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THE CANTERBURY
PUZZLES
AND OTHER CURIOUS PROBLEMS
BY
HENRY ERNEST DUDENEY:
ILLUSTRATED BY PAUL HARDY,
THE AUTHOR AND OTHERS
NEW YORK
mo DUTTON AND COMPANY
1908
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_ Printed in England.
IN expressing his acknowledgments to the periodical Press, the
author of this book desires in particular to thank the proprietors
of The London Magazine, The Strand Magazine, The Royal
Magazine, C. B. Fry’s Magazine, The Captain, The World and
His Wife, The Penny Pictorial Magazine, Tit-Bits, The Daily
Mail, The Tribune, and The Weekly Dispatch, for their courtesy —
in allowing him to reprint articles and problems of his that have
appeared in their pages. Though the volume contains a consider-
able quantity of new matter, it sometimes happens that “the old
is better.” .
COO7 C7
CONTENTS
INTRODUCTION. : : : ; : x1
THE CANTERBURY PUZZLES ; ; |
PUZZLING TIMES AT SOLVAMHALL CASTLE ; 33
THE Merry Monks oF RIDDLEWELL ; 42
THE STRANGE ESCAPE OF THE KING’s JESTER . , 5|
THE SQUIRE'S CHRISTMAS PuzzLE PARTY . ; 58
THE ADVENTURES OF THE PuzzLeE CLuB . 66
THE PRoFEssor’s PUZZLES . ; 8]
MISCELLANEOUS PUZZLES. 89
SOLUTIONS . : : i A Babee
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 |
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
man ——— SEN X1
INTRODUCTION
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. 3
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 along
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
Xil
INTRODUCTION
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 so
often make the mistake of confining themselves to one little corner of
the realm, and thereby missing 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 1 it restricts one’s
nee ee are eens
INTRODUCTION
pleasures, and neglects that variety which is so good for the
brain- soso SS SSDs msssecepeeptenumueseaenemmnemnenstsinta . one Smee ee er antares r
- And there is really a practical utility in puzzle-solving. Regular
exercise 1s 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,
XIV
INTRODUCTION
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,
“T 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 some-
thing 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, |, 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
XV
INTRODUCTION
receiving an afhirmative 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 practically impossible
to divide them into distinct classes. “They often so merge in
character that the best we-can-dois-to—sortthem_into_a few broad
types. Let us. take three or four-examples-in-ittustration of what 1
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 Boeotia 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 >”
XV1
INTRODUCTION
It was explained by (Edipus, 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 honeycomb
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.”
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.
XVI b
INTRODUCTION
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 1,111,111,111,0101,111,111, and then ask for a number
(other than | 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.
The number composed of seventeen ones, II,11I,III,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
XVII
INTRODUCTION
have no divisor is 11. Such a number is, of course, called a prime
number.
The maxim that there is 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—precess———
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, so 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, “1 was working
it out in haddocks !”
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 >
XIX
INTRODUCTION
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 séeing
all sides was not essential, but you went in such a way that, given
XX 7
INTRODUCTION
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 | 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 asquare 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
XX1
INTRODUCTION
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
| 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, | 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, | 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, |, 2, 3, 4, etc., and
it has been done with the exception of the two diagonals, which so
XX11
INTRODUCTION
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 analyses. E-ven in those cases where | have given a
general formula for the solution of a puzzle, he will find great interest
in verifying it for himself.
XXIll
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A CHANCE-GATHERED company of pilgrims on their way to the
shrine of Saint Thomas a 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
| B
. THE CANTERBURY PUZZLES
problems .and ideas. that passed through his active brain. When the
pilgrims-twere stepping: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 1 did once set before my fellow towns-
men 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 CANTERBURY PUZZLES
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 repre-
sented sixty-four towns
through which he had to pass dae 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 apparent 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
3 B 2
THE CANTERBURY PUZZLES
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
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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
4
THE CANTERBURY PUZZLES
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
5
THE-CANTERBURY.PUZZLES
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
, ey
Pr FY
A yi 2 R 3
y :
eur Wf -
4 w
cd
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 a by no means easy task. Yet it can
be done. -
/.—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 witha
6
THE CANTERBURY PUZZLES
little patience.” He then pro-
duced the square shown in the | y iy 3 Af
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 col-
umns, 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,
Gl [Sl kel feo] Kel 1
A el SS
RRR!
er] SK fey eI
Coo A CSS
a kel ey lel lel lel fe. ley le
v1 6 lf eee Sl
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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
/
THE CANTERBURY PUZZLES
into taree 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
8
THE CANTERBURY PUZZLES
“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
f
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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 1s
meant one that adjoins, either laterally or diagonally.
11.—The Nun’s Puzzle.
“T 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
9
THE CANTERBURY PUZZLES
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
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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 alway 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 itso 2? Here then is a riddle in numbers that 1 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
10
THE CANTERBURY PUZZLES
two, or three and three, or five and five, or six and six, or ten and
ten, or fifteen and fifteen, or all thirty ina row. Inno other way may
we ride so that there be no lack of equal numbers in the rows.
Now, a party of pilgrims were able to thus 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 | 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,
11
THE CANTERBURY PUZZLES —
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 |, 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
Q@)
op (qm
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 embroid-
ered with lions
and castles, as de-
picted in the illus-
tration, the pil-
grims disputed
among themselves
as to the meaning
of these orna-
ments. The
Knight, however,
who was skilled in e
THE CANTERBURY PUZZLES
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
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pie.” Onenight 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.
13
THE CANTERBURY PUZZLES
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 akind ; 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.”
14
Trice CANTERBURY PUZZLES
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, ina clockwise direction, and the person on whom that number
fell was to immediately 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.
15
THE CANTERBURY PUZZLES
‘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 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
a a ee creek in Brittany and
ee Spain: His barque
ve & ve Ba aS ycleped was the Mag-
’ S- Ro Nae Saini wa os dalen. | The strange
i ; ‘ : \ : puzzle in navigation that
ae c \ ' | he propounded was as
4 Hee ax } follows.
ey \ ; a ** Here be a chart,”
Xo \ K Herd quoth the Shipman, “ of
yen ‘ ‘ aS five islands, with the
' ~~ crs ee A res ' inhabitants of which |
¥ - Ue ! do trade. In each year
. ay rh my good ship doth sail
ae os over every one of the
eae eae ten courses depicted
CHART oF ye MAGDALB N. 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 ?
18.—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 :—
16
THE CANTERBURY PUZZLES
“ 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 |
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 |
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.
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
LZ Cc
HE CANTERBURY PUZZLES
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
18
‘
THE CANTERBURY PUZZLES
by these two.” To find exact dimensions in the smallest possible
numbers is one of the toughest nuts | 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 Rp
were not for simple minds : e
like his, but he would
show the good pilgrims,
if they willed it, one that GB, Bo
he had frequently heard ya ae se a
certain clever folk in his ei eee feces
own neighbourhood dis- es : ees
cuss. “ The lord of the : a i ee 2
-manor in the part of os xe eed i.
