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Full text of "The Canterbury puzzles and other curious problems. 2d ed., with some fuller solutions and additional notes"

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THE CANTERBURY PUZZLES 



By the same Author 
'AMUSEMENTS IN MATHEMATICS" 
3s. 6d. 



First Ediiion, igoy 



I THE 

CANTERBURY PUZZLES 

AND OTHER CURIOUS PROBLEMS 



BY 

HENRY ERNEST DUDENEY 

AUTHOR OF 
"AMUSEMENTS IN MATHEMATICS," ETC. 



Second Edition 
(With Some Fuller Solutions and Additional Notes) 






THOMAS NELSON AND SONS, LTD. 

LONDON, EDINBURGH, AND NEW YORK 

1919 



CONTENTS 

Preface . . . 9 

Introduction 11 

The Canterbury Puzzles 23 

Puzzling Times at Solvamhall Castle ... 58 

The Merry Monks of Riddlewell ... 68 

The Strange Escape of the King's Jester . . 78 

The Squire's Christmas Puzzle Party ... 86 

Adventures of the Puzzle Club .... 94 

The Professor's Puzzles no 

Miscellaneous Puzzles 118 

Solutions 163 

Index 251 



PREFACE 

When preparing this new edition for the press, my first inclina- 
tion was to withdraw a few puzzles that appeared to be of in- 
ferior interest, and to substitute others for them. But, on second 
thoughts, I decided to let the book stand in its original form and 
add extended solutions and some short notes to certain problems 
that have in the past involved me in correspondence with inter- 
ested readers who desired additional information. 

I have also provided — what was clearly needed for reference — 
an index. The very nature and form of the book prevented any 
separation of the puzzles into classes, but a certain amount of 
classification will be found in the index. Thus, for example, if the 
reader has a predilection for problems with Moving Counters, or for 
Magic Squares, or for Combination and Group Puzzles, he will find 
that in the index these are brought together for his convenience. 

Though the problems are quite different, with the exception 
of just one or two little variations or extensions, from those in 
my book Amusements in Mathematics, each work being complete 
in itself, I have thought it would help the reader who happens 
to have both books before him if I made occasional references 
that would direct him to solutions and analyses in the later book 
calculated to elucidate matter in these pages. This course has 
also obviated the necessity of my repeating myself. For the sake 
of brevity. Amusements in Mathematics is throughout referred to 
as A . in M. 

HENRY E. DUDENEY. 

The Authors* Club, 
July 2, 1919. 



INTRODUCTION 

Readers of The Mill on the Floss will remember that when- 
ever Mr. Tulliver found himself confronted by any little dif&culty 
he was accustomed to make the trite remark, ** It's a puzzling 
world." There can be no denying the fact that we are surrounded 
on every hand by posers, some of which the intellect of man has 
mastered, and many of which may be said to be impossible of 
solution. Solomon himself, who may be supposed to have been 
as sharp as most men at solving a puzzle, had to admit " there 
be three things which are too wonderful for me ; yea, four which 
I know not : the way of an eagle in the air ; the way of a serpent 
upon a rock ; the way of a ship in the midst of the sea ; and the 
way of a man with a maid." 

Probing into the secrets of Nature is a passion with all men ; 
only we select different lines of research. Men have spent long 
lives in such attempts as to turn the baser metals into gold, to 
discover perpetual motion, to find a cure for certain malignant 
diseases, and to navigate the air. 

From morning to night we are being perpetually brought face 
to face with puzzles. But there are puzzles and puzzles. Those 
that are usually devised for recreation and pastime may be roughly 
divided into two classes : Puzzles that are built up on some inter- 
esting 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. 

11 



12 INTRODUCTION 

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 indi- 
vidual 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 puzzle?, while every game, sport, and pastime 
is built up of problems of greater or less difficulty. The spontane- 
ous question asked by the child of his parent, by one cyclist of 
another while taking a brief rest on a stile, by a cricketer during 
the luncheon hour, or by a yachtsman lazily scanning the horizon, 
is frequently a problem of considerable difficulty. In short, we 
are all propounding puzzles to one another every day of our lives — • 
without always knowing it. 

A good puzzle should demand the exercise of our best wit and 
ingenuity, and although a knowledge of mathematics and a certain 
familiarity with the methods of logic are often of great service 
in the solution of these things, yet it sometimes happens that a 
kind of natural cunning and sagacity is of considerable value. 
For many of the best problems cannot be solved by any familiar 
scholastic methods, but must be attacked on entirely original 
lines. This is why, after a long and wide experience, one finds 
that particular puzzles will sometimes be solved more readily by 
persons possessing only naturally alert faculties than by the better 
educated. The best players of such puzzle games as chess and 
draughts are not mathematicians, though it is just possible that 
often they may have undeveloped mathematical minds. 

It is extraordinary what fascination a good puzzle has for a 
great many people. We know the thing to be of trivial impor- 
tance, 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 suf- 
ficient reward for our trouble, even when there is no prize to be 
won. What is this mysterious charm that many find irresistible ? 



INTRODUCTION 13 

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 
doit? 

The answer is simply that it gave us pleasure to seek the solution 
— ^that the pleasure was all in the seeling 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 stimu- 
lates the smallest child, in play or education, to keep level with his 
fellows, and in later life it turns men into great discoverers, inven- 
tors, orators, heroes, artists, and (if they have more material aims) 
perhaps millionaires. 

In starting on a tour through the wide realm of Puzzledom we 
do well to remember that we shall meet with points of interest of 
a very varied character. I shall take advantage of this variety. 
People often make the mistake oF confining themselves to one 
little corner of the realm, and thereby miss opportunities of new 
pleasures that lie within their reach around them. One person 
will keep to acrostics and other word puzzles, another to mathe- 
matical brain-rackers, another to chess problems (which are merely 
puzzles on the chess-board, and have little practical relation to the 
game of chess), and so on. This is a mistake, because it restricts 
one's pleasures, and neglects that variety which is so good for the 
brain. 

And there is really a practical utility in puzzle-solving. Reg- 
ular exercise is supposed to be as necessary for the brain as for the 
body, and in both cases it is not so much what we do as the doing 
of it from which we derive benefit. The daily walk recommended 
by the doctor for the good of the body, or the daily exercise for 
the brain, may in itself appear to be so much waste of time ; but 
it is the truest economy in the end. Albert Smith, in one of his 
amusing novels, describes a woman who was convinced that she 
suffered from " cobwigs on the brain." This may be a very rare 



14 INTRODUCTION 

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 iielp 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 recom- 
mend to the encouragement of both the Universities, as it affords 
the easiest and shortest method of conveying some of the most 
useful principles of logic. It was the maxim of a very wise prince 
that * he who knows not how to dissemble knows not how to reign ' ; 
and I desire you to receive it as mine, that ' he who knows not how 
to riddle knows not how to live.' " 

How are good puzzles invented ? I am not referring to acrostics, 
anagrams, charades, and that sort of thing, but to puzzles that 
contain an original idea. Well, you cannot invent a good puzzle 
to order, any more than you can invent anything else in that manner. 
Notions for puzzles come at strange times and in strange ways. 
They are suggested by something we see or hear, and are led up 
to by other puzzles that come under our notice. It is useless to 
say, " I will sit down and invent an original puzzle," because there 
is no way of creating an idea ; you can only make use of it when 
it comes. You may think this is wrong, because an expert in 
these things will make scores of puzzles while another person, 
equally clever, cannot invent one " to save his life," as we say. 
The explanation is very simple. The expert knows an idea when 
he sees one, and is able by long experience to judge of its value. 
Fertility, like facility, comes by practice. 

Sometimes a new and most interesting idea is suggested by the 



INTRODUCTION 15 

blunder of somebody over another puzzle. A boy was given a 
puzzle to solve by a friend, but he misunderstood what he had to 
do, and set about attempting what most likely everybody would 
have told him was impossible. But he was a boy with a will, and 
he stuck at it for six months, off and on, until he actually succeeded. 
When his friend saw the solution, he said, " This is not the puzzle 
I intended — you misunderstood me — ^but you have found out 
something much greater ! " And the puzzle which that boy acci- 
dentally 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 
puz^esJiaVe been made out of the letters of the alphabet, and from 
those nine little digits and cipher, i, 2, 3, 4, 5, 6, 7, 8, 9, and o. 

It should always be remembered that a very simple person may 
propound a problem that can only be solved by clever heads — ^if 
at all. A child asked, " Can God do everything ? " On receiving 
an affirmative reply, she at once said : " Then can He make a 
stone so heavy that He can't lift it ? " Many wide-awake grown- 
up people do not at once see a satisfactory answer. Yet the diffi- 
culty lies merely in the absurd, though cunning, form of the ques- 
tion, which really amounts to asking, " Can the Almighty destroy 
His own omnipotence ? " It is somewhat similar to the other ques- 
tion, " 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 im- 
movable 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. 



i6 INTRODUCTION 

• 

" Because it is transparent ; " which is, of course, simply another 
way of saying, " Because we can see through it." 

Puzzles have such an infinite variety that it is sometimes very 
difficult to divide them into distinct classes. They often so merge 
in character that the best we can do is to sort them into a few 
broad types. Let us take three or four examples in illustration 
of what I mean. 

First there is the ancient Riddle, that draws upon the imagina- 
tion 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 ? " 
It was explained by CEdipus, who pointed out that man walked on 
his hands and feet in the morning of life, at the noon of life he 
walked erect, and in the evening of his days he supported his 
infirmities with a stick. When the Sphinx heard this explanation, 
she dashed her head against a rock and immediately expired. This 
shows that puzzle solvers may be really useful on occasion. 

Then there is the riddle propounded by Samson. It is perhaps 
the first prize competition in this line on record, the prize being 
thirty sheets and thirty changes of garments for a correct solution. 
The riddle was this : " Out of the eater came forth meat, and out 
of the strong came forth sweetness." The answer was, " A honey- 
comb in the body of a dead lion." To-day this sort of riddle sur- 
vives 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 

(2,077) 



INTRODUCTION 17 

the little peculiarities of the language in which they are written — 
such as anagrams, acrostics, word-squares, and charades. In this 
class we also find palindromes, or words and sentences that read 
backwards and forwards alike. These must be very ancient indeed, 
if it be true that Adam introduced himself to Eve (in the English 
language, be it noted) with the palindromic words, " Madam, I'm 
Adam,*' to which his consort replied with the modest palindrome 
" Eve." 

Then we have Arithmetical Puzzles, an immense class, full of 
diversity. These range from the puzzle that the algebraist finds to 
be nothing but a " simple equation," quite easy of direct solution, 
up to the profoundest problems in the elegant domain of the theory 
of numbers. 

Next we have the Geometrical Puzzle, a favourite and very 
ancient branch of which is the puzzle in dissection, requiring some 
plane figure to be cut into a certain number of pieces that will 
fit together and form another figure. Most of the wire puzzles sold 
in the streets and toy-shops are concerned with the geometry of 
position. 

But thes'e 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 under- 
stood 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 

(2.077) 2 



1 8 INTRODUCTION 

i,iii, III, III, III, 111,111, and then ask for a number (other than 
I 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,iii,iii,iii,iii, 
III, has only these two divisors, 2,071,723 and 5,363,222,357, 
and their discovery is an exceedingly heavy task. The only 
number composed only of ones that we know with certainty to 
have no divisor is 11. Such a number is, of course, called a prime 
number. 

The maxim that there are always a right way and a wrong way 
of doing anything applies in a very marked degree to the solving 
of puzzles. Here the wrong way consists in making aimless trials 
without method, hoping to hit on the answer by accident — a process 
that generally results in our getting hopelessly entangled in the trap 
that has been artfully laid for us. 

Occasionally, however, a problem is of such a character that, 
though it may be solved immediately by trial, it is very difficult 
to do by a process of pure reason. But in most cases the latter 
method is the only one that gives any real pleasure. 

When we sit down to solve a puzzle, the first thing to do is to 
make sure, as far as we can, that we understand the conditions. 
For if we do not understand what it is we have to do, we are not 
very likely to succeed in doing it. We all know the story of the 
man who was asked the question, '* If a herring and a half cost 
three-halfpence, how much witi 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 I " exclaimed the other apologetically ; " I was working 
it out in haddocks 1 " 

It sometimes requires more care than the reader might suppose 
so to word the conditions of a new puzzle that they are at once 
♦ See footnote on page 198. 



INTRODUCTION 



19 



clear and exact and not so prolix as to destroy all interest in the 
thing. I remember once propounding a problem that required 
something to be done in the " fewest possible straight lines/' and 
a person who was either very clever or very foolish (I have never 
quite determined which) claimed to have solved it in only one 
straight line, because, as she said, " I have taken care to make all 
the others crooked ! ** Who could have anticipated such a quibble ? 

Then if you give a ** crossing the river " puzzle, in which people 
have to be got over in a boat that will only hold a certain number 
or combination of persons, directly the would-be solver fails to 
master the difficulty he boldly introduces a rope to pull the boat 
across. You say that a rope is forbidden ; and he then falls back 
on the use of a current in the stream. I once thought I had care- 
fully 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 7 " 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. 



20 INTRODUCTION 

He then amended his definition by saying that the actual seeing 
all sides was not essential, but you went in such a way that, given 
sight, you could see all sides. Upon which it was suggested that 
consequently you could not walk around a man who had been shut 
up in a box ! And so on. The whole thing is amusingly stupid, 
and if at the start you, very properly, decline to admit any but 
a simple and correct definition of "to go around," there is no 
puzzle left, and you prevent an idle, and often heated, argument. 

When you have grasped your conditions, always see if you cannot 
simplify them, for a lot of confusion is got rid of in this way. Many 
people are puzzled over the old question of the man who, while 
pointing at a portrait, says, " Brothers and sisters have I none, but 
that man's father is my father's son." What relation did the man 
in the picture bear to the speaker ? Here you simplify by saying 
that " my father's son " must be either " myself " or " my brother." 
But, since the speaker has no brother, it is clearly " myself." The 
statement simplified is thus nothing more than, '* That man's father 
is myself," and it was obviously his son's portrait. Yet people fight 
over this question by the hour I 

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 re- 
mark, '* It is as hard as squaring a circle," though many people 
have a very hazy notion of what it means. If you have a circle of 
given diameter and wish to find the side of a square that shall con- 
tain exactly the same area, you are confronted with the problem 
of squaring the circle. Well, it cannot be done with exactitude 
(though we can get an answer near enough for all practical purposes), 
because it is not possible to say in exact numbers what is the ratio 
of the diameter to the circumference. But it is only in recent times 
that it has been proved to be impossible, for it is one thing not to 
be able to perform a certain feat, but quite another to prove that 
it cannot be done. Only uninstructed cranks now waste their time 
in trying to square the circle. 



INTRODUCTION 21 

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 
I 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, i 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, i 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, i, 2, 3, 4, etc. ; and 
it has been done, with the exception of the two diagonals, which so 
far have baffled all efforts. But it is not certain that it cannot 
be done. 

Though the contents of the present volume are in the main 
entirely original, some very few old friends will be found ; but these 
will not, I trust, prove unwelcome in the new dress that they have 



22 INTRODUCTION 

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




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 astro- 
labe, 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 difiiculty and interest are two qualities of puzzledom that do 
not necessarily go together. 



23 



24 THE CANTERBURY PUZZLES 

I. — The Reve's Puzzle. 

The Reve was a wily man and something of a scholar. As 
Chaucer tells us, " There was no auditor could of him win/' and 
" there could no man bring him in arrear." The poet also noticed 
that " ever he rode the hindermost of the route." This he did that 
he might the better, without interruption, work out the fanciful 
problems and ideas that passed through his active brain. When the 




pilgrims were stopping at a wayside tavern, a number of cheeses of 
varying sizes caught his alert eye ; and calling for four stools, he told 
the company that he would show them a puzzle of his own that 
would keep them amused during their rest. He then placed eight 
cheeses of graduating sizes on one of the end stools, the smallest 
cheese being at the top, as clearly shown in the illustration. " This 
is a riddle," quoth he, *' that I did once set before my fellow towns- 
men at Baldeswell, that is in Norfolk, and, by Saint Joce, there was 



THE CANTERBURY PUZZLES 



25 



no man among them that could rede it aright. And yet it is withal 
full easy, for all that I do desire is that, by the moving of one cheese 
at a time from one stool unto another, ye shall remove all the cheeses 
to the stool at the other end without ever putting any cheese on one 
that is smaller than itself. To him that will perform this feat in the 
least number of moves that be possible will I give a draught of 
the best that our good host can provide." To solve this puzzle in 
the fewest possible moves, first with 8, then with 10, and afterwards 
with 21 cheeses, is an interesting recreation. 

2. — The Pardoner's Puzzle, 

The gentle Pardoner, ** that straight was come from the court 
of Rome," begged to be excused ; but the company would not spare 
him. " Friends and fellow-pilgrims," said he, " of a truth the 
riddle that I have made is but a poor thing, but it is the best that 




I have been able to devise. Blame my lack of knowledge of such 
matters if it be not to your liking." But his invention was very 
well received. He produced the accompanying plan, and said that 
it represented sixty-four towns through which he had to pass 



26 THE CANTERBURY PUZZLES 

during some of his pilgrimages, and the lines connecting them were 
roads. He explained that the puzzle was to start from the large 
black town and visit all the other towns once, and once only, in 
fifteen straight pilgrimages. Try to trace the route in fifteen 
straight lines with your pencil. You may end where you like, but 
note that the omission of a little road at the bottom is intentional, 
as it seems that it was impossible to go that way. 

3. — The Miller's Puzzle. 

The Miller next took the company aside and showed them 
nine sacks of flour that were standing as depicted in the sketch. 
" Now, hearken, all and some,'* said he, " while that I do set ye 
the riddle of the nine sacks of flour. And mark ye, my lords and 
masters, that there be single sacks on the outside, pairs next unto 
them, and three together in the middle thereof. By Saint Benedict, 
it doth so happen that if we do but multiply the pair, 28, by the 
single one, 7, the answer is 196, which is of a truth the number 
shown by the sacks in the middle. Yet it be not true that the other 
pair, 34, when so multiplied by its neighbour, 5, will also make 196. 




