,
you must cease using and return or destroy all copies of Project Gutenberg
electronic works in your possession. If you paid a fee for obtaining a copy
of or access to a Project Gutenberg electronic work and you do not agree to
be bound by the terms of this agreement, you may obtain a refund from the
person or entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be used on
or associated in any way with an electronic work by people who agree to be
bound by the terms of this agreement. There are a few things that you can
do with most Project Gutenberg electronic works even without complying
with the full terms of this agreement. See paragraph 1.C below. There are a
lot of things you can do with Project Gutenberg electronic works if you
follow the terms of this agreement and help preserve free future access to
Project Gutenberg electronic works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the Foundation”
or PGLAF), owns a compilation copyright in the collection of Project
Gutenberg electronic works. Nearly all the individual works in the
collection are in the public domain in the United States. If an individual
work is unprotected by copyright law in the United States and you are

located in the United States, we do not claim a right to prevent you from
copying, distributing, performing, displaying or creating derivative works
based on the work as long as all references to Project Gutenberg are
removed. Of course, we hope that you will support the Project Gutenberg
mission of promoting free access to electronic works by freely sharing
Project Gutenberg works in compliance with the terms of this agreement for
keeping the Project Gutenberg name associated with the work. You can
easily comply with the terms of this agreement by keeping this work in the
same format with its attached full Project Gutenberg License when you
share it without charge with others.

1.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are in a
constant state of change. If you are outside the United States, check the
laws of your country in addition to the terms of this agreement before
downloading, copying, displaying, performing, distributing or creating
derivative works based on this work or any other Project Gutenberg work.
The Foundation makes no representations concerning the copyright status
of any work in any country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other immediate
access to, the full Project Gutenberg License must appear prominently
whenever any copy of a Project Gutenberg work (any work on which the
phrase “Project Gutenberg” appears, or with which the phrase “Project
Gutenberg” is associated) is accessed, displayed, performed, viewed, copied
or distributed:

This eBook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg™ License included with
this eBook or online at www.gutenberg.org. If you are not located in
the United States, you will have to check the laws of the country
where you are located before using this eBook.

1.E.2. If an individual Project Gutenberg electronic work is derived from
texts not protected by U.S. copyright law (does not contain a notice
indicating that it is posted with permission of the copyright holder), the
work can be copied and distributed to anyone in the United States without
paying any fees or charges. If you are redistributing or providing access to a
work with the phrase “Project Gutenberg” associated with or appearing on
the work, you must comply either with the requirements of paragraphs
1.E.1 through 1.E.7 or obtain permission for the use of the work and the
Project Gutenberg trademark as set forth in paragraphs 1.E.8 or 1.E.9.

1.E.3. If an individual Project Gutenberg electronic work is posted with the
permission of the copyright holder, your use and distribution must comply
with both paragraphs 1.E.1 through 1.E.7 and any additional terms imposed
by the copyright holder. Additional terms will be linked to the Project
Gutenberg License for all works posted with the permission of the
copyright holder found at the beginning of this work.

1.E.4. Do not unlink or detach or remove the full Project Gutenberg License
terms from this work, or any files containing a part of this work or any
other work associated with Project Gutenberg.

1.E.5. Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without prominently
displaying the sentence set forth in paragraph 1.E.1 with active links or
immediate access to the full terms of the Project Gutenberg License.

1.E.6. You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including any
word processing or hypertext form. However, if you provide access to or
distribute copies of a Project Gutenberg work in a format other than “Plain
Vanilla ASCII” or other format used in the official version posted on the
official Project Gutenberg website (www.gutenberg.org), you must, at no
additional cost, fee or expense to the user, provide a copy, a means of
exporting a copy, or a means of obtaining a copy upon request, of the work
in its original “Plain Vanilla ASCII” or other form. Any alternate format
must include the full Project Gutenberg License as specified in paragraph
1.E.1.

1.E.7. Do not charge a fee for access to, viewing, displaying, performing,
copying or distributing any Project Gutenberg works unless you comply
with paragraph 1.E.8 or 1.E.9.

1.E.8. You may charge a reasonable fee for copies of or providing access to
or distributing Project Gutenberg electronic works provided that:

• You pay a royalty fee of 20% of the gross profits you derive from the
use of Project Gutenberg works calculated using the method you
already use to calculate your applicable taxes. The fee is owed to the
owner of the Project Gutenberg trademark, but he has agreed to donate
royalties under this paragraph to the Project Gutenberg Literary
Archive Foundation. Royalty payments must be paid within 60 days
following each date on which you prepare (or are legally required to
prepare) your periodic tax returns. Royalty payments should be clearly
marked as such and sent to the Project Gutenberg Literary Archive
Foundation at the address specified in Section 4, “Information about
donations to the Project Gutenberg Literary Archive Foundation.”

• You provide a full refund of any money paid by a user who notifies
you in writing (or by e-mail) within 30 days of receipt that s/he does
not agree to the terms of the full Project Gutenberg™ License. You
must require such a user to return or destroy all copies of the works
possessed in a physical medium and discontinue all use of and all
access to other copies of Project Gutenberg™ works.

• You provide, in accordance with paragraph 1.F.3, a full refund of any
money paid for a work or a replacement copy, if a defect in the
electronic work is discovered and reported to you within 90 days of
receipt of the work.

• You comply with all other terms of this agreement for free distribution
of Project Gutenberg™ works.

1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™
electronic work or group of works on different terms than are set forth in
this agreement, you must obtain permission in writing from the Project
Gutenberg Literary Archive Foundation, the manager of the Project

Gutenberg™ trademark. Contact the Foundation as set forth in Section 3
below.

1.F.