Sussex whence I come eed Rel : i <
hath a plantation of six- ee ee
teen 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
19 ce2
THE CANTERBURY PUZZLES
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 house-
holder 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 |, 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 some-
20
THE CANTERBURY PUZZLES
what 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 nddle.” 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 un-
doubtedly what
in later times
we should call
a dandy, for,
** Embroidered
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 illustra-
tion to No. 26,
while the Hab-
erdasher 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
2\
THE CANTERBURY PUZZLES
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
22
rit CAN EE RBURY PUZZLES
prayer, Men must give silver to the needy friar.” He went by the
name of Hubert. One day he produced four money bags and
spake 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 | trow there nowhere is.” His virtues and charity made him be-
loved 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 HEP
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.
22
THE CANTERBURY PUZZLES
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 > Att 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. Att 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.
24
THE CANTERBURY PUZZLES
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
Zo
THE CANTERBURY FUZZLES
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 | was awakened at dawn by the crowing of cocks—for which
din may our host never thrive
(io > ofp de db ol)s Ab dre —TI have sought an answer
thereto, but by St. Bernard
be by ofp de fp op de dp I have found it not. There
be sixty-and-four flowers-de-
luce, and the riddle is to
show how I may removel'six
ey
Be
dio of these so that there may
+
fe
yet be an even number of
the flowers in every row and
every column.”
ofp obo m de dp de The Dyer was abashed
- > Wo when every one of the com-
HED pany 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!” Al 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
26
THE CANTERBURY PUZZLES
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. |, it is evident that
27
THE CANTERBURY PUZZLES
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 zig-zag 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 zig-zag 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.
A 4 2 In this way, the Friar
251, Iolo argued, we may go on
Se] e .
wv = straightening out that
zig-zag path until we
ultimately reach a per-
fectly straight line, and
] it therefore follows that
E 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
3 4 prove by actual mea-
surement 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.
100 yds.
A
100 yds.
10
29.—Chaucer’s Puzzle.
Chaucer himself accompanied the pilgrims. Being a mathema-
ticlan and a man of a thoughtful habit, the Host made fun of him,
28
THE CANTERBURY PUZZLES
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
Crysis
Sd Be “ih
o ’ inti | i
Neer
¢
‘
1
~ us om
“ +<
’ +
.
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
29
THE CANTERBURY PUZZLES
- supposed to have set out on I7th 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 hath 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 pro-
pounded was the following.
Heshowed them the diamond-
shaped arrangement of letters
presented in the accompany-
ing illustration, and said, “I
do call it the rat-catcher’s
riddle. In how many differ-
ent ways canst thou read the words, ‘ Was it arat] saw ?’” You
may go in any direction backwards and forwards, upwards or down-
wards, only the letters in any reading must always adjoin one
another.
€>WYN-A-WpeE
So) ea = Oe
FdbY-ArFrWriA-wseéE
ZSrwnN-ArPH-wWP sé
ZSPN-+-+-M>yéE
30
THE CANTERBURY PUZZLES
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
aman “ 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
Li
Océ
N
<i!
Wij,
FA
¢
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
3|
THe CANTERBURY PUZZLES
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.
32
Everybody that has heard of Solvamhall Castle and 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
33 D
THE CANTERBURY PUZZLES
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 rmg. 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
34
PUZZLING TIMES AT SOLVAMHALL CASTLE
great Death’s-head Dungeon with the fair but unconscious maiden
in his arms, all exclaimed, “”Iwas marvellous valiant!” but Sir
Hugh said, “I would never
have turned from my pur-
pose, 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 door-
way once, and once only,
you were bound to end at
the cell that was sought. Can you find the cell 2 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.”
35 D2
THE CANTERBURY PUZZLES
To re-arrange 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, Sr 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.”
36
PUZZLING TIMES AT SOLVAMHALL CASTLE
‘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 follow-
ing striking announcement :
‘““T 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
37
dit CANTERBURY PUZZEERS
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 :—
38
PUZZLING TIMES AT SOLVAMHALL CASTLE
A
Be
Iya tek
. AAAA
Cc ©. C Ce
NA A A A
DD bb Db pp
A AAAAAAA
Bobb pb bon 8 BB
I oe ie NIN ck IR
A AAAAAAAAAA
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.
“| pray you, sir clerk,” said he, “shew 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 acne 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
a9
THE CANTERBURY PUZZLES
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 half 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
il
|
AHA AA to LS sy) AMSA
“gases Ty
Sep,
=a ES Ww, ft
~
|
ya
: SS
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, | 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
PUZZLING TIMES AT SOLVAMHALL CASTLE
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 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 determines the size of the top of the
casket.
4]
Sea
= SLRS ND ERIGMAS..
v=
£3
“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 het 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 daily to the
42
THE MERRY MONKS OF RIDDLEWELL
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.
4|.—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
coed pce
Z LUD,
GG yy
th, <4,
7a DNA
Apr edol
Bs
FY, fod
§ {G7
A LA =
j
/
/}
‘tet A ge ly
WA ah NWA
— = tency
/ O
= /
— ae NIL
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.
43
THE CANTERBURY PUZZLES
“ 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
PLAN OF DO; RM IieO Rey
Rooms on Bauer Floor. 8 Rooms on Lower Floor.
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
44
THE MERRY MONKS OF RIDDLEWELL
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.
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 [.7/\:000>
floored with sixteen large «¢ \ Ce )::
ornamental tiles. When [u:capyct:
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 pat-
terns (the Cross, the
Fleur-de-lys, the Lion, stone os: ppetoran |
and the Star), but plain tiles were also available. The Abbot pro-
posed that they should be laid as shown in our sketch, without any
plain tiles at all, but Brother Richard broke n—
“Ttrow, my Lord Abbot, that a riddle is required of me this
45
‘a,
Peer.
THE: CANTERBURY: PUZZLES
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. j
AA he Readic 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. JMark 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 2 Or have | 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.
46
THE MERRY MONKS OF RIDDLEWELL
“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,
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and the cask was so handy, and the wine was so good withal, and—
and [| 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 | 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, | put in
a pint of water in its stead | ”
It is a curious fact that this is the only riddle in the old record
47
coe CANIERSURY PUZZLES
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
~
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Ati Tif Lo Oy wan
AY ial Ui % IK Up WY Wh? VY)
é : y weet : g Y yj YM / WY)
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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
48
THE MERRY MONKS OF RIDDLEWELL
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 | 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
49 E
THE CANTERBURY PUZZLES
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 | 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
\
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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 | must be where the 12 now is
and the 12 in the place now occupied by |. 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.”
50
THE STRANGE ESCAPE OF THE KINGS
Sd Is
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
todeath. 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
5]
a
THE CANTERBURY FUZZLES
dungeon to await my death. How, by the help of my gift in
answering riddles and puzzles, | did escape from captivity | 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
y=, 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 ?
1
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
52
THE STRANGE ESCAPE OF THE KING’S JESTER
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
(gE
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 1 was indeed to
escape | 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
53
THE CANTERBURY PUZZLES
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
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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 | 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
54
THE STRANGE ESCAPE OF THE KING'S JESTER
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 >?
-
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Peet 1 Al
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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
55
THE CANTERBURY FUZZEES
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 need-
ful 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 |
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 after-
wards remembered
that I had of a
truth entered every
one of the 16 gar-
dens 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 | 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,
[ did at last manage to make a bridge across the ditch. How was
this done ?