^c», -^ 



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 multi- 
plied 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 every- 
body 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 



THE CANTERBURY PUZZLES 



27 



mortal battles had he been fifteen." His shield, as he is seen 
showing it to the company at the " Tabard " in the illustration, 
was, in the peculiar language of the heralds, " argent, semee 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. 



^.—The Wife of BatKs 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 



28 



THE CANTERBURY PUZZLES 



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 I ' 
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 CANTERBURY PUZZLES 



29 



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

7. — The Clerk of Oxenford's Puzzle, 

The silent and thoughtful Clerk of Oxenford, of whom it is re- 
corded 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, *' Oft times 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 



30 



THE CANTERBURY PUZZLES 



truly great. But the small riddle that I did make yester eve for 
the purpose of this company is not so hard that any may not find 
it out with a little patience." He then produced the square showi?- 
in the illustration and said that it was desired so to cut it into foui 
pieces (by cuts along the lines) that they would fit together again 
and form a perfect magic square, in which the four columns, the 
four rows, and the two long diagonals should add up 34. It will 
be found that this is a just sufficiently easy puzzle for most people's 
tastes. 

8. — The Tapiser's Puzzle. 

Then came forward the Tapiser, who was, of course, a maker of 
tapestry, and must not be confounded with a tapster, who draws 
and sells ale. 

He produced a beautiful piece of tapestry, worked in a simple 
chequered pattern, as shown in the diagram. " This piece of 





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tapestry, sirs," quoth he, *' hath one hundred and sixty-nine small 
squares, and I do desire you to tell me the manner of cutting the 
tapestry into three pieces that shall fit together and make one 
whole piece in shape of a perfect square. 

" Moreover, since there be divers ways of so doing, I do wish to 



THE CANTERBURY PUZZLES 31 

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 hght, than the outside. How then may 
I with ease satisfy the scholar as to the quantity of wood that hath 
been cut away ? " This at first sight looks a difficult question, but 
it is so absurdly simple that the method employed by the carpenter 
should be known to everybody to-day, for it is a very useful little 
" wrinkle." 

10. — The Puzzle of the Squire's Yeoman. 

Chaucer says of the Squire's Yeoman, who formed one of his 
party of pilgrims, ** A forester was he truly as I guess," and tells us 
that " His arrows drooped not with feathers low. And in his hand 
he bare a mighty bow." When a halt was made one day at a 



3^ 



THE CANTERBURY PUZZLES 



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 Hne with 
any other arrow." The diagram will show exactly how he did 
this, and no two arrows will be found in line, horizontally, vertically. 




or diagonally. Then the Yeoman said : " Here then is a riddle for 
ye. Remove three of the arrows each to one of its neighbouring 
squares, so that the nine shall yet be so placed that none thereof 
may be in line with another." By a " neighbouring square " is 
meant one that adjoins, either laterally or diagonally. 



II. — The Nun's Puzzle. 

" I trow there be not one among ye," quoth the Nun, on a later 
occasion, " that doth not know that many monks do oft pass the 
time in play at certain games, albeit they be not lawful for them. 
These games, such as cards and the game of chess, do they cun- 
ningly hide from the abbot's eye by putting them away in holes 



THE CANTERBURY PUZZLES 



33 



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 Httle riddle game that wo 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 



is. 



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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 shaU 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 it so ? Here then is a riddle in numbers that I will set 
before this merry company when next we do make a halt. There 
be thirty of us in all riding over the common this mom. Truly we 

(2,077) 3 



34 



THE CANTERBURY PUZZLES 



may ride one and one, in what they do call the single file, or two and 
two, or three and three, or five and five, or six and six, or ten and 
ten, or fifteen and fifteen, or all thirty in a row. In no other way 
may we ride so that there be no lack of equal numbers in the rows. 
Now, a party of pilgrims were able thus to ride in as many as sixty- 



"^ ^^^^iM" 




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 



THE CANTERBURY PUZZLES 



35 



forsooth call to my mind that this mom I bethought me of a riddle 
that I will now put forth." He then produced a sHp of vellum, on 
which was drawn the curious plan that is now given. " Here," 
saith he, " be nine dungeons, with a prisoner in every dungeon save 
one, which is empty. These prisoners be numbered in order, 7, 5, 
6, 8, 2, I, 4, 3, and I desire to know how they can, in as few moves 
as possible, put themselves in the order i, 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 



[^j=gg=^ i}=s 



^J=3>=n 



two men be in any dungeon at the same time. How may it be 
done ? " If the reader makes a rough plan on a sheet of paper 
and uses numbered counters, he will find it an interesting pastime 
to arrange the prisoners in the fewest possible moves. As there is 
never more than one vacant dungeon at a time to be moved into, 
the moves may be recorded in this simple way : 3 — 2 — i — 6, and 
so on. 

14. — The Weaver's Puzzle. 

When the Weaver brought out a square piece of beautiful cloth, 
daintily embroidered with lions and castles, as depicted in the 
illustration, the pilgrims disputed among themselves as to the 
meaning of these ornaments. The Knight, however, who was 
skilled in heraldry, explained that they were probably derived from 
the lions and castles borne in the arms of Ferdinand HI., 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 



36 



THE CANTERBURY PUZZLES 



of cloth may be cut into four several pieces, each of the same size 
and shape, and each piece bearing a Hon 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. 

i^.—The Cook's Puzzle. 
We find that there was a cook among the company ; and his 
services were no doubt at times in great request, " For he could 
roast and seethe, and broil and fry, And make a mortress and well 
bake a pie." One night when the pilgrims were seated at a country 
hostelry, about to begin their repast, the cook presented himself 
at the head of the table that was presided over by the Franklin, and 
said, " Listen awhile, my masters, while that I do ask ye a riddle, 
and by Saint Moden it is one that I cannot answer myself withal. 
There be eleven pilgrims seated at this board on which is set a 
warden pie and a venison pasty, each of which may truly be divided 
into four parts and no more. Now, mark ye, five out of the eleven 
pilgrims can eat the pie, but will not touch the pasty, while four 



THE CANTERBURY PUZZLES 



37 



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 hali- 
dame, 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.** 



38 



THE CANTERBURY PUZZLES 



1 6. — The Sompnou/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 indi- 
vidual was a somewhat quaint though worthy man. ** He was 




a gentle hireling and a kind ; A better fellow should a man not 
find." In order that the reader may understand his appearance 
in the picture, it must be explained that his peculiar headgear is 
duly recorded by the poet. *' A garland had he set upon his head, 
As great as if it were for an ale-stake." 

One evening ten of the company stopped at a village inn and 



THE CANTERBURY PUZZLES 39 

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 possessiqn. He therefore gave the Wife 
of Bath a number and directed her to count round and round the 
circle, in a clockwise direction, and the person on whom that number 
fell was immediately to step out of the ring. The count then began 
afresh at the next person. But the lady misunderstood her in- 
structions, and selected in mistake the number eleven and started 
the count at herself. As will be found, this resulted in all the 
women falling out in turn instead of the men, for every eleventh 
person withdrawn from the circle is a lady. 

" Of a truth it was no fault of mine," said the Sompnour next 
day to the company, " and herein is methinks a riddle. Can any 
tell me what number the good Wife should have used withal, and at 
which pilgrim she should have begun her count so that no other 
than the five men should have been counted out 7 " Of course, 
the point is to find the smallest number that will have the desired 
effect. 

ly.—The Monk's Puzzle. 

The Monk that went with the party was a great lover of sport. 
" Greyhounds he had as swift as fowl of flight : Of riding and of 
hunting for the hare Was all his love, for no cost would he spare." 
One day he addressed the pilgrims as follows : — 

" There is a little matter that hath at times perplexed me greatly, 
though certes it is of no great weight ; yet may it serve to try the 
wits of some that be cunning in such things. Nine kennels have I 
for the use of my dogs, and they be put in the form of a square ; 
though the one in the middle I do never use, it not being of a useful 
nature. Now the riddle is to find in how many different ways I 
may place my dogs in all or any of the outside kennels so that the 



40 



THE CANTERBURY PUZZLES 



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. 



i8. — The Shipman's Puzzle. 

Of this person we are told, *' He knew well all the havens, as 
they were, From Gothland to the Cape of Finisterre, And every 
creek in Brittany and Spain : His barque yclep^d was the Mag- 
dalen." The strange puzzle in navigation that he propounded 
was as follows. 

*' Here be a chart," quoth the Shipman, " of five islands, with 
the inhabitants of which I do trade. In each year my good ship 
doth sail over every one of the ten courses depicted thereon, but 
never may she pass along the same course twice in any year. Is 
there any among the company who can tell me in how many dif- 
ferent ways I may direct the Magdalen's ten yearly voyages, 
always setting out from the same island ? " 



THE CANTERBURY PUZZLES 



41 



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19. — r/j^ 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-att6- 
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 crowned A." It is with the brooch that we are 
concerned, for when asked to give a puzzle she showed this jewel 
to the company and said : "A learned man from Normandy did 
once give me this brooch as a charm, saying strange and mystic 
things anent it, how that it hath an affinity for the square, and such 
other wise words that were too subtle for me. But the good Abbot 
of Chert sey 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. 



42 



THE CANTERBURY PUZZLES 



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 be- 
cause 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 loved gold in special." The problem that the Doctor 
propounded to the assembled pilgrims was this. He produced two 
spherical phials, as shown in our illustration, and pointed out that 
one phial was exactly a foot in circumference, and the other two 
feet in circumference. 

" I do wish," said the Doctor, addressing the company, " to 
have the exact measures of two other phials, of a like shape but 
different in size, that may together contain just as much liquid as 
is contained by these two." To find exact dimensions in the 



THE CANTERBURY PUZZLES 
1 



43 




smallest possible numbers is one of the toughest nuts I have at- 
tempted. 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 

/ ! \ 

♦ • • . • .•• ^ 
■'■= ■;«.' i\ 

<j^'.'. -— 'i^ rig,...^.^,-.'Jj; 

and very good was he. Living in perfect peace and charity" — 
protested that riddles were not for simple minds like his, but he 



44 



THE CANTERBURY PUZZLES 



would show the good pilgrims, if they willed it, one that he had 
frequently heard certain clever folk in his own neighbourhood dis- 
cuss. " The lord of the , manor in the part of Sussex whence I 
come hath a plantation of sixteen fair oak trees, and they be so 
set out that they make twelve rows with four trees in every row. 
Once on a time a man of deep learning, who happened to be travel- 
ling in those parts, did say that the sixteen trees might have been 
so planted that they would make so many as fifteen straight rows, 
with four trees in every row thereof. Can ye show me how this 
might be ? Many have doubted that 'twere possible to be done." 
The illustration shows one of many ways of forming the twelve 
rows. How can we make fifteen ? 

22. — The Franklin's Puzzle. 

" A Franklin was in this company ; White was his beard as is 
the daisy.'* We are told by Chaucer that he was a great 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 CANTERBURY PUZZLES 45 

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 i, 2, 3, up to 15, with the 
last one marked o. " 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- 
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 riddle." This is a 
little puzzle that may be conveniently tried with sixteen numbered 
counters. 

23. — The Squire* s Puzzle. 

The young Squire, twenty years of age, was the son of the 
Knight that accompanied him on the historic pilgrimage. He 
was undoubtedly what in later times we should call a dandy, for, 
" 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 illustration to 
No. 26, while the Haberdasher was propounding his problem of 
the triangle, this young Squire was standing in the background 
making a drawing of some kind ; for " He could songs make and 
well indite, Joust and eke dance, and well portray and write." 

The Knight turned to him after a while and said, *' My son, 
what is it over which thou dost take so great pains withal ? " and 
the Squire answered, *' I have bethought me how I might portray 
in one only stroke a picture of our late sovereign lord King Edward 
the Third, who hath been dead these ten years. 'Tis a riddle to 



46 THE CANTERBURY PUZZLES J 

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 twin- 
kling eyes. *' Courteous he was and lowly of service. There was 
a man nowhere so virtuous." Yet he was " the best beggar in all 
his house," and gave reasons why '* Therefore, instead of weeping 
and much prayer. Men must give silver to the needy friar." He 
went by the name of Hubert. One day he produced four money 
bags and spoke as follows : " If the needy friar doth receive in alms 
five hundred silver pennies, prithee tell in how many different 



THE CANTERBURY PUZZLES 



47 




ways they may be placed in the four bags.'* The good man ex- 
plained that order made no difference (so that the distribution 50, 
100, 150, 200 would be the same as 100, 50, 200, 150, or 200, 50, 
100, 150), and one, two, or three bags may at any time be empty. 



25. — The Parson's Puzzle. 

The Parson was a really devout and good man. " A better 
priest I trow there nowhere is." His virtues and charity made 
him beloved by all his flock, to whom he presented his teaching 
with patience and simplicity; "but first he followed it himself." 
Now, Chaucer is careful to tell us that " Wide was his parish, and 



48 



THE CANTERBURY PUZZLES 



houses far asunder. But he neglected nought for rain or thunder ; " 
and it is with his parochial visitations that the Parson's puzzle 
actually dealt. He produced a plan of part of his parish, through 




which a small river ran that joined the sea some hundreds of mil 
to the south. I give a facsimile of the plan. 

" Here, my worthy Pilgrims, is a strange riddle,** quoth th 
Parson. " Behold how at the branching of the river is an island 
Upon this island doth stand my own poor parsonage, and ye may 
all see the whereabouts of the village church. Mark ye, also, that 
there be eight bridges and no more over the river in my parish. 
On my way to church it is my wont to visit sundry of my flock, and 
in the doing thereof I do pass over every one of the eight bridges 
once and no more. Can any of ye find the path, after this manner, 
from the house to the church, without going out of the parish ? 
Nay, nay, my friends, I do never cross the river in any boat, neither 
by swimming nor wading, nor do I go underground like unto the 
mole, nor fly in the air as doth the eagle ; but only pass over by the 



i 



THE CANTERBURY PUZZLES 



49 



bridges." There is a way in which the Parson might have made 
this curious journey. Can the reader discover it ? At first it 
seems impossible, but the conditions offer a loophole. 

26. — The Haberdasher's Puzzle. 

Many attempts were made to induce the Haberdasher, who 
was of the party, to propound a puzzle of some kind, but for a 
long time without success. At last, at one of the Pilgrims' stop- 
ping-places, he said that he would show them something that 



r/C^/ 




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 

(2,077) 4 



50 THE CANTERBURY PUZZLES 

that he set them. He produced a piece of cloth in the shape of a 
perfect equilateral triangle, as shown in the illustration, and said, 
" Be there any among ye full wise in the true cutting of cloth ? I 
trow not. Every man to his trade, and the scholar may learn 
from the varlet and the wise man from the fool. Show me, then, 
if ye can, in what manner this piece of cloth may be cut into four 
several pieces that may be put together to make a perfect square." 
Now some of the more learned of the company found a way of 
doing it in five pieces, but not in four. But when they pressed 
the Haberdasher for the correct answer he was forced to admit, 
after much beating about the bush, that he knew no way of doing 
it in any number of pieces. '* By Saint Francis," saith he, " any 
knave can make a riddle methinks, but it is for them that may 
to rede it aright." For this he narrowly escaped a sound beating. 
But the curious point of the puzzle is that I have found that the 
feat may really be performed in so few as four pieces, and with- 
out 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 Haber- 
dasher ; but the necessary idea would not come, rack his brains as 
he would. All things, however, come to those who wait — and 
persevere — and one morning he announced, in a state of consider- 
able excitement, that he had a poser to set before them. He brought 
out a square piece of silk on which were embroidered a number of 
fleurs-de-lys in rows, as shown in our illustration. 

" Lordings," said the Dyer, " hearken anon unto my riddle. 
Since I was awakened at dawn by the crowing of cocks— for which 



THE CANTERBURY PUZZLES 



51 



din may our host never thrive — I have sought an answer thereto, 
but by St. Bernard I have found it not. There be sixty-and-four 
flowers-de-luce, and the riddle is to show how I may remove six 
of these so that there may yet be an even number of the flowers 
in every row and every column." 

The Dyer was abashed when every one of the company showed 



^^"^'^'^"k'^'k 


<^^44.«g.44^ 


<*^'jfe'<^*S?'^^^ 


^^^^'^#<ife«& 


^'^^«^i^'^^ 


'k^'h^'k^^'k 


4it4#<^4^ 


'k <k%'^'k'k'k'k 



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



28. — The Great Dispute between the Friar and the Sompnour. 

Chaucer records the painful fact that the harmony of the pil- 
grimage 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 



sa 



THE CANTERBURY PUZZLES 



temporary peace. Unfortunately trouble broke out again over a 
very curious dispute in this way. 




At one point of the journey the road lay along two sides of a 
square field, and some of the pilgrims persisted, in spite of trespass, 
in cutting across from comer to comer, 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 I " On my faith, 
then," exclaimed the Sompnour, " thou art a very fool ! " " Nay," 
repHed the Friar, " if the company will but Hsten 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 



THE CANTERBURY PUZZLES 



53 



side of the field to be loo 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. i, it is evident 
that we go the same distance, for every one of the eight straight 
portions of this path measures exactly 25 yards. Similarly in Fig. 
2, the zigzag contains ten straight portions, each 20 yards long : 
that path is also the same length — 200 yards. No matter how many 
steps we make in our zigzag path, the result is most certainly 




always the same. Thus, in Fig. 3 the steps are very small, yet the 
distance must be 200 yards ; as is also the case in Fig. 4, and would 
yet be if we needed a microscope to detect the steps. In this way, 
the Friar argued, we may go on straightening out that zigzag path 
until we ultimately reach a perfectly straight line, and it therefore 
follows that the diagonal of a square is of exactly the same length 
as two of the sides. 