1.F.1. Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread works
not protected by U.S. copyright law in creating the Project Gutenberg™
collection. Despite these efforts, Project Gutenberg™ electronic works, and
the medium on which they may be stored, may contain “Defects,” such as,
but not limited to, incomplete, inaccurate or corrupt data, transcription
errors, a copyright or other intellectual property infringement, a defective or
damaged disk or other medium, a computer virus, or computer codes that
damage or cannot be read by your equipment.

1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except
for the “Right of Replacement or Refund” described in paragraph 1.F.3, the
Project Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg™ trademark, and any other party distributing a Project
Gutenberg™ electronic work under this agreement, disclaim all liability to
you for damages, costs and expenses, including legal fees. YOU AGREE
THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT
EXCEPT THOSE PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE
THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY
DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE
TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL,
PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE
NOTICE OF THE POSSIBILITY OF SUCH DAMAGE.

1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you
discover a defect in this electronic work within 90 days of receiving it, you
can receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from. If you
received the work on a physical medium, you must return the medium with
your written explanation. The person or entity that provided you with the
defective work may elect to provide a replacement copy in lieu of a refund.
If you received the work electronically, the person or entity providing it to

you may choose to give you a second opportunity to receive the work
electronically in lieu of a refund. If the second copy is also defective, you
may demand a refund in writing without further opportunities to fix the
problem.

1.F.4. Except for the limited right of replacement or refund set forth in
paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO OTHER
WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING
BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR
FITNESS FOR ANY PURPOSE.

1.F.5. Some states do not allow disclaimers of certain implied warranties or
the exclusion or limitation of certain types of damages. If any disclaimer or
limitation set forth in this agreement violates the law of the state applicable
to this agreement, the agreement shall be interpreted to make the maximum
disclaimer or limitation permitted by the applicable state law. The invalidity
or unenforceability of any provision of this agreement shall not void the
remaining provisions.

1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg™ electronic works in accordance
with this agreement, and any volunteers associated with the production,
promotion and distribution of Project Gutenberg™ electronic works,
harmless from all liability, costs and expenses, including legal fees, that
arise directly or indirectly from any of the following which you do or cause
to occur: (a) distribution of this or any Project Gutenberg work, (b)
alteration, modification, or additions or deletions to any Project Gutenberg
work, and (c) any Defect you cause.

Section 2. Information about the Mission of Project
Gutenberg

Project Gutenberg is synonymous with the free distribution of electronic
works in formats readable by the widest variety of computers including
obsolete, old, middle-aged and new computers. It exists because of the

efforts of hundreds of volunteers and donations from people in all walks of
life.

Volunteers and financial support to provide volunteers with the assistance
they need are critical to reaching Project Gutenberg’s goals and ensuring
that the Project Gutenberg collection will remain freely available for
generations to come. In 2001, the Project Gutenberg Literary Archive
Foundation was created to provide a secure and permanent future for
Project Gutenberg and future generations. To learn more about the Project
Gutenberg Literary Archive Foundation and how your efforts and donations
can help, see Sections 3 and 4 and the Foundation information page at
www.gutenberg.org.

Section 3. Information about the Project Gutenberg
Literary Archive Foundation

The Project Gutenberg Literary Archive Foundation is a non-profit 501(c)
(3) educational corporation organized under the laws of the state of
Mississippi and granted tax exempt status by the Internal Revenue Service.
The Foundation’s EIN or federal tax identification number is 64-6221541.
Contributions to the Project Gutenberg Literary Archive Foundation are tax
deductible to the full extent permitted by U.S. federal laws and your state’s
laws.

The Foundation’s business office is located at 41 Watchung Plaza #516,
Montclair NJ 07042, USA, +1 (862) 621-9288. Email contact links and up
to date contact information can be found at the Foundation’s website and
official page at www.gutenberg.org/contact

Section 4. Information about Donations to the Project
Gutenberg Literary Archive Foundation

Project Gutenberg™ depends upon and cannot survive without widespread
public support and donations to carry out its mission of increasing the
number of public domain and licensed works that can be freely distributed
in machine-readable form accessible by the widest array of equipment

including outdated equipment. Many small donations ($1 to $5,000) are
particularly important to maintaining tax exempt status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United States.
Compliance requirements are not uniform and it takes a considerable effort,
much paperwork and many fees to meet and keep up with these
requirements. We do not solicit donations in locations where we have not
received written confirmation of compliance. To SEND DONATIONS or
determine the status of compliance for any particular state visit
www.gutenberg.org/donate.

While we cannot and do not solicit contributions from states where we have
not met the solicitation requirements, we know of no prohibition against
accepting unsolicited donations from donors in such states who approach us
with offers to donate.

International donations are gratefully accepted, but we cannot make any
statements concerning tax treatment of donations received from outside the
United States. U.S. laws alone swamp our small staff.

Please check the Project Gutenberg web pages for current donation methods
and addresses. Donations are accepted in a number of other ways including
checks, online payments and credit card donations. To donate, please visit:
www.gutenberg.org/donate.

Section 5. General Information About Project Gutenberg
electronic works

Professor Michael S. Hart was the originator of the Project Gutenberg
concept of a library of electronic works that could be freely shared with
anyone. For forty years, he produced and distributed Project Gutenberg
eBooks with only a loose network of volunteer support.

Project Gutenberg eBooks are often created from several printed editions,
all of which are confirmed as not protected by copyright in the U.S. unless a

copyright notice is included. Thus, we do not necessarily keep eBooks in
compliance with any particular paper edition.

Most people start at our website which has the main PG search facility:
www.gutenberg.org.

This website includes information about Project Gutenberg, including how
to make donations to the Project Gutenberg Literary Archive Foundation,
how to help produce our new eBooks, and how to subscribe to our email
newsletter to hear about new eBooks.