56
THE STRANGE ESCAPE OF THE KING’S JESTER
Being now free | 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 | had trained my wit to the making and answering
of cunning enigmas. And I do hold that the study of such crafty
matters 1s 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 | 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.
57
PoE 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 hunts-
men. 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
goodwill.
He took a curious and intelligent interest in puzzles of every kind,
58
THE SQUIRE'S CHRISTMAS PUZZLE PARTY
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 |
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 [| 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.
j
55.—The Eleven Penis
A guest asked someone to favour him with eleven pennies, and
he passed the coins to the company, as depicted in our illustration.
59
THE CANTERBURY PUZZLES
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.”
56.—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 chal-
lenged anybody
to put ten lumps
of sugar in them
so that there
would bean 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.
57.—The Christmas Geese.
Squire Hembrow, from Weston Zoyland—wherever that may be
—proposed the following little arithmetical puzzle, from which it 1s
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 | was coming home, whom should | meet
60
THE SQUIRE’S CHRISTMAS PUZZLE PARTY
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
feel Ba te : |
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,
6]
THE CANTERBURY PUZZLES
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 four boys in two
new groups so that the 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 to us 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
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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
62
THE SQUIRE’S CHRISTMAS PUZZLE PARTY
puzzle was an allegory of a sort, and he intended to show how we
might manage our pudding-tasting with as much despatch as
possible.” | fail to fully understand this fanciful and rather over-
strained view of the puzzle. But it would appear that the
puddings were arranged regularly, as | 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 in twenty-one straight strokes.
You can go up or down or horizontally, but not diagonally, and you
must never strike out a pudding twice, as that would imply a second
and unnecessary tasting of these 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. 6
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 exceed-
ingly 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 anyone 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
63
THE CANTERBURY PUZZLES
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, | think, interest
those readers who may find some of the above puzzles too easy. It
64
THE SQUIRE’S CHRISTMAS PUZZLE PARTY
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 | 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.
65 F
i ay
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 con-
nection with this celebrated case, so many absurd and untrue stories
have become current respecting their doings that I have been per-
mitted to publish a correct account of some of their more interest-
ing 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 investi-
gating 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—some-
thing to unravel. A\s often as not the circumstances are of no im-
portance 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.
66
ADVENTURES OF THE PUZZLE CLUB
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 Scots-
man, 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 com-
municate. 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.
“Ts it another of those easy cryptograms ?” asked Wilson. “If
so, | 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
67 F 2
THE CANTERBURY PUZZLES
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 con-
tempt, ‘how can anyone swear that this is his lordship, when the
greater part of him, including his head and shoulders, is hidden from
sight 2 And—and’—she scrutinised the photo carefully—‘ why, I
guess it is impossible from this photograph to say whether the gentle-
man 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 go-
ing. Here itis. 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 Marsford 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.
68
ADVENTURES OF THE PUZZLE ‘CLUB
“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 snap-
shot just when the legs happened to be perfectly straight.”
A oe
eon ye OS
‘
es)
“|’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.” |
69
THE CANTERBURY PUZZLES
“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.
A
Hi wy) 5
AN
\7t}
‘*T’ve got it!” shouted Wilson. “ Extraordinary that none of you
have seen it. It is as clear as possible. It all came to me ina
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
70
ADVENTURES OF THE PUZZLE CLUB
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 gentle-
man 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 the 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, everyone 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
7\
THE CANTERBURY PUZZLES
civilities with the two members of the Puzzle Club. A few refer-
ences 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 hand-
writing. 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.” 7
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 carefully, 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.
s Here, gentlemen, you see, said the inspector, “that the foot-
prints 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. [he 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
72
ADVENTURES OF THE PUZZLE CLUB
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 >”
“T am going straight back to communicate the discovery to head-
quarters. We shall withdraw the cordon and search the coast for
the dead bodies.”
“Then you will make a fatal mistake,” said Melville. ‘‘ The
6
ng
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.”
“T 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 foot-
13
THE CANTERBURY PUZZLES
steps, 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 illustra-
tion 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.
74
ADVENTURES OF THE PUZZLE CLUB
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 O 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
ee.
At ag
~
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.
75
THE CANTERBURY PUZZLES
‘ 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, |
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 toa
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,
76
ADVENFURES OF JHE PUZZLE CLUB
stabbed to the heart. A\ll 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 pedes-
trians 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 pro-
vided 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,
77
THE CANTERBURY PUZZLES
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 :
“ T 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. |
was, in fact, beginning to feel desperate. One hot summer day |
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.
** “Tf 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
78
ADVENTURES. OF THE PUZZLE CLUB
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.’
"
if Pi
anh ie aN:
‘“ My dear chap, that only means ae 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, | saw at once
that for a spot to be exactly two, three, and four furlongs from suc-
cessive 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. | will leave you the puzzle
of working out just what that area is.
79
THE CANTERBURY PUZZLES
“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.
80
PAE AAROPESSOR 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. [he 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 >”
“| 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 improve-
81 G
THE CANTERBURY PUZZLES
ment, embellishment, and extension. Take, for example, magic
squares. [hese 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.”
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
67.—The Coinage Puzzle.
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.”
82
THE PROFESSORS PUZZLES
‘* But how can the coins affect the question ?” asked Grigsby.
**’That you will find out when you approach the solution.”
1 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 ina 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 well 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.”
83 ZG
THE CANTERBURY PUZZLES
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
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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
84
‘THE. PROFESSOR’S PUZZLES
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.
“TI 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.
“T 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 those 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.”
“T 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
85
THE CANTERBURY PUZZLES
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.
86
THE PROFESSOR’S PUZZLES
71.—Romeo's Second Journey.
‘Tt 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 >?”
“Tt 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 2?” asked Hawkhurst. “I think I can do that.” A
few minutes later he exclaimed, “ How’s this ? ”
87
THE CANTERBURY PUZZLES
‘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. |
distinctly said that the jumps were to be made upon the table.
Sometimes it passes the wit of man to so 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.”
88
MISCELLANEOUS PUZZEES
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 con-
af N x
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Aes cut
Skeet, es
fed 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
89
THE CANTERBURY PUZZLES
pastime of “ shying at 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 pre-
paration 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.
90
MISCELLANEOUS PUZZLES
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.
Ta /7
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2.
‘“ 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 chessboord, drew blood
9]
THE CANTERBURY PUZZLES
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 illustra-
tion. 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 chessboard, 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, | foot
"A from the ceiling, as at A,
ot a and a fly is on the opposite
wall, | foot from the floor in
ee ee is the centre, as shown at B.
, "++. | What is the shortest distance
30 ¢€. 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.
iz £6.
92
MISCELLANEOUS PUZZLES
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.
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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
25,
THE CANTERBURY PUZZLES
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 >
17.—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 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 | B
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.”
94
MISCELLANEOUS PUZZLES
/8.—Catching the Hogs.
In the illustration Hendrick and Katrun 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 | will now explain.