Now, in the face of it, this must be wrong ; and it is in fact 
absurdly so, as we can at once prove by actual measurement if we 



54 



THE CANTERBURY PUZZLES 



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 pil- 
grims had not interposed the two would have undoubtedly come 
to blows. The reader will perhaps at once see the flaw in the 
Friar's argument. 

29. — Chaucer's Puzzle. 

Chaucer himself accompanied the pilgrims. Being a mathema- 
tician and a man of a thoughtful habit, the Host made fun of him, 
he tells us, saying, " Thou lookest as thou wouldst find a hare. 
For ever on the ground I see thee stare." The poet replied to the 
request for a tale by launching into a long-spun-out and ridiculous 








'i0ff^'if''§j 



."j^fi 



poem, intended to ridicule the popular romances of the day, after 
twenty-two stanzas of which the company refused to hear any 
more, and induced him to start another tale in prose. It is an 
interesting fact that in the " Parson's Prologue " Chaucer actually 



THE CANTERBURY PUZZLES 55 

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 mo- 
ment the moon's altitude (she being in mid-Libra) was steadily 
increasing as we entered at the west end of the village." A cor- 
respondent 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 Une of the Tales 
tells us that the pilgrimage was in April — they are supposed to 
have set out on 17th April 1387, as stated in No. 23. 

Though Chaucer made this little puzzle and recorded it for 
the interest of his readers, he did not venture to propound it to 
his fellow-pilgrims. The puzzle that he gave them was of a simpler 
kind altogether : it may be called a geographical one. " When, 
in the year 1372, I did go into Italy as the envoy of our sovereign 
lord King Edward the Third, and while there did visit Francesco 
Petrarch, that learned poet did take me to the top of a certain 
mountain in his country. Of a truth, as he did show me, a mug 
will hold less liquor at the top of this mountain than in the valley 
beneath. Prythee tell me what mountain this may be that has 
so strange a property withal." A very elementary knowledge of 
geography will suffice for arriving at the correct answer. 



30. — The Puzzle of the Canon's Yeoman. 

This person joined the party on the road. " * God save,' 
quoth he, * this jolly company ! Fast have I ridden,' saith he, 
* for your sake. Because I would I might you overtake. To ride 
among this merry company.* " Of course, he was asked to enter- 
tain the pilgrims with a puzzle, and the one he propounded was 
the following. He showed them the diamond-shaped arrangement 



56 THE CANTERBURY PUZZLES 

of letters presented in the accompanying illustration, and said, 
" I do call it the rat-catcher's riddle. In how many different 
ways canst thou read the words, ' Was it a rat I saw ? ' " You 




may go in any direction backwards and forwards, upwards or 
downwards, only the successive letters in any reading must always 
adjoin one another. 

31. — The Manciple's Puzzle. 

The Manciple was an officer who had the care of buying victuals 
for an Inn of Court — hke the Temple. The particular individual 
who accompanied the party was a wily man who had more than 
thirty masters, and made fools of them all. Yet he was a man 
" whom purchasers might take as an example How to be wise in 
buying of their victual." 

It happened that at a certain stage of the journey the Miller and 
the Weaver sat down to a hght repast. The Miller produced five 
loaves and the Weaver three. The Manciple coming upon the 
scene asked permission to eat with them, to which they agreed. 
When the Manciple had fed he laid down eight pieces of money 
and said with a sly smile, " Settle betwixt yourselves how the 
money shall be fairly divided. Tis a riddle for thy wits." 



THE CANTERBURY PUZZLES 



57 



A discussion followed, and many of the pilgrims joined in it. 
The Reve and the Sompnour held that the Miller should receive 
five pieces and the Weaver three, the simple Ploughman was 
ridiculed for suggesting that the Miller should receive seven and 
the Weaver only one, while the Carpenter, the Monk, and the Cook 




insisted that the money should be divided equally between the two 
men. Various other opinions were urged with considerable vigour, 
jantil it was finally decided that the Manciple, as an expert in such 
matters, should himself settle the point. His decision was quite 
:orrect. What was it ? Of course, all three are supposed to have 
jaten equal shares of the bread. 







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

32. — The Game of Bandy -Ball. 

Bandy-ball, cambuc, or goff (the game so well known to-day by 
the name of golf), is of great antiquity, and was a special favourite 

58 



PUZZLING TIMES AT SOLVAMHALL CASTLE 59 

at Solvamhall Castle. Sir Hugh de Fortibus was himself a master 
of the game, and he once proposed this question. 

They had nine holes, 300, 250, 200, 325, 275, 350, 225, 375, 
and 400 yards apart. If a man could always strike the ball in a 
perfectly straight line and send it exactly one of two distances, so 
that it would either go towards the hole, pass over it, or drop into 
it, what would the two distances be that would carry him in the 
least number of strokes round the whole course ? 

" Beshrew me," Sir Hugh would say, " if I know any who could 
do it in this perfect way ; albeit, the point is a pretty one." 

Two very good distances are 125 and 75, which carry you round 
in 28 strokes, but this is not the correct answer. Can the reader 
get round in fewer strokes with two other distances ? 

33. — Tilting at the Ring. 

Another favourite sport at the castle was tilting at the ring. A 
horizontal bar was fixed in a post, and at the end of a hanging 
supporter was placed a circular ring, as shown in the above illustrated 
title. By raising or lowering the bar .the ring could be adjusted to 
the proper height — generally about the level of the left eyebrow of 
the horseman. The object was to ride swiftly some eighty paces 
and run the lance through the ring, which was easily detached, 
and remained on the lance as the property of the skilful winner. 
It was a very difficult feat, and men were not unnaturally proud 
of the rings they had succeeded in capturing. ^ 

At one tournament at the castle Henry de Gournay beat Stephen 
Malet by six rings. Each had his rings made into a chain — De 
Gournay's chain being exactly sixteen inches in length, and Malet's 
six inches. Now, as the rings were all of the same size and made 
of metal half an inch thick, the little puzzle proposed by Sir Hugh 
was to discover just how many rings each man had won. 

34. — The Noble Demoiselle. 

Seated one night in the hall of the castle, Sir Hugh desired the 
company to fill their cups and listen while he told the tale of his 



6o THE CANTERBURY PUZZLES 

adventure as a youth in rescuing from captivity a noble demoiselL 
who was languishing in the dungeon of the castle belonging to hi 
father's greatest enemy. The story was a thrilling one, and whei 
he related the final escape from all the dangers and horrors of th 
great Death's-head Dungeon with the fair but unconscious maidei 
in his arms, all exclaimed, " 'Twas marvellous valiant I " But Si 
Hugh said, ** I would never have turned from my purpose, not evei 
to save my body from the bernicles/' 






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

35. — The Archery Butt. 

The butt or target used in archery at Solvamhall was not markec 
out in concentric rings as at the present day, but was prepared ir 



PUZZLING TIMES AT SOLVAMHALL CASTLE 6i 

anciful designs. In the illustration is shown a numbered target 
)repared by Sir Hugh himself. It is something of a curiosity, 
)ecause it will be found that he has so cleverly arranged the num- 
)ers that every one of the twelve lines of three adds up to exactly 
wenty-two. 

One day, when the archers were a little tired of their sport, 
)ir Hugh de Fortibus said, " What ho, merry archers ! Of a truth 
t is said that a fool's bolt is soon shot, but, by my faith, I know 








[ot any man among you who shall do that which I will now put 
orth. Let these numbers that are upon the butt be set down 
fresh, so that the twelve lines thereof shall make twenty and 
hree instead of twenty and two." 

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



62 



THE CANTERBURY PUZZLES 



36. — The Donjon Keep Window. 

On one occasion Sir Hugh greatly perplexed his chiel builder. 
He took this worthy man to the walls of the donjon keep and 
pointed to a window there. 

** Methinks," said he, '* yon window is square, and measures, 
on the inside, one foot every way, and is divided by the narrow 
bars into four lights, measuring half a foot on every side." 

*• Of a truth that is so, Sir Hugh." 

" Then I desire that another window be made higher up whose 



Jl 








I> 


if- 






= 


flat;. ..... 









four sides shall also be each one foot, but it shall be divided by bars 
into eight lights, whose sides shall be all equal." 

" Truly, Sir Hugh," said the bewildered chief builder, " I know 
not how it may be done." 

" By my halidame ! " exclaimed De Fortibus in pretended rage, 
" let it be done forthwith. I trow thou art but a sorry craftsman 
if thou canst not, forsooth, set such a window in a keep wall." 

It will be noticed that Sir Hugh ignores the thickness of the bars, 



PUZZLING TIMES AT SOLVAMHALL CASTLE 63 

37. — The Crescent atid the Cross. 

When Sir Hugh's kinsman, Sir John de Collingham, came back 
from the Holy Land, he brought with him a flag bearing the sign 
of a crescent, as shown in the illustration. It was noticed that 
De Fortibus spent much time in examining this crescent and 
comparing it with the cross borne by the Crusaders on their own 
banners. One day, in the presence of a goodly company, he made 
the following striking announcement : — 

** I have thought much of late, friends and masters, of the 
conversion of the crescent to the cross, and this has led me to the 







^Jl^mr^':^ 



7:^- 



^4^0/'M'J^BM0m> 



'"^^m^^ 



finding of matters at which I marvel greatly, for that which I shall 
now make known is mystical and deep. Truly it was shown to me 
in a dream that this crescent of the enemy may be exactly converted 
into the cross of our own banner. Herein is a sign that bodes good 
for our wars in the Holy Land.'* 

Sir Hugh de Fortibus then explained that the crescent in one 
banner might be cut into pieces that would exactly form the perfect 
cross in the other. ;^ It is certainly rather curious ; and I show 
how the conversion from crescent to cross may be made in ten 



64 THE CANTERBURY PUZZLES 

pieces, using every part of the crescent. The flag was alike or 
both sides, so pieces may be turned over where required. 

38. — The Amulet. 

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

A 
B B 
R R R, 
A A A A^ 
C C C C C 
A A A A A A 
D D D D D D D^ 
AAAAAAAA^ 
BBBBBBBBB 
RRRRRRRRRR. 
AAAAAAAAAAA 

To-day we know that Abracadabra was the supreme deity oi 
the Assyrians, and this curious arrangement of the letters of the 
word was commonly worn in Europe as an amulet or charm against 
diseases. But Sir Hugh had never heard of it, and, regarding the 
document rather seriously, he sent for a learned priest. 

" I pray you, Sir Clerk," said he, ** show me the true intent ol 
this strange writing." 

" Sir Hugh," repUed the holy man, after he had spoken in a 
foreign tongue with the stranger, "it is but an amulet that this 
poor wight doth wear upon his breast to ward off the ague, the 
toothache, and such other afflictions of the body." 

" Then give the varlet food and raiment and set him on his 
way," said Sir Hugh. " Meanwhile, Sir Clerk, canst thou tell me in 



PUZZLING TIMES AT SOLVAMHALL CASTLE 65 

how many ways this word * Abracadabra ' may be read on the 
amulet, always starting from the A at the top thereof ? " 

Place your pencil on the A at the top and count in how many 
different ways you can trace out the word downwards, always 
passing from a letter to an adjoining one. 

39. — The Snail on the Flagstaff. 

It would often be interesting if we could trace back to their 
origin many of the best known puzzles. Some of them would be 
found to have been first propounded in very ancient times, and 
there can be very little doubt that while a certain number may 
have improved with age, others will have deteriorated and even 




lost their original point and bearing. It is curious to find in the 

Solvamhall records our familiar friend the climbing snail puzzle, 

and it will be seen that in its modem form it has lost its original 

subtlety. 

On the occasion of some great rejoicings at the Castle, Sir Hugh 
(2,077) 5 



66 



THE CANTERBURY PUZZLES 



was superintending the flying of flags and banners, when somebody 
pointed out that a wandering snail was climbing up the flagstaff. 
One wise old fellow said : — 

*' They do say, Sir Knight, albeit I hold such stories as mere 
fables, that the snail doth climb upwards three feet in the daytime, 
but shppeth back two feet by night." 




** Then," replied Sir Hugh, " tell us how many days it will take 
this snail to get from the bottom to the top of the pole." 

** By bread and water, I much marvel if the same can be done 
unless we take down and measure the staff." 

" Credit me," repUed the knight, " there is no need to measure 
the staff." 

Can the reader give the answer to this version of a puzzle that 
wc all know so well ? 



PUZZLING TIMES AT SOLVAMHALL CASTLE 67 

40. — Lady Isabel's Casket. 

Sir Hugh's young kinswoman and ward. Lady Isabel de Fitz- 
amulph, 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 J-inch ; and the rest of the 
surface was exactly inlaid with pieces of wood, each piece being a 
perfect square, and no two pieces of the same size. Many young 
men failed, but one at length succeeded. The puzzle is not an easy 
one, but the dimensions of that strip of gold, combined with those 
other conditions, absolutely determine the size of the top of the 
casket. 







Taeir>Quai7it Puzzles a^d Er^KiMA.s. 



" 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 hps, but alas ! they were silenced for ever. Thus passed 
away the hfe of the jovial and greatly beloved Abbot of the old 
monastery of Riddle well. 

The monks of Riddlewell Abbey were noted in their day for 
the quaint enigmas and puzzles that they were in the habit of 
propounding. The Abbey was built in the fourteenth century, 
near a sacred spring known as the Red-hill Well. This became 
in the vernacular Reddlewell and Riddlewell, and under the Lord 
Abbot David the monks evidently tried to justify the latter form 
by the riddles they propounded so well. The solving of puzzles 
became the favourite recreation, no matter whether they happened 
to be of a metaphysical, philosophical, mathematical, or mechanical 
kind. It grew into an absorbing passion with them, and as I have 
shown above, in the case of the Abbot this passion was strong 
even in death. 

It would seem that the words " puzzle," *' problem," *' enigma," 
etc., did not occur in their vocabulary. They were accustomed to 
call every poser a " riddle," no matter whether it took the form of 
** Where was Moses when the light went out ? " or the Squaring of 
the Circle. On one of the walls in the refectory were inscribed 



THE MERRY MONKS OF RIDDLEWELL 69 

the words of Samson, " I will now put forth a riddle to you," to 
remind the brethren of what was expected of them, and the rule 
was that each monk in turn should propose some riddle weekly to the 
community, the others being always free to cap it with another if 
disposed to do so. Abbot David was, undoubtedly, the puzzle 
genius of the monastery, and everybody naturally bowed to his 
decision. Only a few of the Abbey riddles have been preserved, 
and I propose to select those that seem most interesting. I shall 
try to make the conditions of the puzzles perfectly clear, so that 
the modern reader may fully understand them, and be amused 
in tr3dng to find some of the solutions. 

41. — The Riddle of the Fish-pond. 

At the bottom of the Abbey meads was a small fish-pond where 
the monks used to spend many a contemplative hour with rod and 




line. One day, when they had had very bad luck and only caught 
twelve fishes amongst them. Brother Jonathan suddenly declared 



70 THE CANTERBURY PUZZLES 

that as there was no sport that day he would put forth a riddle 
for their entertainment. He thereupon took twelve fish baskets 
and placed them at equal distances round the pond, as shown in 
our illustration, with one fish in each basket. 

" Now, gentle anglers," said he, " rede me this riddle of the 
Twelve Fishes. Start at any basket you like, and, always going in 
one direction round the pond, take up one fish, pass it over two 
other fishes, and place it in the next basket. Go on again ; take up 
another single fish, and, having passed that also over two fishes, 
place it in a basket ; and so continue your journey. Six fishes only 
are to be removed, and when these have been placed, there should 
be two fishes in '^ach of six baskets, and six baskets empty. Which 
of you merry wights will do this in such a manner that you shall go 
round the pond as few times as possible ? " 

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



42. — The Riddle of the Pilgrims. 

One day, when the monks were seated at their repast, the Abbot 
announced that a messenger had that morning brought news that 
a number of pilgrims were on the road and would require their 
hospitality. 

" You will put them," he said, '* in the square dormitory that 
has two floors with eight rooms on each floor. There must be 
eleven persons sleeping on each side of the building, and twice as 
many on the upper floor as on the lower floor. Of course every 
room must be occupied, and you know my rule that not more 
than three persons may occupy the same room." 

I give a plan of the two floors, from which it will be seen that 
the sixteen rooms are approached by a well staircase in the centre. 
After the monks had solved this little problem and arranged for 



THE MERRY MONKS OF RIDDLEWELL 71 

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 

Plan of Dormitory. 





Eight Rooms on Upper Floor. 



Eight Rooms on Lower Floor. 



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 floored with 
sixteen large ornamental tiles. When these became cracked and 
burnt with the heat of the great fire, it was decided to put down 
new tiles, which had to be selected from four different patterns 
(the Cross, the Fleur-de-lys, the Lion, and the Star) ; but plain tiles 
were also available. The Abbot proposed that they should be 
laid as shown in our sketch, without any plain tiles at all ; but 
Brother Richard broke in, — 

" I trow, my Lord Abbot, that a riddle is required of me this 
day. Listen, then, to that which I shall put forth. Let these 



72 



THE CANTERBURY PUZZLES 



sixteen tiles be so placed that no tile shall be in line with another of 
the same design "—(he meant, of course, not in line horizontally, 
vertically, or diagonally)— " and in such manner that as few plain 




tiles as possible be required.'* When the monks handed in their 
plans it was found that only Friar Andrew had hit upon the correct 
answer, even Friar Richard himself being wrong. All had used 
too many plain tiles. 



44. — The Riddle of the Sack Wine. 

One evening, when seated at table. Brother Benjamin was called 
upon by the Abbot to give the riddle that was that day demanded 
of him. 