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Copy the simple diagram on a conveniently large sheet of card-
board 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
Katrun, and the other the hogs. The first player moves the Dutch-
man and his wife one square each in any direction|(but not diagonally),
23
THE CANTERBURY PUZZLES
and then the second player moves both pigs one square each ; and so
on, in turns, until Hendrick catches one hog and Katriin 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 cardsharpers at
racecourses and in railway-carriages.
at. As, on its own merits, however,
rv, 2a) © the game is particularly interesting,
I will make no apology for presenting
it to my readers.
Y ° y yi The cardsharper lays down the
9 9g o v twenty-four cards shown in the illus-
tration, and invites the innocent way-
: Pa . oa farer to try his luck or skill by seeing
~) ® 0 ep which of them can first score thirty-
one, or drive his opponent beyond,
ool |@4% 10 O| |&%) in the following manner :
ool lee lool lee One player turns down a card, say
a 2, and counts ‘two’; the second
ool f@ @| [66] [#4] player turns down a card, say a 5,
oO} | @ 0) & and, adding this to the score, counts
seven”; the first player turns down
Sol ee 0 Ol lee. another card, say a |, and counts
ool lael lool lae@ eight”; and so the play proceeds
ool lee! 10 0] le] 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
96
MISCELLANEOUS PUZZLES
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 im vain for another 3 with which to wi, 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 ar-
ranging the details. The
letters in the diagram show
the different nationalities,
and indicate not only just 3
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
97 H
THE CANTERBURY PUZZLES
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 | once saw on the
Continent. Each clown bore one of the numbers | to 9 on his
body. After going through
the usual tumbling, juggling,
and other antics, they gene-
rally 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. |
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. | 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
98
MISCELLANEOUS PUZZLES
after the man of magic had foretold the most favourable issues, and
concocted a love-potion that was certain to help his cause, the con-
versation 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.”
99 H 2
THE CANTERBURY PUZZLES
“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,
A 5, 6,7 6.0L
we get exactly the
same number in the
circle, starting from a
different place. For
example, multiply by
4, and the product is
2352941 176470588,
which starts from the
dart in the circle. So,
if we multiply by 3,
we get the same re-
sult 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.
100
MISCELLANEOUS: PUZZLES"
84.—The Japanese’ Ladies ard the Carpet.
Three Japanese ladies possessed a square ancestral carpet of con-
siderable 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
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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 feet square, 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
101
“THE ©ANTERBURY PUZZLES
the other twe sisters, who insisted that the square carpet should be
so cut chat edch’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 corre-
spondent 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.—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 resi-
dent 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 histourat 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 per-
form 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.
102
MISCELLANEOUS PUZZLES
86.—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
103
THE CANTERBURY PUZZLES
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 ?
87.—The Chifu-Chemulpo Puzzle.
Here is a puzzle that was recently on sale in the London shops.
It represents a military tram—an engine and eight cars. The
puzzle is to reverse the cars, so that they shall be in the order
S7.-0;.9;-4, 3, 2, |, mstead of 1, 2,-3, 4, 5,6; 7,6) with tne
engine left, as at first, on the side track. Do this in the fewest
possible moves. Every
time the engine or a
S car is moved from the
i main to the side track,
| = or vice-versa, it counts
1 3 bok S a) OH a move for each car
or engine passed over
“42 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 ?
104
MISCELLANEOUS PUZZLES
88.—The Eccentric Market-woman.
Mrs. Covey, who keeps a little poultry farm in Surrey, is one of
the most eccentric women | ever met. Her manner of doing
business is always original, and sometimes quite weird and wonderful.
In our illustration she is seen 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
it is such an improvement on it that I have no hesitation in pre-
senting 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 >?
105
THE CANTERBURY PUZZLES
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. Con-
tinue 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 condi-
tions, 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. |
106
MISCELLANEOUS PUZZLES
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 | 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 in-
stinctively 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,” &c. If the child’s
mother has occasion to add up the numbers |, 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, 25x 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, {TRY OUR ties
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 pic-
tures in all; but
one picture on
No. | is repeated on No. 4, and two other pictures on No. 4 are
repeated on No. 3. There are, therefore, only twenty-seven differ-
107
THE CANTERBURY PUZZLES
ent pictures. [he owner always keeps No. | 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 2. 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
7 - thirty-six sties Is in a
Ye. | | straight line (either
horizontally, vertic-
Ye. ally, or diagonally),
with at least one of
| the pigs, yet no pig is
in line with another.
= In how many differ-
ent ways may the
four pigs be placed
to fulfil these con-
» ditions > If you turn
> this page round you
yy get three more ar-
Ene rangements, and if
i "wD 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
108
MISCELLANEOUS PUZZLES
blocks ; that is, blocks bearing the ten digits, or Arabic figures—l1,
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 | is the same.
You will find it _
quite impossible to a Td
get any smaller re- li 2 !
sult. em ’
Now, my puzzle
is to find the largest | Spaae
possible result. . 4iSnay_ 61 STL t ®) |
Divide the blocks | 2 go, AL \
into any two groups U3 +(e = RS
of five that you like, : ;
and arrange them
to form two multi-
plication 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
109
THE CANTERBURY PUZZLES
—three marked to represent foxes and three to represent geese.
Place the geese on the discs |, 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 I, 2, and 3, and the geese on 10, 11, and 12—
that is, make them
exchange places—
in the fewest possi-
ble moves.
But you must be
careful never to let
a fox and goose get
within reach of
each other, or there
wilk 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
110
MISCELLANEOUS PUZZLES
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 | 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, | told
him that it was my
wish to make the
table from the tim-
ber 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 to-
gether, 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, | 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 proportions 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 ?
111
THE CANTERBURY PUZZLES
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 re~
sult 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 com-
paring 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
112
MISCELLANEOUS PUZZLES
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 com-
plexion of one of
the cook’s follow-
ers), | had consid-
erable regard for
him.
‘There he is,”
said Mrs. Solders,
when I called to
inqure. ° That's
how he’s been for
three weeks. He
hardly eats any-
thing, and takes no
rest, whilst his busi-
ness 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 Mr. Solders to explain matters to me. [t 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
113 I
THE CANTERBURY PUZZLES
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 [ 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 youdreaming about >”
I said at last.
“Two feet ” he murmured.
‘“Somebody’s Trilbys >” I in-
quired.
_“* Five times round
“Two feet, five times round !
What on earth are you saying >”
“Wait a minute,” he said, begin-
ning 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.
114
99
MISCELLANEOUS PUZZLES
“ 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 through-
out. If it were tapering, the puzzle would be more difficult.
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
115 PZ
THE CANTERBURY PUZZLES
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 trades-
men’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
—"
= - Hi> pes
~ a i
nee, \ :
= (Ses AO;
— oS ee aw} iS Ls
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 remark-
able coincidences—little things against the probability of the occur-
rence of which the odds are immense—that fill us with bewilder-
ment. 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 | to 9 and 0, and, what is more remarkable, that if the
numbers on the first and second cars are multiplied together they _
116
MISCELLANEOUS PUZZLES
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
117
THE CANTERBURY PUZZLES
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, a9 a6,a/7 a2,
and a2a/. The |, 8 and O 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 varia-
tions 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 there-
fore 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.
“| suppose,’ said Mr. Softleigh one morning toa jovial, weather-
beaten skipper, “‘ you have seen many wonderful sights on the rolling
seas > |
‘“ Bless you, sir,.yes,” said the skipper. “P’r’aps you've never
118
MISCELLANEOUS PUZZLES
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 uncer-
tain whether they existed.”
“ Uncertin! You wouldn’t say there was anything uncertin
about a sea-sarpint
if once youd 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 flash-
ing 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 along-
side 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.” | |
119
“-_
THE CANTERBURY PUZZLES
“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
(CE
Lp
S
DY, aa ~’ 7 L
=
3
escort home. There were just a gross of osculations altogether.
How much longer would the ladies have taken over their needle-
work task if the sisters of the curate referred to had played lawn
tennis instead of attending the meetings > Of course we must
120
MISCELLANEOUS PUZZLES
assume that the ladies attended regularly, and I am sure that they
all worked equally well. A mutual kiss 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 |
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 calcu-
121
ANA
Shinai
THE CANTERBURY PUZZLES
lated 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 trian-
gular territory of equal area, beyond
the boundaries of which they would
pass at the cost of their lives. Now,
the royal surveyor found himself con-
fronted 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 dis-
tricts 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.
122
MISCELLANEOUS PUZZLES
108.—Flato 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 won-
ders, and that a
cat has nine lives
—and sometimes
nine tails.
Most people are
acquainted with
some of the curious
properties of the « ‘ = fee BZ 4 |
BS
number nine in
ordinary arith-
metic. For exam-
ple, 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
123
THE CANTERBURY PUZZLES
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 | 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, to so 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. Whichis correct ?
124
MISCELLANEOUS PUZZLES
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 infact the parent of “ Nine Men’s Morris,” referred to
by Shakespeare in ““A Midsummer Night’s Dream” (Act II.,
sc. 2). Each player
has three counters,
which they play alter-
nately on to the nine
points shown in the |
diagram, with the ob-
ject of getting three in
a line and so winning, va Bs
But after the six : am
counters are played
they then proceed to
move (always to an
adjacent’ unoccupied |
point) with the same
object. In the above & a
rr tf
Ce
S
example White played as 3
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
125
THE CANTERBURY PUZZLES
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
g ,3 Nii
RES ig
~ NY p 4
NY FSS ee
SQ
N
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 oclock
at night he dismissed the servants, saying that he had some important
126
MISCELLANEOUS PUZZLES
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.
ps /
oH
7, 3 : \\ (,
2 wna
4y) i
ei oe es )) Cy
i cK
awe j yyy, f
iY Y, ,
{ i v;
¢ -
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
AZZ
*
THE CANTERBURY PUZZLES
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 equili-
brium. 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 dis-
covered, 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 ?
128
MISCELLANEOUS PUZZLES
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,
aes’ =
Cs
— BSS a5 aN
1% a 0 4 9,
he i, i ms 4
POR
dR OX
eg
\
s
yy
Ny
NS
Yitta,
eeds
Mine
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
122 K
THE CANTERBURY PUZZLES
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
“ |
\
Ngee ‘
eh SS eS
<
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. E:very 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 >
130
SOLUTIONS
hae 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.
3 2 3 +4 5 6 7
4 I | 6 IO 15 oi 28
5 4 10 20 35 56 84
Number of Moves.
3 a a eames 63 2127
4. I Se 7 AQ. =120% 62145700
5 Tos III 5b L023 2orG
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
13] Kaz
THE CANTERBORY PUZZLES
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 solu-
tion—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 re-
moved in 49, and 21 in 321 moves. Also, with five stools, we find
from the third and sixth rows that 20 cheeses require I 11 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 6
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 | move; then replace the 3 in 7 moves; and finally replace the
6 in 17—making im 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,
pa at and in the case of five stools
if a ii T in pyramidal, then there will
poe ld
hp od
Mp—pb—tpp—t
Fe F-A-HHMH{) Ss 2.-The Pardoner’s Puzzle.
: es gig ny LE ch The diagram will show
tee how the popes started
from the large black town and visited all the other towns once,
and once only, in fifteen straight pilgrimages.
132
| be more than one way of
making the piles and sub-
| sidiary tables will be re-
quired. This is the case
with the Reeves 6 cheeses,
]
]
]
But I will leave the reader
to work out for himself the
uae ECW soa Saat wine eta ren
extension of the problem.
SOLUTIONS
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 (counting the reversals as different, of course), but they all
require the moving of more 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 realised an
by reference to the ac- ABODNDOBDABDOOO
companying diagram :—
Join A, B, C, and D, OOQOD
and there are 66 squares CIOKG
of this size to be formed; | |
the size A, E, F, G,
gives 48; A, H, I, J,
B77 beh, Me Loe
oO O@
OOOO © ©
GOre @ Ole
a
wn
4;
OOOO O080®
COW OOo ee e ©
OOOO OO®OO
OXOK TOROIOROX®
<<
%
nr
7 WN
1S |
“XJ
ica
<
i 220
aoe CS
DAKO
SCNT
te Gens Aa. 7h. 24: K hom b. 14-6. ©. 5, Do.
K, n, p, G, 10; Kgl 6. ©; to, G4 Ou, et 2. te
total number is thus 575. These groups have been treated as
if each of them represented a different sized square. ‘This ts
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
Ke hyn, b.
133
THE CANTERBURY PUZZLES
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 com-
mand 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 |
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 whatsoever.”
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.
134
SOLUTIONS
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
i 75, 522
apa
16(3,
iS J
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 pat-
tern properly matched. It was also stipulated in effect that one of
SESE SESES a] [ey jel jet [ele
Ql les] ler] [lL [@] {oe HES] [SE IO] IG IS] fe
Sl Jel le Jel ley |e “ke I@L_/@_ le _lerié@ct
S| eT ia lel Tap Ie a! iQ Iel [él ley lel le
&| [Gl IT esl ley (eh SC ISyT [Skater lel [Ah .
Oe fe) ler le, (eT ie. og |. ro] IG) lal 1eL [el 1é
ESE mE a fol le, lol ial
“| EUs ler ierier ie Re erlelécelleryer ie
S| IL jel jel fel lel jel iS S] Jet fa] le} fal: lela
&) KY Fel i, eT ISL ef let ie &) IGl it I@P hel [el |e
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.
135
THE CANTERBURY PUZZLES
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 | foot by | 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.
ee] ei
A a Squire’s Yeoman.
a
————
——="
The illustration will show
how three of the arrows were
removed each to a neighbour-
ing square on the signboard of
a
still no arrow was in line with
another. The black dots indi-
cate the squares on which the
three arrows originally stood.
11.—The Nun’s Puzzle.
As there are eighteen cards bearing
the letters “CANTERBURY PIL-
GRIMS,” write the numbers | 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 com-
pletes the process by placing Y against 2, P against a I against 10,
and so on, he will get the letters all placed in the following
136
the “ Chequers” Inn, so that —
SOLUTIONS
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 | 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 !
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,
5 1.2 4°64. 1, 2. 4.8 7, 4, 5.6 As there ts never more
than one vacant dungeon to be moved into, there can be no ambiguity
in the notation. |
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 con-
tamed 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.