" Forsooth," said he, " I am no good at the making of riddles, 
as thou knowest full well ; but I have been teasing my poor brain 
over a matter that I trust some among you will expound to me, 
for I cannot rede it myself. It is this. Mark me take a glass of 
sack from this bottle that contains a pint of wine and pour it into 
that jug which contains a pint of water. Now, I fill the glass with 
the mixture from the jug and pour it back into the bottle holding 



THE MERRY MONKS OF RIDDLEWELL 73 

the sack. Pray tell me, have I taken more wine from the bottle 
than water from the jug ? Or have I taken more water from the 
jug than wine from the bottle ? " 

I gather that the monks got nearer to a great quarrel over this 
little poser than had ever happened before. One brother so far 
forgot himself as to tell his neighbour that " more wine had got into 
his pate than wit came out of it," while another noisily insisted that 
it all depended on the shape of the glass and the age of the wine. 
But the Lord Abbot intervened, showed them what a simple 
question it really was, and restored good feeling all round. 

45. — The Riddle of the Cellarer, 

Then Abbot David looked grave, and said that this incident 
brought to his mind the painful fact that John the Cellarer had 




been caught robbing the cask of best Malvoisie that was reserved 
for special occasions. He ordered him to be brought in. 

*' Now, varlet/' said the Abbot, as the ruddy-faced Cellarer 



74 THE CANTERBURY PUZZLES 

came before him, ** thou knowest that thou wast taken this morn- 
ing in the act of stealing good wine that was forbidden thee. What 
hast thou to say for thyself ? " 

" Prithee, my Lord Abbot, forgive me ! " he cried, falling on 
his knees. " Of a truth, the Evil One did come and tempt me, 
and the cask was so handy, and the wine was so good withal, and 
— and I had drunk of it ofttimes without being found out, and — " 

" Rascal ! that but maketh thy fault the worse ! How much 
wine hast thou taken ? " 

" Alack-a-day 1 There were a hundred pints in the cask at the 
start, and I have taken me a pint every day this month of June — 
it being to-day the thirtieth thereof — and if my Lord Abbot can 
tell me to a nicety how much good wine I have taken in all, let 
him punish me as he will." 

*' Why, knave, that is thirty pints." 

" Nay, nay ; for each time I drew a pint out of the cask, I put 
in a pint of water in its stead ! " 

It is a curious fact that this is the only riddle in the old record 
that is not accompanied by its solution. Is it possible that it proved 
too hard a nut for the monks ? There is merely the note, " John 
suffered no punishment for his sad fault." 



46. — The Riddle of the Crusaders. 

On another occasion a certain knight, Sir Ralph de Bohun, was 
a guest of the monks at Riddlewell Abbey. Towards the close of 
a sumptuous repast he spoke as follows : — 

" My Lord Abbot, knowing full well that riddles are greatly to 
thy liking. I will, by your leave, put forth one that was told unto 
me in foreign lands. A body of Crusaders went forth to fight the 
good cause, and such was their number that they were able to. 
form themselves into a square. But on the way a stranger took 
up arms and joined them, and they were then able to form exactly 
thirteen smaller squares. Pray tell me, merry monks, how many 
men went forth to battle ? " 



THE MERRY MONKS OF RIDDLEWELL 75 

Abbot David pushed aside his plate of warden pie, and made 
a few hasty calculations. 

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




Crusaders each. But which of you can tell me how many men 
there would have been if, instead of 13, they had been able to form 
113 squares under exactly the like conditions ? " 

The monks gave up this riddle, but the Abbot showed them the 
answer next morning. 



47. — The Riddle of St. Edmondsbury. 

" It used to be told at St. Edmondsbury," said Father Peter on 
one occasion, " that many years ago they were so overrun with 
mice that the good abbot gave orders that all the cats from the 
country round should be obtained to exterminate the vermin. A 
record was kept, and at the end of the year it was found that every 
cat had killed an equal number of mice, and the total was exactly 
i,iii,iii mice. How many cats do you suppose there were ? " 



76 



THE CANTERBURY PUZZLES 



" Methinks one cat killed the lot," said Brother Benjamin. 

** Out upon thee, brother ! I said ' cats.' " 

** Well, then," persisted Benjamin, " perchance i,iii,iii cats 
each killed one mouse." 

** No," rephed Father Peter, after the monks' jovial laughter 
had ended, " I said ' mice ; ' and all I need add is this — ^that each 
cat killed more mice than there were cats. They told me it was 
merely a question of the division of numbers, but I know not the 
answer to the riddle." 

The correct answer is recorded, but it is not shown how they 
arrived at it. 



t#-^r f^ 




48,— The Riddle of the Frogs' Ring. 

One Christmas the Abbot offered a prize of a large black jack 

mounted in sUver, 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 



THE MERRY MONKS OF RIDDLEWELL 77 

after gave out anything that did not excite the ridicule ol his 
brethren. It was called the ** Frogs' Ring." 

A ring was made with chalk on the floor of the nail, and divided 
into thirteen compartments, in which twelve discs of wood (called 
*' frogs ") were placed in the order shown in our illustration, one 
place being left vacant. The numbers i to 6 were painted white 
and the numbers 7 to 12 black. The puzzle was to get all the 
white numbers where the black ones were, and vice versa. The 
white frogs move round in one direction, and the black ones the 
opposite way. They may move in any order one step at a time, or 
jumping over one of the opposite colour to the place beyond, just as 
we play draughts to-day. The only other condition is that when 
all the frogs have changed sides, the i must be where the 12 now is 
and the 12 in the place now occupied by i. The puzzle was to 
perform the feat in as few moves as possible. How many moves 
are necessary ? 

I will conclude in the words of the old writer : " These be some of 
the riddles which the monks of Riddlewell did set forth and expound 
each to the others in the merry days of the good Abbot David." 



THE STRANGE ESCAPE OF THE 
KING'S JESTER. 

A PUZZLING ADVENTURE. 

At one time I was greatly in favour with the king, and his 
Majesty never seemed to weary of the companionship of the court 
fool. I had a gift for making riddles and quaint puzzles which 
ofttimes caused great sport ; for albeit the king never found the right 
answer of one of these things in all his hfe, yet would he make 
merry at the bewilderment of those about him. 

But let every cobbler stick unto his last ; for when I did set out 
to learn the art of performing strange tricks in the magic, wherein 
the hand doth ever deceive the eye, the king was affrighted, and 
did accuse me of being a wizard, even commanding that I should 
be put to death. Luckily my wit did save my life. I begged that 
I might be slain by the royal hand and not by that of the execu- 
tioner. 

" By the saints," said his Majesty, " what difference can it 
noake 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 hfe 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 Uke way save me. I was cast into the 

78 



STRANGE ESCAPE OF THE KING'S JESTER 79 

dungeon to await my death. How, by the help of my gift in 
answering riddles and puzzles, I did escape from captivity I will 
now set forth ; and in case it doth perplex any to know how some 
of the strange feats were performed, I will hereafter make the 
manner thereof plain to all. 



49. — The Mysterious Rope. 

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

50. — The Underground Maze. 

The only way out of the yard that I now was in was to descend 
a few stairs that led up into the centre (A) of an underground 




8o THE CANTERBURY PUZZLES 

maie. through the winding of which I must pass before I could 
take my leave by the door (B). But I knew full well that in the 
great darkness of this dreadful place I might well wander for hours 
and yet return to the place from which I set out. How was I then 

B 




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 
•iiding 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 need- 
ful 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 
ioooeed on the thousandth attempt withal. If I was indeed to 
escape I must waste not a moment. 

Now, once had I heard the learned monk who did invent the 
lock say that he feared that the king's servants, having such bad 



STRANGE ESCAPE OF THE KING'S JESTER 8i 

memories, would mayhap forget the right letters ; so perchance, 
thought I, he had on this account devised some way to aid their 
memories. And what more natural than to make the letters 




form some word ? I soon found a word that was English, made of 
three letters — one letter being on each of the three dials. After 
that I had pointed the hands properly to the letters the door opened 
and I passed out. What was the secret word ? 



52. — Crossing the Moat. 

I was now face to face with the castle moat, which was, indeed, 
very wide and very deep. Alas ! I could not swim, and my chance 
of escape seemed of a truth hopeless, as, doubtless, it would have 
been had I not espied a boat tied to the wall by a rope. But after 
I had got into it I did find that the oars had been taken away, and 

(2,077) 6 



82 THE CANTERBURY PUZZLES 

that there was nothing that I could use to row me across. When 
I had untied the rope and pushed off upon the water the boat lay 




quite still, there being no stream or current to help me. 
then, did I yet take the boat across the moat ? 



How, 



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 
lenses, but was allowed to amuse himself therein. They were 
iqaare, and divided into 16 parts by high walls, as shown in the 
plan thereof, so that there were openings from one garden to an- 



STRANGE ESCAPE OF THE KING'S JESTER 83 

other, 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, I had 
to slip with agility from one garden to another, so that I might not 

B 




be seen, but escape unobserved. I did succeed in so doing, but 
afterwards remembered that I had of a truth entered every one 
of the 16 gardens once, and never more than once. This was, 
indeed, a curious thing. How might it have been done ? 



54. — Bridging the Ditch. 

I now did truly think that at last was I a free man, but I had 
quite forgot that I must yet cross a deep ditch before I might get 
right away. This ditch was 10 feet wide, and I durst not attempt 
to jump it, as I had sprained an ankle in leaving the garden. Look- 
ing around for something to help me over my difficulty, I soon 



84 



THE CANTERBURY PUZZLES 



found eight narrow planks of wood lying together in a heap. With 
these alone, and the planks were each no more than 9 feet long. 
I did at last manage to make a bridge across the ditch. How was 
this done ? 

Being now free I did hasten to the house of a friend who pro- 



^ 



i 



y^' 



f ^///////////////Mi 

PLAN O^ OlTCM 



^^J ^////////////^ 




vided me with a horse and a disguise, with which I soon succeeded 
in placing myself out of all fear of capture. 

Through the goodly offices of divers persons at the king's court 
I did at length obtain the royal pardon, though, indeed, I was never 
restored to that full favour that was once my joy and pride. 

Ofttimes have I been asked by many that do know me to 
set forth to them the strange manner of my escape, which more 
than one hath deemed to be of a truth wonderful, albeit the feat 
was nothing astonishing withal if we do but remember that from 
my youth upwards I had trained my wit to the making and answer- 
ing of cunning enigmas. And I do hold that the study of such 
crafty matters is good, not alone for the pleasure that is created 
thereby, but because a man may never be sure that in some sudden 
and untoward difficulty that may beset him in passing through this 
life of ours such strange learning may not serve his ends greatly, 
and, mayhap, help him out of many difficulties. 

I am now an aged man, and have not quite lost all my taste 



STRANGE ESCAPE OF THE KING'S JESTER 85 

for quaint puzzles and conceits ; but, of a truth, never have I found 
greater pleasure in making out the answers to any of these things 
than I had in mastering them that did enable me, as the king's 
jester in disgrace, to gain my freedom from the castle dungeon and 
so save my Ufe. 



THE SQUIRE'S CHRISTMAS PUZZLE PARTY 

A FINE specimen of the old English country gentleman was 
Squire Davidge, of Stoke Courcy Hall, in Somerset. When the 
last century was yet in its youth, there were few men in the west 
country more widely known and more generally respected and 
beloved than he. A born sportsman, his fame extended to Exmoor 
itself, where his daring and splendid riding in pursuit of the red 
deer had excited the admiration and envy of innumerable younger 
huntsmen. But it was in his own parish, and particularly in his 
own home, that his genial hospitality, generosity, and rare jovial 
humour made him the idol of his friends — and even of his relations, 
which sometimes means a good deal. 

At Christmas it was always an open house at Stoke Courcy 
Hall, for if there was one thing more than another upon which 
Squire Davidge had very pronounced views, it was on the question 
of keeping up in a royal fashion the great festival of Yule-tide. 
" Hark ye, my lads," he would say to his sons : " our country will 
begin to fall on evil days if ever we grow indifferent to the claims of 
those Christmas festivities that have helped to win us the proud 
name of Merrie England." Therefore, when I say that Christmas 
at Stoke Courcy was kept up in the good old happy, rollicking, 
festive style that our grandfathers and great-grandfathers so dearly 
loved, it will be unnecessary for me to attempt a description. We 
have a faithful picture of these merry scenes in the Bracebridge 
Hall of Washington Irving. I must confine myself in this sketch 
to one special feature in the Squire's round of jollification during 
the season of peace and good will. 



THE SQUIRE'S CHRISTMAS PUZZLE PARTY 87 

He took a curious and intelligent interest in puzzles of every 
kind, and there was always one night devoted to what was known 
as " Squire Davidge's Puzzle Party." Every guest was expected 
to come armed with some riddle or puzzle for the bewilderment and 
possible delectation of the company. The old gentleman always 
presented a new watch to the guest who was most successful in his 
answers. It is a pity that all the puzzles were not preserved ; but I 
propose to present to my readers a few selected from a number that 
have passed down to a surviving member of the family, who has 
kindly allowed me to use them on this occasion. There are some 
very easy ones, a few that are moderately difficult, and one hard 
brain-racker, so all should be able to find something to their taste. 

The little record is written in the neat angular hand of a young 
lady of that day, and the puzzles, the conditions of which I think it 
best to give mainly in my own words for the sake of greater clearness, 
appear to have been all propounded on one occasion. 

55. — The Three Teacups. 

One young lady — of whom our fair historian records with 
delightful inconsequence : " This Miss Charity Lockyer has since 
been married to a curate from Taunton Vale " — ^placed three empty 




teacups on a table, and challenged anybody to put ten lumps of 
sugar in them so that there would be an odd number of lumps in 
every cup. ** One young man, who has been to Oxford University, 
and is studying the law, declared with some heat that, beyond a 
doubt, there was no possible way of doing ij;, 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. 



88 



THE CANTERBURY PUZZLES 
56.— -The Eleven Pennies. 



A guest asked some one to favour him with eleven pennies, and 
he passed the coins to the company, as depicted in our illustration. 
The writer says : " He then requested us to remove five coins from 




the eleven, add four coins and leave nine. We could not but think 
there must needs be ten pennies left. We were a good deal amused 
at the answer hereof." 

57. — The Christmas Geese. 
Squire Hcmbrow, from Weston Zoyland — wherever that may 
be — proposed the following little arithmetical puzzle, from which 
it is probable that several somewhat similar modem 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 con- 
sidered 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 



THE SQUIRE'S CHRISTMAS PUZZLE PARTY 89 

in a way not desired. " Well, first of all I sold Mr. Jasper Tyler 
half of the flock and half a goose over ; then I sold Farmer Avent 
a third of what remained and a third of a goose over ; then I sold 
Widow Foster a quarter of what remained and three-quarters of 
a goose over ; and as I was coming home, whom should I meet 
but Ned Collier : so we had a mug of cider together at the Barley 
Mow, where I sold him exactly a fifth of what I had left, and gave 
him a fifth of a goose over for the missus. These nineteen that 
I have brought back I couldn't get rid of at any price." Now, how 
many geese did Farmer Rouse send to market ? My humane 
readers may be relieved to know that no goose was divided or put 
to any inconvenience whatever by the sales. 




58. — The Chalked Numbers. 

{ " We laughed greatly at a pretty jest on the part of Major 
\ Trenchard, a merry friend of the Squire's. With a piece of chalk 



90 THE CANTERBURY PUZZLES 

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, i, 2, 3, 4 being on one side, and 5, 7, 8, 
9 on the other. It will be seen that the numbers of the left-hand 
group add up' to 10, while the numbers in the other group add up to 
29. The Major's puzzle was to rearrange the eight boys in two new 
groups, so that the four numbers in each group should add up alike. 
The Squire's niece asked if the 5 should not be a 6 ; but the Major 
explained that the numbers were quite correct if properly regarded, 

c^ ^ Q a a m m Q 

a a a a^j^ a 4 Q 
a a a a% a & -^ 

S^'—Tasting the Plum Puddings. 
" Everybody, as I suppose, knows well that the number of 
dificrent Christmas plum puddings that you taste will bring you 



THE SQUIRE'S CHRISTMAS PUZZLE PARTY 91 

the same number of lucky days in the new year. One of the guests 
(and his name has escaped my memory) brought with him a sheet 
of paper on which were drawn sixty-four puddings, and he said the 
puzzle was an allegory of a sort, and he intended to show how 
we might manage our pudding-tasting with as mjich dispatch 
as possible." I fail to fully understand this fanciful and rather 
overstrained view of the puzzle. But it would ^ppear that the 
puddings were arranged regularly, as I have sifcwn 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 comer, bearing a sprig of holly, 
and strike out all the sixty-four puddings through their centres 
in twenty-one straight strokes. You can go up or down or hori- 
zontally, but not diagonally or obliquely; and you must never 
strike out a pudding twice, as that would imply a second and un- 
necessary tasting of those indigestible dainties. But the peculiar 
part of the thing is that you are required to taste the pudding that 
is seen steaming hot at the end of your tenth stroke, and to taste 
the one decked with holly in the bottom row the very last of all. 



60. — Under the Mistletoe Bough, 

" At the party was a widower who has but lately come into these 
parts,** says the record; **and, to be sure, he was an exceedingly 
melancholy man, for he did sit away from the company during 
the most part of the evening. We afterwards heard that he had 
been keeping a secret account of all the kisses that were given and 
received under the mistletoe bough, Truly, I would not have 
suffered any one to kiss me in that manner had I known that so 
unfair a watch was being kept. Other maids beside were in a like 
way shocked, as Betty Marchant has since told me." But it seems 
that the melancholy widower was merely collecting material for the 
following little osculatory problem. 

The company consisted of the Squire and his wife and six other 
married couples, one widower and three widows, twelve bachelors 



92 



THE CANTERBURY PUZZLES 



and boys, and ten maidens and little girls. Now, everybody was 
found to have kissed everybody else, with the following exceptions 
and additions : No male, of course, kissed a male. No married 
man kissed a married woman, except his own wife. All the 
bachelors and boys kissed all the maidens and girls twice. The 
widower did not kiss anybody, and the widows did not kiss each 
other. The puzzle was to ascertain just how many kisses had been 




thus given under the mistletoe bough, assuming, as it is charitable 
to do, that every kiss was returned—the double act being counted 
as one kiss. 