137
THE CANTERBURY PUZZLES
But five out of 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, (iu.) = 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 (ii.)
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 rmg. 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 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.
138
SOLUTIONS
18.—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.
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.
~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
139
THE GANTERBURY PUZZLES
the published answers, by his method, that I have seen inaccurate,
but nobody has ever published the much smaller result that I now
print. Thecubes of $42072é82686 and 318577684688 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 has taken the trouble to cube out these
numbers and finds 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 2772523
28340511
and 217426828 added together make exactly 28. See also No. 61,
‘The Silver Cubes.”
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 know-
ledge I cannot rigor-
ously demonstrate that
fifteen rows cannot be
beaten, I have a strong “pious opinion” that it is the highest
number of rows obtainable.
22.—The Franblin’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
| 140
SOLUTIONS
straight directions. The trick.
consists in the fact that, al-
though the six bottles (3, 5,
6, 9, 10 and 15) in which
the flowers have been placed
are not removed, yet the six-
teen 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 asingle line because it con-
tains 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 con-
struct, 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
14]
THE CANTERBURY PUZZLES
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 illustra-
tion. That this
was not pro-
hibited we shall
$O00n: firtds
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
“ED 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 !
oe
oe? meer
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 A B in D and B C in E;; produce the line
A E to F making E F equal to E B; bisect A F in G and
describe the arc A H F; produce EB to H, and E H is the
— 142
SOLUTIONS
length of the side of the required square; from E with distance
E. H, describe the arc H J, and make J K- equal to B E; now,
from the points D and K drop perpendiculars on E. J 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 pee as
general form :—“A New ae
Problem on Superposition :
a demonstration that an equi-
lateral triangle can be cut
into four pieces that may be
reassembled to form a square,
with some examples of a
general method for trans-
forming all rectilinear tri-
angles into squares by dis-
section.” It was also issued as a challenge to the readers of the
Daily Mail (see issues of Ist and 8th February, 1905), but though
many hundreds of attempts were sent in there was not a single
solver.
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 2? 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 rn is even the remainders
in rows and columns will be even, and where n is odd the remainders
will be odd.
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
143
THE CANTERBURY PUZZLES
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 zig-zag 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 zig-zag 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 false
to say we might go on dropping marbles into a basket until they
became 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
144
SOLUTIONS
is 2n + |, and it is a palindrome without diagonal readings, is
ae DE.
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 $ of the eight loaves.
Therefore, as the Miller provided 4° and ate 8, he contributed
% to the Manciple’s meal, whereas the Weaver provided 8, ate §,
and contributed only 1. 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.
145 L
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
Hs
- a ee — 6
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.”
146
SOLUTIONS
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 | drive, the fifth in 3
drives and | backward approach, the sixth in 2 drives and | approach,
the seventh in | drive and | 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 de-
serve, 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 fee 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 13 in. x 3 + | in. = 6in,,
and 12? in. X 9 + |in. = 16in. 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
147 pez
THE CANTERBURY PUZZLES
a link multiplied by the number of links and added to twice the thick-
ness 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.
“Tt 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
148
SOLUTIONS
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 ee
number we may choose to select o seca BY, LL if ree
from 22 to 38 inclusive, except- <3.
ing 30. In some cases there are
several different solutions, but in
the case of 23 there are only two.
I give one of these, and leave the
reader to discover the other for
himself. In every instance there
must be an even number in the
central place, and any such num-
ber from 2 to 18 may occur.
Every solution has its comple-
mentary. Thus, if for every
number in the accompanying
drawing we substitute the differ- . y
ence between it and 20 we get a
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.
Sy
Se Sse
eae’? |: Mi
SSS: en
frog 5 \
5
<=
SSS
SS
.
4
SS
. ROS
~)
ts
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
149
THE CANTERBURY PUZZLES
'
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 | marvelled at the thing,
]
seeing its exceed-
ing difficulty. My
masters and kins-
men, it is done in
this wise.
The worthy
knight then pointed
out that the cres-
cent was of a par-
ticular and some-
what irregular
form, the two distances a to b and c to d being straight lines, and
the arcs ac and b d 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
150
Cc F
G e ret
SOLUTIONS
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 adjoming one.
A
> Sagal 8:
Ror RK
A AAA
CoC «Ce
AAAAAA
woo b> DD 4D
AAAAAAAA
BS Boe Bb be 8
me on ROR ROR Re ie
AAAAAAAAAAA
“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, 1,024, is the answer
thou dost seek.”
15]
THE CANTERBURY PUZZLES
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 flag-
staff 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 in-
side 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
152
SOLUTIONS
squares can be calculated direct from 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
20
ae WR
. i
VV
20
hE —>
oe »
x
| pe
Oo
Soe
20
20
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 al-
beit such as these we may never bring within our understanding, yet
153
THE CANTERBURY PUZZLES
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 unfore-
seen for them that do come after us.” |
154
Tie MERRY. MONKS OF RiIDDEEWEIEE
41.—The Riddle of the Fishpond.
Number the fish baskets in the illustration from | 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. | and transfer it to basket No. 4 :—
1 to 4, 5 to 8, 9 to 12, 3 to 6, 7 to 10, Il to 2, and complete
the last revolution to 1, making three revolutions in all. Or you can
proceed this way :—
A167 06 10-1 1012 to 5.-2 to). Oto 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
155
THE CANTERBURY PUZZLES
By eed h oe)
have been 27, and
that, since three
more than _ this
number were ac-
tually provided
with beds, the
total number of
pilgrims was 30.
3,2 4°
The accompany-
ing diagram shows
how they might
be arranged, and
if in each instance
we regard the
upper floor as
placed above the
8 Rooms on Upper Floor 8 Rooms on Lower Floor. -
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. 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.
Y
ASB
SOLUTIONS
44.—The Riddle of the Sack 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
“oo pints (the square of 99 divided by 100); after the third theft
there would remain “1ovv0 (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
73100 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
157
THE CANTERBURY PUZZLES
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, | have ascertained that the
exact quantity of wine stolen would be
26°02996266117195772699849076832850577473237376473235 -
55652999
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 | 13 squares
of 5,329 men—that is, 73 by 73.
47.—The Riddle of St. Edmondsbury.
The reader is aware that there are prime numbers and composite
whole numbers. Now, I,1 11,111 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.
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
158
SOLUTIONS
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,
O10 6, 5 4 3, 2-7, 8. 9710 NG, 5. 4: 3. 2-1), 6,0, 4 3.2:
12,078.99 (1071 12)-7, 6.9.10. 11,6 5. 4.3, 212 7 8.
D102 1156.5, 45-2 6-9: 10.114 3. 2,10, 1b 22. We thus
have made 118 moves within the conditions, the black frogs have
changed places with the white ones, and | and 12 are side by side
in the positions stipulated.
159
THE SPRANGE ESCAPE OF LAE. RINGS
STS
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
160
SOLUTIONS
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.
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. Att 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
16] M
THE CANTERBURY 2U7Z/7LES
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,
B and no garden may be entered twice.
oe rT T | The trick consists in the fact that you
tein Se = may enter that starred garden without
snail + ‘ae necessarily leaving the other. If,
ger nmreeeee =e ae ow’
-
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 door-
way, upon the star, he discovered it
was a false alarm and withdrew, he
A 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 | made my exit at B.” This is the only answer
possible, and it was doubtless that
which the jester intended.
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.
162
THE] SOUIRE 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 Eleven Pennies.
lt 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 |!
56.—The Three Tea-cups.
Miss Charity Lockyer clearly wanted to “‘ get level” with the
propounder of the last puzzle, for she had a trick up her sleeve quite
as good as his own. She
proposed that ten lumps
of sugar should be placed
in three tea-cups, so that
there should be an odd
number of lumps in
every cup. The illustration shows Miss Charity’s answer, and the
figures on the cups indicate the number of lumps that have been
163 M 2
THE CANTERBURY PUZZLES
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.
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
50s + 4, 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% + 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 84 + #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, 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.
164
SOLUTIONS
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 | 2 7 8 and 5 3 4 6, the figures
of which, in each case, add up to 18. There are just three other
ways in which the boys may be grouped: | 368—2457, 1467
755 06- and. 25060 /-—1.4 56.
59.—Tasting the Plum Puddings.
My diagram will show how this puzzle is to be solved. It is the
only way within the conditions laid down. Starting at the pudding
> Gd
® @ & PB
QO @—<2 an
O-2—-2
£2
=
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.
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
165
THE CANTERBURY PUZZLES
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 perform-
ance 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.
/ Married couples. 3. 3 2 IA
Be WW IGOWSs eb een ee ee
(2 Gachelors.and Boys, .oe7 4, 7 2
10 Maidens and Girls . . . . . 10
A Olal eae ee 39 Persons
Now, if everyone 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 17]
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 276351 inches along each edge, and the
other must measure +0837 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.”
166
tHe ADVENTURES OF THE, POZZEs
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
167
THE CANTERBURY PUZZLES
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 manoeuvre 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 foot-
prints 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 set 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 I, 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.
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
168
SOLUTIONS
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 ra
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. There are several
different ways of attacking this problem, but I will leave the reader
the pleasure of working out his own solution.
169
THEVPROFESSOR s PUZZLES
67.—The Coinage Puzzle.
Ats. 4s. 2 6
6a Ais, ai
Qs, “ Js
Is. Os. Qs,
ds. ds.
Os 6d. Gd.
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 2 must be placed
in one of the corners. Other-
wise 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 1s 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
4s
is based on a similar prin-
ciple, 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
170
SOLUTIONS
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 104 [2a]
diagonals add up Is. 6d.
There is no stamp on one
square and the conditions |
did not forbid this omission. ee Is. oy
The stamps at present in ies ts EI
circulation are these :—#4d.,
id. led. 2d, Zed. 3d:,-4d.,. 5d.) 6d.; 9d.)--IOdig Vs: 25300;
5s., 10s., £1, and £5.
wi
he
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 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
17]
THE CANTERBURY PUZZLES
solution, with its pretty, symmetrical arrangement, is seen, it looks
ridiculously simple.
It will be found that Romeo reaches Nahe s balcony after visiting
every house once and only
Jp Va Va
WY; ZZ Wh; zy once, and making fourteen
CAH turnings, not counting the
y AR SZ VZAS Le, g turn he makes at starting.
MV, Y U4, Zu = These are the fewest turn-
Yy Y, ings possible, and the pro-
Ln Z Le blem can only be solved by
Yy Vy, Vy 1 the route shown or its re-
y versal.
Le Ae UA,
LZ) WZ.
Vt LAL Z@ 71.—Romeo’s Second
a hoes a go i Journey.
HBD 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 turn-
ings are required to perform y Vi, LF
the feat. The Professor 5 Yy :
informs me that the Helix LW YH a Livy GY
Asspersa, or common or gar- Lp | Y Y, y
den snail, has a peculiar Z
aversion to making turn- V AY y
ings: so much so, that one Yy GY YY Y
specimen with which he oy ; |
, Ys
made experiments went off Liz
in a straight line one night Y Y Z )
and has never come back Y, Y Y yer)
since.
AED
WS
\N
N
\
WS
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
172
SOLUTIONS
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 occu-
pied. 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 Pro-
fessor forgot to state, goes direct South.
173
MISCELLANEOUS PUZZLES
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: 0 . 0 . 000 . 000.
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
0 . OOOOD000000, 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 0 . 000 . 0000000,
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. Leet us suppose that he knocks down the
single one, then we play to leave him 00 . 0000000. Now, what-
ever he does we can afterwards leave him either 000 . 000 or
0.000.000. We know why the former wins, and the latter
gum : ; wins also, because, however he
; Vp may play, we can always leave
AVA GZ a him either 0 . 0, or 0.0.0.0,
Y or 00 . 00, as the case may
Lh Ws be. The complete analysis |
: GY, can now leave for the amuse-
vi wy ment of the reader.
ee ZA 74.—The Broken Chessboard.
Zaz The illustration will show
oes
how the thirteen pieces can
be put together so as to con-
SOLUTIONS
struct the perfect board, and the reverse problem of cutting these
particular pieces out will be found equally entertaining.
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 bea cardboard box. Then the box may be cut
in various different ways so that the cardboard may be laid flat on the
30 4€.
1 e A
Toe ie
‘< apr Wit. -
24 ors get
4: 42 ft. e
+ PR goog meen ne oe Hee ee 5 7 FLOOR
x B FLOOR
A
An .
ox % +
Nod is ei
\d wae “i oO: ‘i
3 por Se
se" FLooR ef” FLOOR
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
175
THE CANTERBURY PUZZLES
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 the obviously shortest route (that shown in No. 1), he would
have gone 42 feet! Route No. 2 1s 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 fly are 2, 3, 4,5, and 6 feet from
the ceiling and 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 bottles. 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.
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 Katrun.
Each hog merely runs in and out of one of the nearest corners,
176
SOLUTIONS
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 Dutch-
woman 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 Katrin 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.
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 makel0 or 17,
and win. The first player may also win by leading a I, but the play is
complicated. Itis, 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. The illustration shows the required directions for
the five systems of lines, so that no line shall ever cross another.
177 N
THE CANTERBURY PUZZLES
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 | 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 «6, Now, since the re-
curring decimal ‘9 is equal to 2, and therefore to |, it is evident
that, although the clown who bears the figure | 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
De AEG
Seas cae a,
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 16393442622950819672131147540983-
60655737704918032787. If you want to multiply this by 71, all
178
SOLUTIONS
you have to do is to place another | 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.
83.—The Ribbon Problem.
The solution is as follows: Place this rather lengthy number
on the ribbon, 02127659574468085 | 063829787 234042553 191-
4393617. It may be multiplied by any number up to 46 inclu-
sive to give the same order of figures in the ring. The number
previously given can be multiplied by any number up to 16. |
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
one-seventeenth and one-forty-seventh respectively. Multiply the
one by seventeen and the other by forty-seven and you will get all
nines in each case.
84.—The Japanese Ladies and the Carpet.
If the squares had not to be all the [a Bi
same size, the carpet could be cut in four |
pleces 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 Bee
>
have the squares exactly equal in size, 4 5
we shall require six pieces, as shown in 5
the larger diagram. No. | is acomplete 6
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.