6i. — The Silver Cubes, 
The last extract that I wiU give is one that will, I think, interest 
U106C readers who may find some of the above puzzles too easy. 



THE SQUIRE'S CHRISTMAS PUZZLE PARTY 93 

It is a hard nut, and should only be attempted by those who flatter 
themselves that they possess strong intellectual teeth. 

" Master Herbert Spearing, the son of a widow lady in our 
parish, proposed a puzzle in arithmetic that looks simple, but 
nobody present was able to solve it. Of a truth I did not venture 
to attempt it myself, after the young lawyer from Oxford, who 
they say is very learned in the mathematics and a great scholar, 
failed to show us the answer. He did assure us that he believed 
it could not be done, but I have since been told that it is possible, 
though, of a certainty, I may not vouch for it. Master Herbert 
brought with him two cubes of solid silver that belonged to his 




mother. He showed that, as they measured two inches every way, 
each contained eight cubic inches of silver, and therefore the two 
contained together sixteen cubic inches. That which he wanted 
to know was — * Could anybody give him exact dimensions for two 
cubes that should together contain just seventeen cubic inches of 
silver ? * " Of course the cubes may be of different sizes. 

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




myEmm 



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 pubhc 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 Hght 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 pubUsh 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 excel- 
lent work of a high and dry kind. But the main body soon took 
to investigating the problems of real Hfe 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 puzzUng character. They seek perplexity for its own 
sake— something to unravel. As often as not the circumstances 
are of no importance to anybody, but they just form a little puzzle 
in real life, and that is sufficient. 

62. — The Ambigitous Photograph. 

A good example of the lighter kind of problem that occasionally 
before them is that which is known amongst them by the 



ADVENTURES OF THE PUZZLE CLUB 95 

name of " The Ambiguous Photograph." Though it is perplexing 
to the inexperienced, it is regarded in the club as quite a trivial 
thing. Yet it serves to show the close observation of these sharp- 
witted fellows. The original photograph hangs on the club wall, 
and has baffled every guest who has examined it. Yet any child 
should be able to solve the mystery. I will give the reader an 
opportunity of trying his wits at it. 

Some of the members were one evening seated together in their 
clubhouse in the Adelphi. Those present were : Henry Melville, 
a barrister not overburdened with briefs, who was discussing 
a problem with Ernest Russell, a bearded man of middle age, 
who held some easy post in Somerset House, and was a Senior 
Wrangler and one of the most subtle thinkers of the club ; Fred 
Wilson, a journalist of very buoyant spirits, who had more real 
capacity than one would at first suspect ; John Macdonald, a 
Scotsman, whose record was that he had never solved a puzzle 
himself since the club was formed, though frequently he had put 
others on the track of a deep solution ; Tim Churton, a bank clerk, 
full of cranky, unorthodox ideas as to perpetual motion ; also 
Harold Tomkins, a prosperous accountant, remarkably familiar 
with the elegant branch of mathematics — the theory of numbers. 

Suddenly Herbert Baynes entered the room, and everybody 
saw at once from his face that he had something interesting to 
communicate. Baynes was a man of private means, with no 
occupation. 

" Here's a quaint little poser for you all," said Baynes. " I 
have received it to-day from Dovey." 

Dovey was proprietor of one of the many private detective 
agencies that found it to their advantage to keep in touch with the 
club. 

'* Is it another of those easy cryptograms ? " asked Wilson. 

" If so, I would suggest sending it upstairs to the bilHard-marker." 

; " Don't be sarcastic, Wilson," said Melville. " Remember, we 

i are indebted to Dovey for the great Railway Signal Problem that 

a gave us all a week's amusement in the solving." 



96 THE CANTERBURY PUZZLES 

" 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 httle Yankee who married Lord Marksford two years 
ago? Lady Marksford and her husband have been in Paris for 
two or three months. Well, the poor creature soon got under the 
influence of the green-eyed monster, and formed the opinion that 
Lord Marksford was flirting with other ladies of his acquaintance. 

" Now, she has actually put one of Dovey's spies on to that 
excellent husband of hers ; and the myrmidon has been shadowing 
him about for a fortnight with a pocket camera. A few days ago 
he came to Lady Marksford in great glee. He had snapshotted 
his lordship while actually walking in the public streets with a 
lady who was not his wife.'* 

** ' What is the use of this at all ? * asked the jealous woman. 

" ' Well, it is evidence, your ladyship, that your husband was 
walking with the lady. I know where she is staying, and in a few 
days shall have found out all about her.' 

" * But, you stupid man,' cried her ladyship, in tones of great 
contempt, * how can any one swear that this is his lordship, when 
the greater part of him, including his head and shoulders, is hidden 
from sight ? And — and ' — she scrutinized the photo carefully — 
* why, I guess it is impossible from this photograph to say whether 
the gentleman is walking with the lady or going in the opposite 
direction 1 * 

" Thereupon she dismissed the detective in high dudgeon. 
Dovey has himself just returned from Paris, and got this account 
of the incident from her ladyship. He wants to justify his man, 
if possible, by showing that the photo does disclose which way 
the man is going. Here it is. See what you fellows can make 
of it." 

Our illustration is a faithful drawing made from the original 
photograph. It will be seen that a slight but sudden summer 
tbower is the real cause of the difficulty. 

All agreed that Lady Marksford was right-— that it is impossible 
to determine whether the man is walking with the lady or not. 



ADVENTURES OF THE PUZZLE CLUB 97 

** 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 ! 

(2,077) 7 



98 



THE CANTERBURY PUZZLES 



Then he has his overcoat over his arm, but which way his arm 
goes it is impossible to see." 

" How about the bend of the legs ? " asked Churton 
" Bend I 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." 




"I'm thinking that perhaps " began Macdonald, adjusting 

his eye-glasses. 

'* Don't think, Mac," advised Wilson. " It might hurt you. 
Besides, it is no use you thinking that if the dog would kindly 
pass on things would be easy. He won't." 

" The man's general pose seems to me to imply movement to 
the left." Tomkins thought. 

" On the contrary." Melville declared, " it appears to me 
clearly to suggest movement to the right." 

" Now. look here, you men," said Russell, whose opinions 



ADVENTURES OF THE PUZZLE CLUB 99 

ilways carried respect in the club. " It strikes me that what we 
lave to do is to consider the attitude of the lady rather than that 
)f the man. Does her attention seem to be directed to somebody 
)y her side ? '* 

Everybody agreed that it was impossible to say. 

" I've got it 1 " shouted Wilson. " Extraordinary that none of 
/ou have seen it. It is as clear as possible. It all came to me in 
L flash I " 

" Well, what is it ? '* asked Baynes. 

** Why, it is perfectly obvious. You see which way the dog is 
[oing — to the left. Very well. Now, Baynes, to whom does the 
log belong ? " 

*' To the detective I " 

The laughter against Wilson that followed this announcement 
vas simply boisterous, and so prolonged that Russell, who had at 
he time possession of the photo, seized the opportunity for making 
L most minute examination of it. In a few moments he held up 
lis hands to invoke silence. 

" Baynes is right," he said. " There is important evidence 
here which settles the matter with certainty. Assuming that the 
gentleman is really Lord Marksford — and the figure, so far as it is 
dsible, 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 
lid," said Melville. " Recollect that * no member shall openly 
lisclose his solution to a puzzle unless all present consent.' " 

** You need not have been alarmed," explained Russell. '* I 
vas simply going to say that I have no hesitation in declaring that 
x)rd Marksford is walking in one particular direction. In which 
lirection I will tell you when you have all * given it up.' " 



63.— T/t^ Cornish Cliff Mystery, 

Though the incident known in the Club as " The Cornish Cliff 
lystery " has never been published, every one remembers the case 



too THE CANTERBURY PUZZLES 

with which it was connected— an embezzlement at Todd's Bank in 
Comhill 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 Comisl: 
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 exchangee 
civilities with the two members of the Puzzle Club. A few refer- 
ences to some of the leading London detectives, and the productior 
of a confidential letter Melville happened to have in his pockei 
from one of them, soon established complete confidence, and th( 
inspector opened out. 

He said that he had just been to examine a very important clu( 
a quarter of a mile from there, and expressed the opinion thai 
Messrs. Lamson and Marsh would never again be found alive. A 
the suggestion of Melville the four men walked along the roac 
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." 

Arrived at the spot, they left the hard road and got over the 



ADVENTURES OF THE PUZZLE CLUB loi 

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

** Here, gentlemen, you see," said the inspector, ** that the 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. The conclusion is obvious.** 

" That, knowing they were unable to escape capture, they de- 
cided 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 an3rwhere. Go round there to the left, 
and you will be satisfied that the most experienced mountaineer 



102 THE CANTERBURY PUZZLES 

that ever lived could not make a descent, or even anywhere got 
over the edge of the cliff. There is no ledge or foothold within 
fifty feet." 

" Utterly impossible," said Melville, after an inspection. *' What 
do you propose to do ? " 

" I am going straight back to communicate the discovery to 
headquarters. We shall withdraw the cordon and search the coast 
for the dead bodies." 

" Then you will make a fatal mistake," said Melville. " The 
men are aUve 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 httle fellow." 

" I thought as much," said Melville. " And yet you will find 
that Lamson takes a shorter stride than Marsh. Notice, also, the 
pecuharity 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- 
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 i 
some two hundred yards in that way with such exactitude. You I 
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 i 
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 j 
they were caught, hiding under some straw in a bam, within two • 
miles of the spot. How did they get away from the edge of the 
difi? 



ADVENTURES OF THE PUZZLE CLUB 103 

64. — The Runaway Motor-Car, 

The little af!air 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 comer, 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 i. 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 every- 
thing else. 

He is always working out sums in his head ; and all he could say 
was that there were five figures in the number, and that he found 
that when he multiplied the first two figures by the last three they 
made the same figures, only in different order — ^just as 24 multiphed 
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 
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 i. We have therefore to find 
these numbers. It may conceivably happen that there is only 



,04 THE CANTERBURY PUZZLES 

one such number, in which case the thing is solved. But even if 
there are several cases, the owner of the actual car may easily be 

found. 

" How will you manage that ? " somebody asked. 

•• Surely," repUed Russell, " the method is [quite obvious. By 




the process of elimination. Every owner except the one in fault 
will be able to prove an alibi. Yet, merely guessing offhand, I 
think it quite probable that there is only one number that fits the 
case. Wc 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? 



ADVENTURES OF THE PUZZLE CLUB 105 



65. — The Mystery of Ravensdene Park. 

The mystery of Ravensdene Park, which I will now present, 
was a tragic affair, as it involved the assassination of Mr. Cyril 
Hastings at his country house a short distance from London. 

On February 17th, at 11 p.m., there was a heavy fall of snow, 
and though it lasted only half an hour, the ground was covered to 
a depth of several inches. Mr. Hastings had been spending the 
evening at the house of a neighbour, and left at midnight to walk 
home, taking the short route that lay through Ravensdene Park — 




that is, from D to A in the sketch-plan. But in the early morning 
he was found dead, at the point indicared by the star in our diagram, 
stabbed to the heart. All the seven gates were promptly closed, 
and the footprints in the snow examined. These were fortunately 
very distinct, and the police obtained the following facts : — 

The footprints of Mr. Hastings were very clear, straight from 
D to the spot where he was found. There were the footprints of 
the Ravensdene butler — who retired to bed five minutes before 
midnight — from E to EE. There were the footprints of the game- 
keeper from A to his lodge at A A. Other footprints showed that 



io6 



THE CANTERBURY PUZZLES 



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 par- 
ticularly noticed that no track ever crossed another track. Of this 
the poUce 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 




poanble to discover who committed the crime ? Was it the butler ? 
Or the gamekeeper ? Or the man who came in at B and went 
out at BB ? Or the man who went in at C and left at CC ? They 
provided themselves with diagrams — sketch-plans, hke the one we 
have reproduced, which simplified the real form of Ravensdene 
Park without destroying the necessary conditions of the problem. 

Our friends then proceeded to trace out the route of each person. 
in accordance with the positive statements of the police that we have 
given. It was soon evident that, as no path ever crossed another, 



ADVENTURES OF THE PUZZLE CLUB 107 

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 poUce luckily knew whose footprints this route represented, an 
arrest was made that led to the man's conviction. 

Can our readers discover whether A, B, C, or E committed the 
deed ? Just trace out the route of each of the four persons, and 
the key to the mystery will reveal itself. 



66. — The Buried Treasure. 

The problem of the Buried Treasure was of quite a different 
character. A young fellow named Dawkins, just home from 
Australia, was introduced to the club by one of the members, in 
order that he might relate an extraordinary stroke of luck that 
he had experienced " down under," as the circumstances involved 
the solution of a poser that could not fail to interest all lovers of 
puzzle problems. After the club dinner, Dawkins was asked to 
tell his story, which he did, to the following effect : — 

** I have told you, gentlemen, that I was very much down on 
my luck. I had gone out to Australia to try to retrieve my for- 
tunes, but had met with no success, and the future was looking 
very dark. I was, in fact, beginning to feel desperate. One hot 
summer day I happened to be seated in a Melbourne wineshop, 
when two fellows entered, and engaged in conversation. They 
thought I was asleep, but I assure you I was very wide awake. 

" * If only I could find the right field,' said one man, ' the 
treasure would be mine ; and as the original owner left no heir, I 
have as much right to it as anybody else.* 

" ' How would you proceed ? * asked the other. 

" * Well, it is Uke 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 comer, three furlongs 
from the next comer, and four furlongs from the next comer to 
that. You see, the worst of it is that nearly all the fields in the 



lo8 THE CANTERBURY PUZZLES 

district are square ; and I doubt whether there are two of exactly 
the same size. If only I knew the size of the field I could soon 
discover it, and. by taking these simple measurements, quickly 
secure the treasure/ 

" • But you would not know which corner to start from, nor 
which direction to go to the next comer/ 

'• • My dear chap, that only means eight spots at the most to 




dig over ; and as the paper says that the treasure is three feet 
deep, you bet that wouldn't take me long/ 

" Now, gentlemen," continued Dawkins, " I happen to be a bit 
of a mathematician ; and hearing the conversation, I saw at once 
that for a spot to be exactly two, three, and four furlongs from 
tnooewive comers of a square, the square must be of a particular 
area. You can't get such measurements to meet at one point in 
any »quarc you choose. They can only happen in a field of one 



ADVENTURES OF THE PUZZLE CLUB 109 

size, and that is just what these men never suspected. I will 
leave you the puzzle of working out just what that area is. 

" Well, when I found the size of the field, I was not long in 
discovering the field itself, for the man had let out the district in 
the conversation. And I did not need to make the eight digs, for, 
as luck would have it, the third spot I tried was the right one. The 
treasure was a substantial sum, for it has brought me home and 
enabled me to start in a business that already shows signs of being 
a particularly lucrative one. I often smile when I think of that 
poor fellow going about for the rest of his life saying : * If only I 
knew the size of the field ! ' while he has placed the treasure safe 
in my own possession. I tried to find the man, to make him some 
compensation anonymously, but without success. Perhaps he stood 
in little need of the money, while it has saved me from ruin." 

Could the reader have discovered the required area of the field 
from those details overheard in the wineshop ? It is an elegant 
little puzzle, and furnishes another example of the practical utility, 
on unexpected occasions, of a knowledge of the art of problem- 
solving. 



THE PROFESSOR'S PUZZLES 

*' Why, here is the Professor ! " exclaimed Grigsby. *' We'll make 
him show us some new puzzles." 

It was Christmas Eve, and the club was nearly deserted. 
Only Grigsby, Hawkhurst, and myself, of all the members, 
seemed to be detained in town over the season of mirth and mince- 
pies. The man, however, who had just entered was a welcome 
addition to our number. " The Professor of Puzzles," as we had 
nicknamed him, was very popular at the club, and when, as on the 
present occasion, things got a little slow, his arrival was a positive 
blessing. 

He was a man of middle age, cheery and kind-hearted, but 
inclined to be cynical. He had all his life dabbled in puzzles, 
problems, and enigmas of every kind, and what the Professor 
didn't know about these matters was admittedly not worth know- 
ing. His puzzles always had a charm of their own, and this was 
mainly because he was so happy in dishing them up in palatable 
form. 

*• You are the man of all others that we were hoping would 
drop in," said Hawkhurst. " Have you got anything new ? " 

" I have always something new," was the reply, uttered with 
failed conceit — for the Professor was really a modest man — " I'm 
rimply glutted with ideas." 

" Where do you get all your notions ? " I asked. 

** Everywhere, anywhere, during all my waking moments. In- 
deed, two or three of my best puzzles have come to me in my 
dftams/' 

110 



THE PROFESSOR'S PUZZLES 



III 



" Then all the good ideas are not used up ? " 

" Certainly not. And all the old puzzles are capable of im- 
provement, embellishment, and extension. Take, for example, 
magic squares. These were constructed in India before the Chris- 
tian 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." 











® 






^M. 


fc\ 






u*iM 


« 


^ 



67. — The Coinage Puzzle. 

He made a rough diagram, and placed a crown and a florin in 
two of the divisions, as indicated in the illustration. 

" Now," he continued, " place the fewest possible current English 



,,2 THE CANTERBURY PUZZLES 

coins in the seven empty divisions, so that each of the three 
columns, three rows, and two diagonals shall add up fifteen shillings. 
Of course, no division may be without at least one coin, and no two 
divisions may contain the same value." 

" But how can the coins affect the question ? " asked Grigsby. 

" That you will find out when you approach the solution." 

" I shall do it with numbers first," said Hawkhurst, " and then 
substitute coins." 

Five minutes later, however, he exclaimed, " Hang it all ! I 
caait help getting the 2 in a corner. May the florin be moved from 
Hs present position ? " 

" Certainly not." 

" Then I give it up." 

But Grigsby and I decided that we would work at it another 
time, so the Professor showed Hawkhurst the solution privately, 
and then went on with his chat. 



68. — The Postage Stamps Puzzles. 

" Now, instead of coins we'll substitute postage-stamps. Take 
ten current English stamps, nine of them being all of different 
values, and the tenth a duplicate. Stick two of them in one divi- 
sion 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 I " 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." 



THE PROFESSOR'S PUZZLES 



113 



" 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." 



© © a S B .@ ,® B 



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 



(2,077) 



114 THE CANTERBURY PUZZLES 

—very grotesque in form and brilliant in colour. While we were 
looking at them he asked the waiter to place sixty-four tumblers 
on the club table. When these had been brought and arranged in 
the form of a square, as shown in the illustration, he placed eight 
of the little green frogs on the glasses as shown. 

" Now," he said, " you see these tumblers form eight horizontal 
and eight vertical lines, and if you look at them diagonally (both 
ways) there are twenty-six other lines. If you run your eye along 
all these forty-two lines, you will find no two frogs are anywhere in 
a line. 

" The puzzle is this. Three of the frogs are supposed to jump 
from their present position to three vacant glasses, so that in their 
new relative positions still no two frogs shall be in a line. What 
are the jumps made ? " 

" I suppose " began Hawkhurst. 

** I know what you are going to ask," anticipated the Professor. 
" No ; the frogs do not exchange positions, but each of the three 
jumps to a glass that was not previously occupied." 

" But surely there must be scores of solutions ? " I said. 

*' I shall be very glad if you can find them," replied the Pro- 
fessor with a dry smile. ** I only know of one— or rather two, 
counting a reversal, which occurs in consequence of the position 
being symmetrical." 

70. — Romeo and Juliet. 

For some time we tried to make these little reptiles perform the 
feat allotted to them, and failed. The Professor, however, would 
not give away his solution, but said he would instead introduce 
to us a little thing that is childishly simple when you have once seen 
it, but cannot be mastered by everybody at the very first attempt. 

" Waiter 1 " he called again. " Just take away these glasses, 
J, and bring the chessboards." 

** I hope to goodness," exclaimed Grigsby, " you are not going 
to show us some of those awful chess problems of yours. * White 
to mate Black in 427 moves without moving his pieces.' * The 



THE PROFESSOR'S PUZZLES 



"S 



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 









;. rn ,lJ ,1- 




i- ^ V •i''-'' 




Ii - 


M 








;.>i:r:J,jr. 










ii'i ' "i. ! 


ROMEO 


■'?i^f!, 




'.1! ..1!': ;■, 










'f-i:A 








^ 






", '.il' 




,) 












1 




•i .>«•,• 


JULIET 


, '.' 










i!#" 
WM 




'■•H 




■'■!. 1 ". ' 






'ill- 1' 








y:;4 





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 1 that does it." 

" Yes," said the Professor : " Romeo has got there, it is true. 



ii6 THE CANTERBURY PUZZLES 

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." 

Hawklmrst, curiously enough, hit on the solution at once, and the 
Professor remarked that this was just one of those puzzles that a 
person might solve at a glance or not master in six months. 

71. — Romeo's Second Journey. 

** It was a sheer stroke of luck on your part, Hawkhurst," he 
added. , ** Here is a much easier puzzle, because it is capable of 
more systematic analysis ; yet it may just happen that you will not 
do it in an hour. Put Romeo on a white square and make him 
crawl into every other white square once with the fewest possible 
turnings. This time a white square may be visited twice, but the 
snail must never pass a second time through the same corner of a 
square nor ever enter the black squares." 

** May he leave the board for refreshments ? " asked Grigsby. 

*' No ; he is not allowed out until he has performed his feat." 

72. — The Frogs who would a-wooing go. 

While we were vainly attempting to solve this puzzle, the 
Professor arranged on the table ten of the frogs in two rows, as they 
will be found in the illustration. 

1^ 1^ %. ^ 



'' Th * is entertaining," I said. " What is it ? " 
'* It - puzzle I made a year ago, and a favourite with the 

lew people who have seen it. It is called ' The Frogs who would 



THE PROFESSOR'S PUZZLES 117 

a-wooing go.* Four of them are supposed to go a-wooing, and 
after the four have each made a jump upon the table, they are in 
such a position that they form five straight rows with four frogs in 
every row." 

" What's that ? " asked Hawkhurst. " I think I can do that." 
A few minutes later he exclaimed, *' How's this ? " 

" They form only four rows instead of five, and you have moved 
six of them," explained the Professor. 

" Hawkhurst," said Grigsby severely, " you are a duffer. I see 
the solution at a glance. Here you are ! These two jump on their 
comrades* backs." 

" No, no," admonished the Professor ; " that is not allowed. 
I distinctly said that the jumps were to be made upon the table. 
Sometimes it passes the wit of man so to word the conditions of a 
problem that the quibbler will not persuade himself that he has 
found a flaw through which the solution may be mastered by a 
child of five." 

After we had been vainly puzzHng 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 beUeves 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 h^me." 



MISCELLANEOUS PUZZLES 

73. — The Ganie 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 modem game of ninepins. Kayle-pins were not con- 
fined in those days to any particular number, and they were gen- 
erally made of a conical shape and set up in a straight row. 

At first they were knocked down by a club that was thrown at 
them from a distance, which at once suggests the origin of the 
pastime of ** shying for cocoanuts '* that is to-day so popular on 
Bank Holidays on Hampstead Heath and elsewhere. Then the 
players introduced balls, as an improvement on the club. 

In the illustration we get a picture of some of our fourteenth- 
century ancestors playing at kayle-pins in this manner. 

Now, I will introduce to my readers a new game of parlour 
kayle-pins, that can be played across the table without any prepara- 
tion whatever. You simply place in a straight row thirteen domi- 
noes, chcss-f>awns, draughtsmen, counters, coins, or beans — 
anything will do — all closejtogether, 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. 

118 



MISCELLANEOUS PUZZLES 



1x9 



Therefore, in playing our table-game, all you have to do is to 
imock 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 L, 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, 
pubUshed in 1613 : — 

** Towards the end of his reigne he appointed his two sonnes 
Robert and Henry, with joynt authoritie, govemours of Normandie ; 



120 



THE CANTERBURY PUZZLES 



the one to suppresse either the insolence or levitie of the other. 
These went together to visit the French king lying at Constance : 
where, entertaining the time with varietie of disports, Henry played 







with Louis, then Daulphine of France, at chesse, and did win of him 
very much. 

** Hereat Louis beganne to growe warme in words, and was 
therein little respected by Henry. The great impatience of the one 
tnd the sniall forbearance of the other did strike in the end such a 
heat between them that Louis threw the chessmen at Henry's face. 



MISCELLANEOUS PUZZLES 



121 



** Henry again stroke Louis with the chessboard, drew blood 
with the blowe, and had presently slain him upon the place had he 
not been stayed by his brother Robert. 

" Hereupon they presently went to horse, and their spurres 
claimed so good haste as they recovered Pontoise, albeit they were 
sharply pursued by the French." 

Now, tradition — on this point not trustworthy — says that the 
chessboard broke into the thirteen fragments shown in our 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 chess-board, and 
the puzzle is simply to cut them out and fit them together, so as 
to make a perfect board properly chequered. The pieces may be 
easily cut out of a sheet of " squared " paper, and, if mounted on 
cardboard, they will form a source of perpetual amusement in the 
home. 

If you succeed in constructing the chessboard, but do not record 
the arrangement, you will find it just as puzzling the next time you 
feel disposed to attack it. 

Prince Henry himself, with all his skill and learning, would have 
found it an amusing pastime. 



75. — The Spider and the Fly. 

Inside a rectangular room, measuring 30 feet in length and 
12 feet in width and height, a spider is at a point on the middle of 



"n.^ 


« 
1 
1 


^ 


•-.^ 




30 a 



one of the end walls, i foot from the ceiling, as at A ; and a fly is 
on the opposite wall, i foot from the floor in the centre, as shown 



laa 



THE CANTERBURY PUZZLES 



at B. What is the shortest distance that the spider must crawl in 
Older to reach the fly, wliich remains stationary ? Of course the 
gilder never drops or uses its web, but crawls fairly. 

76. — The Perplexed Cellarman. 

Here is a little puzzle culled from the traditions of an old mon- 
astery 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. 




A 



kmu 



Tl»e Abbot, moreover, had a fine taste in wines. On one occa- 
•ion 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" 



MISCELLANEOUS PUZZLES 



123 



" A fair dozen in large ])ottles, my lord abbot, and the like 
in the small," replied the cf llarman, " whereof five of each have 
been drunk in the refectory.'* 

" So be it. There be three varlets waiting at the gate. Let the 
two dozen bottles be given unto them, both full and empty ; and see 
that the dole be fairly made, so that no man receive more wine than 
another, nor any difference in bottles." 

Poor John returned to his cellar, taking the three men with him, 
and then his task began to perplex him. Of full bottles he had 
seven large and seven small, and of empty bottles five large and five 
small, as shown in the illustration. How was he to make the 
required equitable division ? 

He divided the bottles into three groups in several ways that at 
first sight seemed to be quite fair, since two small bottles held just 
the same quantity of wine as one large one. But the large bottles 
themselves, when empty, were not worth two small ones. 

Hence the abbot's order that each man must take away the same 
number of bottles of each size. 

Finally, the cellarman had to consult one of the monks who was 
good at puzzles of this kind, and who showed him how the thing 
was done. Can you find out 
just how the distribution was 
made ? 



77. — Making a Flag. 

A good dissection puzzle 
in so few as two pieces is 
rather a rarity, so perhaps 
the reader will be interested 
in the following. The dia- 
gram represents a piece of 
bunting, and it is required to 
cut it into two pieces (without any waste) that will fit together and 
form a perfectly square flag, with the four roses symmetrically 




124 



THE CANTERBURY PUZZLES 



placed. This would be easy enough if it were not for the four 
roses, as we should merely have to cu*. from A to B, and insert the 
piece at the bottom of the flag. But we are not allowed to cut 
through any of the roses, and therein Hes the difficulty of the puzzle. 
Of course we make no allowance for " turnings." 

78. — Catching the Hogs, 

In the illustration Hendrick and Katriin are seen engaged in the 
exhilarating sport of attempting the capture of a couple of hogs. 
Why did they fail ? 




^r-V/..-. 



































w 


— 








)t 




p 














_ 

















Strange as it may seem, a complete answer is afforded in the 
little puzzle game that I will now explain. 



MISCELLANEOUS PUZZLES 125 

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 Katriin, and the other the hogs. The first player moves the 
Dutchman and his wife one square each in any direction (but not 
diagonally), and then the second player moves both pigs one square 
each (not diagonally) ; and so on, in turns, until Hendrick catches 
one hog and 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 card- 
sharpers at racecourses and in railway carriages. 

As, on its own merits, however, the game is particularly interest- 
ing, I will make no apology for presenting it to my readers. 

The cardsharper lays down the twenty-four cards shown in the 
illustration, and invites the innocent wayfarer to try his luck or 
skill by seeing which of them can first score thirty-one, or drive 
his opponent beyond, in the following manner : — • 

One player turns down a card, say a 2, and counts " two " ; 
the second player turns down a card, say a 5, and, adding this to 
the score, counts " seven " ; the fijrst player turns down another 
card, say a i, and counts " eight " ; and so the play proceeds 
alternately until one of them scores the " thirty-one," and so 
wins. 

Now, the question is, in order to win, should you turn down the 
first card, or courteously request your opponent to do so ? And 
how should you conduct your play ? The reader will perhaps say : 
" Oh, that is easy enough. You must play first, and turn down a 
3 ; then, whatever your opponent does, he cannot stop your making 



126 



THE CANTERBURY 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 





Q 



ov 

90 
00 

00 
00 







00 

00 





00 
00 





tt 



plays a 4 and counts " twenty-one " ; you play 3 and make your 
" twenty-four." 

Now the sharper plays the last 4 and scores " twenty-eight." 
You look in vain for another 3 with which to win, for they are 
all turned down I 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." 



MISCELLANEOUS PUZZLES 127 

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 fortihcations. 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 I " So the Celestial Govern- 




ment officials were kept busy arranging the details. The letters in 
the diagram show the different nationalities, and indicate not only 
just where each line must enter the city, but also where the station 
belonging to that line must be located. As it was agreed that 
the line of one company must never cross the line of another, 
the representatives of the various countries concerned were 
engaged so many weeks in trying to find a solution to the problem, 
that in the meantime a change in the Chinese Government was 
brought about, and the whole scheme fell through. Take your 
pencil and trace out the route for the line A to A, B to B, C to 
C, and so on, without ever allowing one line to cross another or 
pass through another company's station. 



128 



THE CANTERBURY PUZZLES 



Si.— The EigM Clowns. 

This illustration represents a troupe of clowns I once saw on the 
Continent. Each clown bore one of the numbers i to 9 on his 
body. After going through the usual tumbling, juggling, and other 
antics, they generally concluded with a few curious little numerical 




tricks, one of which was the rapid formation of a number of magic 
squares. It occurred to me that if clown No. i 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. i that 
must be absent. 



MISCELLANEOUS PUZZLES 



129 



82. — The Wizard's Arithmetic, 

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







" And art thou learned also in the magic of numbers ? '* asked 
the knight. " Show me but one sample of thy wit in these matters." 

The old wizard took five blocks bearing numbers, and placed 
them on a shelf, apparently at random, so that they stood in 
the order 41096, as shown in our illustration. He then took 
in his hands an 8 and a 3, and held them together to form the 
number ^z- 

' (2.077) O 



130 THE CANTERBURY PUZZLES 

" Sir Knight, tell me," said the wizard, " canst thou multiply 
one number into the other in thy mind ? " 

" Nay, of a truth," the good knight replied. " I should need 
to set out upon the task with pen and scrip." 

•• Yet mark ye how right easy a thing it is to a man learned in 
the lore of far Araby, who knoweth all the magic that is hid in 
the philosophy of numbers 1 " 

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. 




S3.— The Ribbon Problem. 

If wc take the ribbon by the ends and pull it out straight, we 
have the number 05882352941 17647. This number has the peculi- 
arity that, if we multiply it by any one of the numbers, 2, 3, 4, 5, 



MISCELLANEOUS PUZZLES 



131 



6, 7, 8, or 9, we get exactly the same number in the circle, starting 
from a different place. For example, multiply by 4, and the pro- 
duct is 2352941 176470588, which starts from the dart in the circle. 
So, if we multiply by 3, we get the same result starting from the 
star. Now, the puzzle is to place a different arrangement of figures 
on the ribbon that will produce similar results when so multiplied ; 
only the o and the 7 appearing at the ends of the ribbon must not 
be removed. 

84. — The Japanese Ladies and the Carpet. 

Three Japanese ladies possessed a square ancestral carpet of 
considerable intrinsic value, but treasured also as an interesting 




heirloom in the family. They decided to cut it up and make three 
square rugs of it, so that each should possess a share in her own 
house. 



r32 THE CANTERBURY PUZZLES 

One lady suggested that the simplest way would be for her to 
take a smaller share than the other two, because then the carpet 
need not be cut into more than four pieces. 

There are three easy ways of doing this, which I will leave the 
reader for the present the amusement of finding for himself, merely 
saying that if you suppose the carpet to be nine square feet, then one 
lady may take a piece two feet square whole, another a two feet 
square in two pieces, and the third a square foot whole. 

But this generous offer would not for a moment be entertained 
by the other two sisters, who insisted that the square carpet should 
be so cut that each should get a square mat of exactly the same 
size. 

Now, according to the best Western authorities, they would 
have found it necessary to cut the carpet into seven pieces ; but a 
correspondent in Tokio assures me that the legend is that they 
did it in as few as six pieces, and he wants to know whether such 
a thing is possible. 

Yes ; it can be done. 

Can you cut out the six pieces that will form three square 
mats of equal size ? 

85. — Captain Longbow and the Bears. 

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



MISCELLANEOUS PUZZLES 



133 




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 ? 



«34 



THE CANTERBURY PUZZLES 



86.— The English Tour. 

This puzzle has to do with railway routes, and in these days 
of much travelling should prove useful. The map of England shows 
twenty-four towns, connected by a system of railways. A resident 
at the town marked A at the top of the map proposes to visit every 




one of the towns once and only once, and to finish up his tour 
at Z. This would be easy enough if he were able to cut across 
country by road, as well as by rail, but he is not. How does he 
perform the feat ? Take your pencil and, starting from A, pass 
from town to town, making a dot in the towns you have visited, 
and sec if you can end at Z. 

87. — The Chifu-ChemtUpo Puzzle. 
Here is a puzzle that was once on sale in the London shops. 
It represents a military train — an engine and eight cars. The 



MISCELLANEOUS PUZZLES 



135 



puzzle is to reverse the cars, so that they shall be in the order 
8. 7. 6, 5, 4, 3, 2, I, instead of i, 2, 3, 4, 5, 6, 7, 8, with the 
engine left, as at first, on the side track. Do this in the fewest 
possible moves. Every time the engine or a car is moved from the 
main to the side track, or vice versa, it counts a move for each car 
or engine passed over one of the points. Moves along the main 




track are not counted. With 8 at the extremity, as shown, there 
is just room to pass 7 on to the side track, run 8 up to 6, and bring 
down 7 again ; or you can put as many as five cars, or four and the 
engine, on the siding at the same time. The cars move without 
the aid of the engine. The purchaser is invited to *' try to do it 
in 20 moves." How many do you require ? 



88. — The Eccentric Market-woman. 