179 Nz
THE CANTERBURY PUZZLES
85.—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 de-
monstrated to his
satisfaction that it
cannot be done in
accordance with
the conditions as
he at first under-
stood them, he
should carefully ex-
amine 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.
86.—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.
180
SOLUTIONS
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 Cap-
tain 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 im- -
paled animal would complete S.
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 be settled by
the captain’s pocket compass.
GO MILES TO
CO ee oa
ORTH POLE
y
— \
\
ne
>
D
v
p
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 :-—
056/78 = 10 moves
1234
056 = 2 moves
i733 67 4
36 = 5 moves
0312 87 4
0 = 9 moves
87654321
Twenty-six moves in all.
181
THE CANTERBURY PUZZLES
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 trans-
action 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 I1, 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 I, and write it down on 1; and soon. The second
word can be inserted in the same order. The solution depends on find-
ing those words in which the second and eighth letters are the same,
and also the fourth and sixth the same, because these letters inter-
change without destroying the words. MARITIMA (or sea-pink)
would also solve the puzzle if it were an English word.
90.—The Round Table.
Here is the way of arranging the seven men :—
Jag) wheal D hee Dia daa ©
A CD BY G “br
Aj Do BC srs Gar
AG, Bolen Coa)
Avwk: Cb GoD. B
Pooks Do Gee By
Ph Ce BO bw
Ro GC hr kB
Ao Bk De EG. 6
A he CGB
AG BD ste
Ake GC Be aD
A EB, CC. DzG
yee, Oa Gye Salad Bs eyes ol
Aer DG BC. Ee
182
SOLUTIONS
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 (cs ne - 2) ways. [he 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. Recently, 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 method is far too complex for me
to explain here, and the arrangements alone would take up too
much space. The case of I] 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.
91.—The Five Tea-Tins.
There are twelve ways of arranging the boxes without consider-
ing 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, includ-
183
THE CANTERBURY PUZZLES
ing the example that I gave, not counting the reversals and reflec-
tions of these arrangements as different. Jaenisch, in his ““ Analyse
Mathématique au jeu des Echecs” (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, |
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 J. 206104
B. 136002 K. 241005
C. 140502 L. 250014
D. 140520 M. 250630
Es22560025 N. 260015
F. 160304 O. 261005
G. 201405 P. 261040
H. 201605 Q. 306104
I. 205104 —
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 x 2) + (4 x 1) + (8x 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 Gf we do not count
reversals as different), and I will leave the reader to find it for him-
self,
184
SOLUTIONS
93.—The Number Blocks.
Arrange the blocks so as to form the two multiplication sums
915 x 64 and 732 x 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 :—
(056 116 127 5126 7 e
te ee a ey
7 1-6 6 1 72 BS
pee 0s 10 ee
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.
95.—Robinson se | By
Crusoe’s Table. :
The diagram
shows how the
piece of wood
should be cut in G77@# Y,
WLLL
two pieces to form Le Li
the square table-
top. A, B, C, D y WYttht. Yl. jlo
are the corners of
the table. The ZA. LDL
way lin which the UGY jl fh Yo
piece FE. fits into Cid Li
the piece F will be obvious to the eye of the reader. The shaded
part is the wood that is discarded.
185
THE CANTERBURY PUZZLES
96.—The Fifteen Orchards.
The number must be the least common multiple of 1, 2, 3, &c.,_
up to 15, that, when divided by 7, leaves the remainder 1, by 9
leaves 3, by I] 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. x 10 ft. x 10ft., 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. x 12°6 ft. x 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 com-
plete 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 eee are 40ft. (one-fifth of 200ft.) and 16ft. 8in. There-
fore the hypotenuse is 43ft. 4in. The length of the garland is
186
SOLUTIONS
therefore five times as long, 216ft. 8in. 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.
100.—On the Ramsgate Sands.
Just six different rings may be formed without breaking the
conditions. Here is one way of effecting the arrangements.
Ao DE Gel a lee
Pe S rG@ T- kK M B-D bon). i
Ab Ga) Mec bl) Bb Ek
AoE ol MoD TE CC G KB kg
Poe he Cor ME fe Ge ie eal
Vs Cd") a cree Se anes Sire De eet Gites 4 sigs | a |
Join the ends and you have the six rings.
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 x 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 x 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.
187
THE CANTERBURY PUZZLES
641
ct
62
29
22
1%
Al
6 |
7%
12
19
Z|
67
There are 640 different routes.
this kind is not practicable.
i fe
103.—The Tube Railway.
HED
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.
A general formula for puzzles of
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 follow-
ing table by “ directions” is meant the order of stations irrespective
of “routes.” Thus, the “direction,” BCD E 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” B D C E admits of
no variation ; therefore yields only one route.
2 two-line directions of 3 routes — 6
| three-line __,, ay eat ee ey
| = aE ah! Weaee ane
2 four-line - Se ea Bena 4
Z i : e1Ge.. RS 3
6 five-line - els wate ess 9:
Carried forward oy FOU
|
SOLUTIONS
Brought forward a S00
2 five-line directions of 18 routes — 36
2 six-line tt Ge ey
f2 seyven-me 90 aS ASL
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 tal! 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. There-
fore 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
189 :
THE CANTERBURY PUZZLES
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 might
it clearly overcomes the tendency to slip 2 ft. during the daytime, in
climbing up. Inrowing up ariver 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 over-
looked the obvious fact that if you give all your sides a common
190
SOLUTIONS
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 m?+n?, m*—n’, and 2 mn 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.
2
9
mn + mor th a
2 whee
mo n=
2mn + n? =
(eee > aie |
Now, if we form three triangles from the following pairs of
generators: a and b, a and c, aand 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 foundtwo..... 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 z? and 4a,
and give each side the denominator 2z(b? — h?), 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 ince ea OG 1320 1418
Second Prince . . 280 2442 2458
Whicd ince 5 2391 2960 2969
hourtn Emcee: 2... 1/1 6160 6161
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
19]
Eris CANTERBURY PUsZLES
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, a°b — b’a = 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 To divide 18 by ‘9 (or nine-tenths) we, of course,
‘9. 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, pro-
vided he plays to the centre on his first move. But a good variation
ie
SOLUTIONS
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 atonce. This
should always end in a draw, but to ensure it, the first player must
play to two adjoining corners (such as | 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
12 eat the grass on 10 acres in 8 weeks,
487. et ee Oneee
2, ee tay
64 aU Oe
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
eB O
THE CANTERBURY PUZZLES
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 clearly 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 lh. 5m. 27 3-11sec., we shall get all these eleven times, the
last being 2h. 54min. 32 8-I Isec. Another addition brings us back
to four oclock. 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 2h. 54min. 32 8-1 Isec.
exactly, or, put the other way, at 5min. 27 3-1 Isec. 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
194
SOLUTIONS
finds favour in the nursery. The general solution is that for n men
the number must be m(n”t!)-(n-—1), where m is any integer.
Each man will receive m(n—-1)”-1 biscuits at the final division,
though in the case of two men, when m=], 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.
THE END
R. CLAY AND SONS, LTD., BREAD ST. HILL, E.C., AND BUNGAY, SUFFOLK.
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