Mrs. Covey, who keeps a little poultry farm in Surrey, is one 
of the most eccentric women I ever met. Her manner of doing 
business is always original, and sometimes quite weird and won- 
derful. She was once found explaining to a few of her choice 
friends how she had disposed of her day's eggs. She had evidently 
got the idea from an old puzzle with which we are all familiar ; but 
as it is an improvement on it, I have no hesitation in presenting 
it to my readers. She related that she had that day taken a 
certain number of eggs to market. She sold half of them to one 
customer, and gave him half an egg over. She next sold a third of 
what she had left, and gave a third of an egg over. She then sold 
a fourth of the remainder, and gave a fourth of an egg over. Finally, 



L 
136 THE CANTERBURY PUZZLES 

she disposed of a fifth of the remainder, and gave a fifth of an 



over. Then what she had left she divided equally among 
thirteen of her friends. And, strange to say, she had not through- 
out all these transactions broken a single egg. Now, the puzzle 
is to find the smallest possible number of eggs that Mrs, Covey 
could have taken to market. Can you say how many ? 

{ 89. — The Primrose Puzzle. 

Select the name of any flower that you think suitable, and that 
contains eight letters. Touch one of the primroses with your 
pencil and jump over one of the adjoining flowers to another, on 




which you mark the first letter of your word. Then touch another 
vacant flower, and again jump over one in another direction, and 
write down the second letter. Continue this (taking the letters in 
tbdr 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 



MISCELLANEOUS PUZZLES 137 

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 frieniis, 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 couW 
the reader have prepared an arrangement for every sitting ? The 
hotel proprietor was asked to draw up a scheme, but he miserably 
failed. 

91. — The Five Tea Tins. 

Sometimes people will speak of mere counting as one of the 
simplest operations in the world ; but on occasions, as I shall show, 
it is far from easy. Sometimes the labour can be diminished by the 
use of little artifices ; sometimes it is practically impossible to make 
the required enumeration without having a very clear head indeed. 
An ordinary child, buying twelve postage stamps, will almost 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, " i, 2, 3, 4," etc. If the child's 
mother has occasion to add up the numbers i, 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, 25x51=1,275. But his smart son of 
twenty may go one better and say, ** Why multiply by 25 ? Just 
add two o's to the 51 and divide by 4, and there you are ! " 

A tea merchant has five tin tea boxes of cubical shape, which 
he keeps on his counter in a row, as shown in our illustration. 
Every box has a picture on each of its six sides, so there are thirty 



138 



THE CANTERBURY PUZZLES 



pictures in all ; but one picture on No. i is repeated on No. 4, and 
two other pictures on No. 4 are repeated on No. 3. There are, 
therefore, only twenty-seven different pictures. The owner always 
keeps No. I 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 pic- 
tures in front shall never be twice alike. He found the making 
of the count a tough Httle nut. Can you work out the answer 
without getting your brain into a tangle ? Of course, two similar 
pictures may be in a row, as it is all a question of their order. 



92. — The Four Porkers. 
The four pigs are so placed, each in a separate sty, that although 
every one of the thirty-six sties is in a straight Hne (either hori- 
lontally, vertically, or diagonally), with at least one of the pigs, 



MISCELLANEOUS PUZZLES 



139 



yet no pig is in line with another. In how many different ways 
may the four pigs be placed to fulfil these conditions? If you 



%. 




















-%, 






































Ik 




V 











turn this page round you get three more arrangements, and if you 
turn it round in front of a mirror you get four more. These are 
not to be counted as different arrangements. 



93. — The Number Blocks. 

The children in the illustration have found that a large number 
of very interesting and instructive puzzles may be made out of 
number blocks ; that is, blocks bearing the ten digits or Arabic 
figures — i, 2, 3, 4, 5, 6, 7, 8, 9, and o. 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 multipUcation 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 multipUed 



M 



THE CANTERBURY PUZZLES 



by I is the same. You will find it quite impossible to get any 
smaller result. 

Now, my puzzle is to find the largest possible result. Divide 
the blocks into any two groups of five that you like, and arrange 




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



94. — Foxes and Geese. 

Here is a little puzzle of the moving counters class that my 
readers will probably find entertaining. Make a diagram of any 
convenient size similar to that shown in our illustration, and pro- 
vide six counters — three marked to represent foxes and three to 



MISCELLANEOUS PUZZLES 



141 



represent geese. Place the geese on the discs i, 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 possible 
moves. 

But you must be careful never to let a fox and goose get within 
reach of each other, or there will be trouble. This rule, you will 




find, prevents you moving the fox from 11 on the first move, as on 
either 4 or 6 he would be within reach of a goose. It also prevents 
your moving a fox from 10 to 9, or from 12 to 7. If you play 
10 to 5, then your next move may be 2 to 9 with a goose, which 
you could not have played if the fox had not previously gone from 
10. It is perhaps unnecessary to say that only one fox or one 
goose can be on a disc at the same time. Now, what is the 
smallest number of moves necessary to make the foxes and geese 
change places ? 



u^ 



THE CANTERBURY PUZZLES 



95. — Robinson Crusoe's Table. 

Here is a curious extract from Robinson Crusoe's diary. It is 
not to be found in the modem editions of the Adventures, and 
fe omitted in the old. This has always seemed to me to be a pity. 

*• The third day in the morning, the wind having abated during 
the night, I went down to the shore hoping to find a typewriter and 
other useful things washed up from the wreck of the ship ; but all 




that fell in my way was a piece of timber with many holes in it. 
My man Friday had many times said that we stood sadly in need 
of a square table for our afternoon tea, and I bethought me how 
this piece of wood might be used for that purpose. And since 
during the long time that Friday had now been with me I was not 
wanting to lay a foundation of useful knowledge in his mind, I told 
him that it was my wish to make the table from the timber I had 
found, without there being any holes in the top thereof. 

" Friday was sadly put to it to say how this might be, more 



MISCELLANEOUS PUZZLES 



M3 



especially as I said it should consist of no more than two pieces 
joined together ; but I taught him how it could be done in such a 
way that the table might be as large as was possible, though, to 
be sure, I was amused when he said, * My nation do much better ; 
they stop up holes, so pieces sugars not fall through/ " 

Now, the illustration gives the exact proportion of the piece 
of wood with the positions of the fifteen holes. How did Robinson 
Crusoe make the largest possible square table-top in two pieces, so 
that it should not have any holes in it ? 



96. — The Fifteen Orchards. 

In the county of Devon, where the cider comes from, fifteen of 
the inhabitants of a village are imbued with an excellent spirit of 
friendly rivalry, and a few years ago they decided to settle by 






Vv:r^ 







^■•t 



'->^- 



actual experiment a little difference of opinion as to the cultiva- 
tion of apple trees. Some said they want plenty of light and air, 
while others stoutly maintained that they ought to be planted 



144 THE CANTERBURY PUZZLES 

pretty closely, in order that they might get shade and protection 
from cold winds. So they agreed to plant a lot of young trees, a 
different number in each orchard, in order to compare results. 

One man had a single tree in his field, another had two trees, 
another had three trees, another had four trees, another five, and 
so on, the last man having as many as fifteen trees in his little 
orchard. Last year a very curious result was found to have come 
about. Each of the fifteen individuals discovered that every tree 
in his own orchard bore exactly the same number of apples. But, 
what was stranger still, on comparing notes they found that the 
total gathered in every allotment was almost the same. In fact, 
if the man with eleven trees had given one apple to the man who 
had seven trees, and the man with fourteen trees had given three 
each to the men with nine and thirteen trees, they would all have 
had exactly the same. 

Now, the puzzle is to discover how many apples each would 
have had (the same in every case) if that little distribution had 
been carried out. It is quite easy if you set to work in the right 
way. 

97. — The Perplexed Plumber, 

When I paid a visit to Peckham recently I found everybody 
asking. " What has happened to Sam Solders, the plumber ? *' He 
seemed to be in a bad way, and his wife was seriously anxious about 
the state of his mind. As he had fitted up a hot-water apparatus 
for me some years ago which did not lead to an explosion for at 
least three months (and then only damaged the complexion of 
one of the cook's followers), I had considerable regard for him. 

" There he is," said Mrs. Solders, when I called to inquire. 
" That's how he's been for three weeks. He hardly eats anything, 
and takes no rest, whilst his business is so neglected that I don't 
know what is going to happen to me and the five children. All 
day long— and night too — there he is, figuring and figuring, and 
tearing his hair like a mad thing. It's worrying me into an early 
grave." 



MISCELLANEOUS PUZZLES 



145 



I persuaded Mrs. Solders to explain matters to me. It seems 
that he had received an order from a customer to make two rect- 
angular zinc cisterns, one with a top and the other without a top. 
Each was to hold exactly 1,000 cubic feet of water when filled to 
the brim. The price was to be a certain amount per cistern, in- 
cluding cost of labour. Now Mr. Solders is a thrifty man, so he 
naturally desired to make the two cisterns of such dimensions that 



/%. f 


L 


^h f- 


i 


\ 




}' 


n 




t 


m 


Ir 


*•••• «••• 


/^<3. 


Im- 




^y 



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." 

(3,077) 10 



146 



THE CANTERBURY PUZZLES 



98. — The Nelson Column. 

During a Nelson celebration I was standing in Trafalgar Square 
with a friend of puzzling proclivities. He had for some time been 
gazing at the column in an abstracted way, and seemed quite 
unconscious of the casual remarks that I addressed to him. 

" What are you dreaming about ? " I said at last. 




•* Two feet " he murmured. 

" Somebody's Trilbys ? " I inquired. 

" Five times round " 

" Two feet, five times round ! What on earth are you saying ? " 

" Wait a minute," he said, beginning to figure something out 
on the back of an envelope. I now detected that he was in the 
throes of producing a new problem of some sort, for I well knew 
llit methods of working at these things. 



MISCELLANEOUS PUZZLES 147 

" 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 diffi- 
cult, but is really remarkably easy." 

He was right. The puzzle is quite easy if properly attacked. 
Of course the height and circumference are not correct, but chosen 
for the purposes of the puzzle. The artist has also intentionally 
drawn the cylindrical shaft of the column of equal circumference 
throughout. If it were tapering, the puzzle would be less easy. 

99. — The Two Errand Boys. 

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

100. — On the Ramsgate Sands. 

Thirteen youngsters were seen dancing in a ring on the Rams- 
gate sands. Apparently they were pla5ang ** Round the Mulberry 
Bush." The puzzle is this. How many rings may they form 
without any child ever taking twice the hand of any other child — 
right hand or left ? That is, no child may ever have a second 
time the same neighbour. 

loi. — 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 re- 



148 



THE CANTERBURY PUZZLES 



markable coincidences — little things against the probability of the 
occurrence of which the odds are immense — ^that fill us with be- 
wildennent. 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 i to 9 and o, and, what is more remarkable, that if 
the numbers on the first and second cars are multiplied together 
they will make the number on the third car. That is, yS, 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 



MISCELLANEOUS PUZZLES 149 

would themselves fulfil this extra condition. What are the three 
numbers that we want ? Remember that they must have two, 
three, and five figures respectively. 

102. — A Reversible Magic Square. 

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

103. — The Tube Railway. 

The above diagram is the plan of an underground railway. The 
fare is uniform for any distance, so long as you do not go twice 
along any portion of the line during the same journey. Now a 
certain passenger, with plenty of time on his hands, goes daily 
from A to F. How many different routes are there from which 




he may select ? For example, he can take the short direct route, 
A, B, C, D, E, F, in a straight line ; or he can go one of the long 
routes, such as A, B, D, C, B, C, E, D, E, F. It will be noted that 
he has optional lines 'between certain stations, and his selections 
of these lead to variations of the complete route. Many readers 
will find it a very perplexing little problem, though its conditions 
are so simple. 



I$0 



THE CANTERBURY PUZZLES 



104.'-The Skipper and the Sea-Serpent. 

Mr. Simon Softleigh had spent most of his life between Tooting 
Bee 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. 

" I suppose," said Mr. Softleigh one morning to a jovial, weather- 




beaten skipper. " you have seen many wonderful sights on the rolling 
seas ? " 

" Bless you, sir, yes," said the skipper. " P'raps you've never 
•ccn a vanilla iceberg, or a mermaid a-hanging out her things to dry 
cm 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 un- 
certain whether they existed." 

" Unccrtin ! You wouldn't say there was anything uncertin 



MISCELLANEOUS PUZZLES 151 

about a sea-sarpint if once you'd seen one. The first as I seed was 
when I was skipper of the Saucy Sally. We was a-coming round 
Cape Horn with a cargo of shrimps from the Pacific Islands when 
I looks over the port side and sees a tremenjus monster like a snake, 
with its 'ead out of the water and its eyes flashing fire, a-bearing 
down on our ship. So I shouts to the bo'sun to let down the boat, 
while I runs below and fetches my sword — the same what I used 
when I killed King Chokee, the cannibal chief as eat our cabin- 
boy — and we pulls straight into the track of that there sea-sarpint. 
Well, to make a long story short, when we come alongside o* the 
beast I just let drive at him with that sword o* mine, and before 
you could say * Tom Bowling ' I cut him into three pieces, all of 
exactually the same length, and afterwards we hauled 'em aboard 
the Saucy Sally, What did I do with 'em ? Well, I sold 'em to 
a feller in Rio Janeiro. And what do you suppose he done with 
'em ? He used 'em to make tyres for his motor-car — ^takes a lot to 
puncture a sea-sarpint's skin." 

" What was the length of the creature ? " asked Simon. 

" Well, each piece was equal in length to three-quarters the 
length of a piece added to three-quarters of a cable. There's a 
little puzzle for you to work out, young gentleman. How matiy 
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 Httle 
puzzle, for it may confidently be said that out of a ^Jiousand readers 
who attempt the solution not one will get it exactly right. 



105. — The Dorcas Society. 

At the close of four and a half months' hard work, the ladies of 
a certain Dorcas Society were so delighted with the completion of 
a beautiful silk patchwork quilt for the dear curate that everybody 
kissed everybody else, except, of course, the bashful young man 
himself, who only kissed his sisters, whom he had called for, to 
escort home. There were just a gross of osculations altogether. 



152 



THE CANTERBURY PUZZLES 



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 
assume that the ladies attended regularly, and I am sure that they 
all worked equally well. A mutual kiss here counts as two oscula- 

ttOQS. 

io6. — The Adventurous Snail. 

A simple version of the puzzle of the cUmbing 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 




craids up a pole 12 feet high, ascending 3 feet every day and slip- 
ping 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 i foot in every 
twenty-four hours. But the modem infant in arms is not taken 
in in this way. He says, correctly enough, that at the end of the 



MISCELLANEOUS PUZZLES 153 

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 HeUx Aspersa 
family, a pioneer in mountaineering, in the act of making the 
perilous ascent of a wall 20 feet high. Judging by the trail, the 
gentleman calculated that the snail ascended 3 feet each day, 
sleeping and slipping back 2 feet every night. 

" Pray tell me," said the philosopher to his friend, who was in 
the same Une 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 cUmbing up, and 
sleeping and shpping 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 con- 
sulting with his advisers he decided not to exile the princes, but to 
confine them to the four comers of the country, where each should 
be given a triangular territory of equal area, beyond the boundaries 
of which they would pass at the cost of their lives. Now, the 
royal surveyor found himself confronted by great natural diffi- 
culties, owing to the wild character of the country. The result 
was that while each was given exactly the same area, the four tri- 



,54 THE CANTERBURY PUZZLES 

angular districts were all of different shapes, somewhat in the manner 
shown in the illustration. The puzzle is to give the three measure- 




ments 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. 

io8. — Plato and the Nines. 

Both in ancient and in modem times the number nine has been 
considered to possess pecuHarly mystic qualities. We know, for 
instance, that there were nine Muses, nine rivers of Hades, and 
that Vulcan was nine days falHng down from heaven. Then it 
has been confidently held that nine tailors make a man ; while 
we know that there are nine planets, nine days* wonders, and that 
a cat has nine fives — and sometimes nine tails. 

Most people are acquainted with some of the curious properties 
of the number nine in ordinary arithmetic. For example, write 
do^^Ti a number containing as many figures as you Hke, 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 



MISCELLANEOUS PUZZLES 



155 



strolling out to the groves of Academia to bother poor old Plato 
with hisnonsensical ideas about what he called his " lucky number.'* 
But Plato devised a way of getting rid of him. When the seer one 
day proposed to inflict on him a lengthy disquisition on his favourite 
topic, the philosopher cut him short with the remark, '* Look here, 
old chappie " (that is the nearest translation of the original Greek 
term of familiarity) : *' when you can bring me the solution of this 
little mystery of the three nines I shall be happy to listen to your 




treatise, and, in fact, record it on my phonograph for the benefit 
of posterity." 

Plato then showed, in the manner depicted in our illustration, 
that three nines may be arranged so as to represent the number 
eleven, by putting them into the form of a fraction. The puzzle he 
then propounded was so to arrange the three nines that they will 
represent the number twenty. 

It is recorded of the old crank that, after working hard at the 
problem for nine years, he one day, at nine o'clock on the morning 
of the ninth day of the ninth month, fell down nine steps, knocked 



156 THE CANTERBURY PUZZLES 

out nine teeth, and expired in nine minutes. It will be remem- 
bered 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 cell^ 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 geta three 
in a hnc 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 per- 
fect knowledge of it, one of three things must always happen, 
(i) The first player should win ; (2) the first player should lose ; 
or (3) the game should always be drawn. Which is correct ? 

no. — Ovid's Game. 

Having examined " Noughts and Crosses," we will now con- 
sider an extension of the game that is distinctly mentioned in the 
works of Ovid. It is. in fact, the parent of " Nine Men's Morris," 
referred to by Shakespeare in A Midsummer Night's Dream (Act ii., 
Scene 2). Each player has three counters, which they play alternately 
00 to the nine points shown in the diagram, with the object of 
getting three in a line and so winning. But after the six counters 



MISCELLANEOUS PUZZLES 



«57 



are played they then proceed to move (always to an adjacent 
unoccupied point) with the same object. In the example below 
White played first, and Black has just played on point 7. It is now 
White's move, and he will undoubtedly play from 8 to 9, and then. 




whatever Black may do, he will continue with 5 to 6, and so win. 
That is the simple game. Now, if both players are equally perfect 
at the game what should happen ? Should the first player always 
win ? Or should the second player win ? Or should every game 
be a draw ? One only of these things should always occur. Which 
is it? 

III. — The Farmers Oxen. 

A child may propose a problem that a sage cannot answer. 
A farmer propounded the following question : " That ten-acre 
meadow of mine will feed twelve bullocks for sixteen weeks or 
eighteen bullocks for eight weeks. How many bullocks could I 
feed on a forty-acre field for six weeks, the grass growing regularly 
all the time ? " 

It will be seen that the sting lies in the tail. That steady 



158 THE CANTERBURY PUZZLES 

groNvth of the grass is such a reasonable point to be considered, and 
yet to some readers it will cause considerable perplexity. The 
grass is, of course, assumed to be of equal length and uniform thick- 
ness in every case when the cattle begin to eat. The difficulty is 
not so great as it appears, if you properly attack the question. 



112. — The Great Grangemoor Mystery. 

Mr. Stanton Mowbray was a very wealthy man, a reputed 
millionaire, residing in that beautiful old mansion that has figured 
so much in English history, Grangemoor Park. He was a bachelor, 
spent most of the year at home, and lived quietly enough. 

According to the evidence given, on the day preceding the night 
of the crime he received by the second post a single letter, the 
contents of which evidently gave him a shock. At ten o'clock at 
night he dismissed the servants, saying that he had some important 
business matters to look into, and would be sitting up late. He 
would require no attendance. It was supposed that after all had 
gone to bed he had admitted some person to the house, for one 
of the servants was positive that she had heard loud conversation 
at a very late hour. 

Next morning, at a quarter to seven o'clock, one of the man- 
servants, on entering the room, found Mr. Mowbray lying on the 
floor, shot through the head, and quite dead. Now we come to 
the curious circumstance of the case. It was clear that after the 
bullet had passed out of the dead man's head it had struck the tall 
clock in the room, right in the very centre of the face, and actually 
welded together the three hands ; for the clock had a seconds hand 
that revolved round the same dial as the hour and minute hands. 
But although the three hands had become welded together exactly 
as they stood in relation to each other at the moment of impact, 
yet they were free to revolve round the swivel in one piece, and 
had been stupidly spun round several times by the servants before 
Mr. Wiley Slyman was called upon the spot. But they would not 
move separately. 



MISCELLANEOUS PUZZLES 



159 



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 morn- 
ing he went away by train. If the crime took place after his de- 
parture, 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 
dock 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 : 



i6o THE CANTERBURY PUZZLES | 

** 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 instan- 
taneously, as they are free on the swivel, they would swing round 
of themselves into equilibrium. Give me a few moments, and I 
can tell you beyond any doubt the exact time that the pistol was 
fired." 

Mr. Wiley Slyman took from his pocket a notebook, and began 
to figure it out. In a few minutes he handed the police inspector 
a slip of paper, on which he had written the precise moment of 
the crime. The stranger was proved to be an old enemy of Mr. 
Mowbray's, was convicted on other evidence that was discovered ; 
but before he paid the penalty for his wicked act, he admitted that 
Mr. Slyman's statement of the time was perfectly correct. 

Can you also give the exact time ? 

113. — Cutting a Wood Block. 

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

114. — The Tramps and the Biscuits, 

Four merry tramps bought, borrowed, found, or in some othi 
manner obtained possession of a box of biscuits, which they agree 
to divide equally amongst themselves at breakfast next morning. 
In the night, while the others were fast asleep under the greenwood 



MISCELLANEOUS PUZZLES 



i6i 



tree, one man approached the box, devoured exactly a quarter of 
the number of biscuits, except the odd one left over, which he 
threw as a bribe to their dog. Later in the night a second man 
awoke and hit on the same idea, taking a quarter of what remained 
and giving the odd biscuit to the dog. The third and fourth men 
did precisely the same in turn, taking a quarter of what they found 




and giving the odd biscuit to the dog. In the morning they divided 
what remained equally amongst them, and again gave the odd 
biscuit to the animal. Every man noticed the reduction in the 
contents of the box, but, believing himself to be alone responsible, 
made no comments. What is the smallest possible number of 
biscuits that there could have been in the box when they first 
acquired it ? 



(2,077) 



11 



SOLUTIONS 



THE CANTERBURY PUZZLES 

I. — 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 

4 

5 


1234567 
I 3 6 10 15 21 28 
I 4 10 20 35 56 84 


Natural Numbers. 
Triangular Numbers. 
Triangular Pyramids. 




Number of Moves. 




3 
4 

5 


I 3 7 15 31 63 127 
I 5 17 49 129 321 769 
X 7 31 III 351 1023 2815 





The first row contains the natural numbers. The second row is 
found by adding the natural numbers together from the beginning. 
The numbers in the third row are obtained by adding together the 
numbers in the second row from the beginning. The fourth row 
contains the successive powers of 2, less i. 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 tow is obtained in the same way. This table will at once 
give solutions for any number of cheeses with three stools, for 



i64 THE CANTERBURY PUZZLES 

triangular numbers with four stools, and for pyramidal numbers 
with five stools. In these cases there is always only one method 
of solution — ^that is, of piling the cheeses. 

In the case of three stools, the first and fourth rows tell us that 
4 cheeses may be removed in 15 moves, 5 in 31, 7 in 127. The 
second and fifth rows show that, with four stools, 10 may be 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 iii moves, 
and 35 cheeses 351 moves. But we also learn from the table the 
necessary method of piling. Thus, with four stools and 10 cheeses, 
the previous column shows that we must make piles of 6 and 3, 
which will take 17 and 7 moves respectively — that is, we first pile 
the six smallest cheeses in 17 moves on one stool ; then we pile 
the next 3 cheeses on another stool in 7 moves ; then remove the 
largest cheese in i move ; then replace the 3 in 7 moves ; and 
finally replace the 6 in 17 : making in all the necessary 49 moves. 
Similarly we are told that with five stools 35 cheeses must form 
piles of 20, 10, and 4, which will respectively take iii, 49, and 15 
moves. 

If the number of cheeees in the case of four stools is not tri- 
angular, and in the case of five stools pyramidal, then there will 
be more than one way of making the piles, and subsidiary tables 
will be required. This is the case with the Reve's 8 cheeses. But 
I will leave the reader to work out for himself the extension of 
the problem. 

2. — The Pardoner's Puzzle, 

The diagram on page 165 will show how the Pardoner started 
from the large black town and visited all the other towns once, 
and once only, in fifteen straight pilgrimages. 

Sec No. 320, " The Rook's Tour," in A. in M. 

Z.—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 



[ >-[}-«-« 



[MUMB— H- 



SOLUTIONS 

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a-B 



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e B-£i-a-ffl 



[MM] 



be moved. There are just three other ways in which they might 
have been arranged (4, 39, 156, 78, 2 ; or 3, 58, 174, 29, 6 ; or 6, 29, 
174, 58, 3), but they all require the moving of seven sacks. 



®®©®0(S)®®000 

®©o®ooooooo 
©®®®ooooooo 

®®0®® 0® 0000 




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 comer. How this 



i66 THE CANTERBURY PUZZLES 

result is achieved may be realized by reference to the accompany- 
ing diagram :— Join A, B, C, and D, and there are 66 squares of 
this size to be formed ; the size A, E, F, G gives 48 ; A, H, I, J, 
32 ; B. K, L, M, 19 ; B, N, O, P, 10 ; B, Q, R, S, 4 ; E, T, F, C, 57 ; 
I. U. V, P, 33 ; H, W, X, J. 15 ; K, Y, Z, M. 3 ; E, a, b, D, 82 ; 
H, d, M, D, 56 ; H, e, f , G, 42 ; K, g. f, C, 32 ; N, h, z, F, 24 ; 
K, h, m, b, 14 ; K, O, S, D, 16 ; K, n, p, G, 10 ; K, q, r, J, 6 ; 
Q, t, p, C, 4 ; Q, u, r, i, 2. The total number is thus 575. These 
groups have been treated as if each of them represented a different 
sized square. This is correct, with the one exception that the 
squares of the form B, N, O, P are exactly the same size as those 
of the form K, h, m, b. 



5.-7^ Wife of Bath's RiddUs. 

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

e.'-The Host's Puzzle. 

The puzzle propounded by the jovial host of the " Tabard " Inn 
of Southwark had proved more popular than any other of the 
whole collection. " I see, my merry masters," he cried, " that I 
have sorely twisted thy brains by my little piece of craft. Yet it is 
but a simple matter for me to put a true pint of fine old ale in each 
of these two measures, albeit one is of five pints and the other of 
three pints, without using any other measure whatever," 

The host of the " Tabard " Inn thereupon proceeded to explain 
to the pilgrims how this apparently impossible task could be done. 
He first filled the 5-pint and 3-pint measures, and then, turning the 
tap, allowed the barrel to run to waste — a proceeding against which 



SOLUTIONS 



167 



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 ; trans- 
ferred the two pints from the 5-pint to the 3-pint ; filled the 5-pint 
from the barrel, leaving one pint now in the barrel ; filled 3-pint 
from 5-pint ; allowed the company to drink the contents of the 
3-pint ; filled the 3-pint from the 5-pint, leaving one pint now in 
the 5-pint ; drank the contents of the 3-pint ; and finally drew off 
one pint from the barrel into the 3-pint. He had thus obtained the 
required one pint of ale in each measure, to the great astonishment 
of the admiring crowd of pilgrims. 

7. — Clerk of Oxenford's Puzzle. 

The illustration shows how the square is to be cut into four 
pieces, and how these pieces are to be put together again to make 



a magic square. It will be found that the four columns, four rows, 
and two long diagonals now add up to 34 in every case. 



8. — The Tapiser's Puzzle. 

The piece of tapestry had to be cut along the lines into three 
pieces so as to fit together and form a perfect square, with the 



i68 



THE CANTERBURY PUZZLES 



pattern properly matched. It was also stipulated in effect that one 
of the three pieces must be as small as possible. The illustration 





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shows how to make the cuts and how to put the pieces together, 
while one of the pieces contains only twelve of the little squares. 

9. — The Carpenter's Puzzle. 

The carpenter said that he made a box whose internal dimensions 
were exactly the same as the original block of wood — ^that is, 3 feet 
by I foot by i 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. 
Tlien he removed the pillar, taking great care not to lose any of 
the sand, which, on being shaken down alone in the box, filled a 
space equal to one cubic foot. This was, therefore, the quantity 
of wood that had been cut away. 



10. — The Puzzle of the Squire's Yeoman. 

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



SOLUTIONS 



iM 





■ 
1 


♦MJil JW J 








t 




■ Ji 






wA 




iiyit 


El 


■ 




pi 


ill J 




illy 


R 


§1 jlllll 


ilii 1 



II. — The Nun's Puzzle. 

As there are eighteen cards bearing the letters "CANTERBURY 
PILGRIMS," write the numbers i to i8 in a circle, as shown in 
the diagram. Then write the first letter C against i, and each 




successive letter against the second number that happens to be 
vacant. This has been done as far as the second R. If the reader 
completes the process by placing Y against 2, P against 6, I against 
10, and so on, he will get the letters all placed in the following 
order :— CYASNPTREIRMBLUIRG, which is the required arrange- 
ment for the cards, C being at the top of the pack and G at the 
bottom. 



170 THE CANTERBURY PUZZLES 

12. — The Merchant's Puzzle. 

This puzzle amounts to finding the smallest possible number that 
has exactly sixty-four divisors, counting i and the number itself as 
divisors. The least number is 7,560. The pilgrims might, there- 
fore, have ridden in single file, two and two, three and three, four 
and four, and so on, in exactly sixty-four different ways, the last 
manner being in a single row of 7,560. 

The Merchant was careful to say that they were going over a 
common, and not to mention its size, for it certainly would not 
be possible along an ordinary road ! 

To find how many different numbers will divide a given number, 
N, let N = 0^ 6^ c*" . . ., where a, b, c , , , are prime numbers. 
Then the number of divisors will be y) -h i) (^ + i) (r + i) . . ., 
which includes as divisors i and N itself. Thus in the case of 
my puzzle — 

7,560 = 2^ X 3S X 5 X 7 
Powers = 3311 
Therefore 4x4x2x2 = 64 divisors. 

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



13. — The Man of Law's Puzzle. 

The fewest possible moves for getting the prisoners into their 
dungeons in the required numerical order are twenty-six. The 
men move in the following order : — i, 2, 3, i, 2, 6, 5, 3, i, 2, 6, 5, 
3, X, 2, 4, 8, 7, I, 2, 4, 8, 7, 4, 5, 6. As there are never more than 



SOLUTIONS 



171 



one vacant dungeon to be moved into, there can be no ambiguity 
in the notation. 

The diagram may be simplified by my " buttons and string " 

A :b 




F\ 


H 


K 


M 


H 


M 


KK 





method, fully explained in A, in M., p. 230. It then takes one 
of the simple forms of A or B, and the solution is much easier. In 
A we use counters ; in B we can employ rooks on a comer of a 



chessboard. In both cases we have to get the order 

fewest possible moves. 

See also solution to No. 94. 



\^v\ 



in the 



14. — The Weavers Puzzle. 

The illustration shows clearly how the Weaver cut his square 
of beautiful cloth into four pieces of ex- 
actly the same size and shape, so that 
each piece contained an embroidered lion 
and castle unmutilated in any way. 



iS^—The Cook's Puzzle. 

There were four portions of warden 
pie and four portions of venison pasty to 
be distributed among eight out of eleven 
guests. But five out of the eleven will only eat the pie, four will only 




172 THE CANTERBURY PUZZLES 

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

16. — The Somfynour's Puzzle. 

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

17. — The Monk's Puzzle. 

The Monk might have placed dogs in the kennels in two thou- 
sand 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 cs long as the Monk kept his animals 
within these limits the thing was always possible. 

The general solution to this puzzle is difficult. I find that 



SOLUTIONS 



173 



for n dogs on every side of the square, the number of different 

ways IS — ! -^-5 — ■ -L-^^ where n is odd, and 

45 

— *-^ ! + I, where n is even, if we count only those 

40 

arrangements that are fundamentally different. But if we count 

all reversals and reflections as different, as the Monk himself did, 

then n dogs (odd or even) may be placed in-^''^^^'+ ^4^'"^ ^?^-f i 

6 

ways. In order that there may be n dogs on every side, the number 

must not be less than 2m nor greater than 4«, but it may be any 

number within these limits. 

An extension of the principle involved in this puzzle is given in 

No. 42, " The Riddle of the Pilgrims." See also " The Eight Villas " 

and " A Dormitory Puzzle " in A. in M, 

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




19. — The Puzzle of the Prioress. 

The Abbot of Chertsey was quite correct. The curiously- 
shaped cross may be cut into four pieces that will fit together and 



174 THE CANTERBURY PUZZLES 

form a perfect square. How this is done is shown in the illustra- 
tion. 

See also p. 31 in i4. in M. 

20. — The Puzzle of the Doctor of Physic. 

Here we have indeed a knotty problem. Our text-books tell 
us that all spheres are similar, and that similar solids are as the 
cubes of corresponding lengths. Therefore, as the circumferences 
of the two phials were one foot and two feet respectively and the 
cubes of one and two added together make nine, what we have to 
find is two other numbers whose cubes added together make nine. 
These numbers clearly must be fractional. Now, this little ques- 
tion has really engaged the attention of learned men for two hun- 
dred and fifty years ; but although Peter de Fermat showed in the 
seventeenth century how an answer may be found in two fractions 
with a denominator of no fewer than twenty-one figures, not only 
are all the published answers, by his method, that I have seen 
inaccurate, but nobody has ever pubHshed the much smaller result 
that I now print. The cubes of Jii|^?|J|U^ and |^;?|f|||08 
added together make exactly nine, and therefore these fractions 
of a foot are the measurements of the circumferences of the two 
phials that the Doctor required to contain the same quantity of 
liquid as those produced. An eminent actuary and another cor- 
respondent have taken the trouble to cube out these numbers, and 
they both find my result quite correct. 

If the phials were one foot and three feet in circumference 
respectively, then an answer would be that the cubes of iTTliKJ 
and ai'lSigg added together make exactly 28. See also No. 61, 
" The Silver Cubes." 

Given a known case for the expression of a number as the sum 
or difference of two cubes, we can, by formula, derive from it an 
infinite number of other cases alternately positive and negative. 
Thus Fermat, starting from the known case i^ -f 2^ = 9 (which we 
will call a fundamental case), first obtained a negative solution in 



SOLUTIONS 



175 



bigger figures, and from this his positive solution in bigger figures 
still. But there is an infinite number of fundamentals, and I found 
by trial a negative fundamental solution in smaller figures than 
his derived negative solution, from which I obtained the result 
shown above. That is the simple explanation. 

We can say of any number up to 100 whether it is possible or 
not to express it as the sum of two cubes, except 66. Students 
should read the Introduction to Lucas's Theorie des Nombres, 

p. XXX. 

Some years ago I published a solution for the case of 



(i)"+(S)" 



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




21. — The Ploughman's Puzzle, 

The illustration shows how the sixteen trees might have been 
planted so as to form as many as fifteen straight rows with four 
trees in every row. This is in excess of what was for a long time 



176 



THE CANTERBURY PUZZLES 



believed to be the maximum number of rows possible ; and though 
with our present knowledge I cannot rigorously demonstrate that 
fifteen rows cannot be beaten, I have a strong " pious opinion " 
that it is the highest number of rows obtainable. 

22. — The Franklin* s Puzzle. 
The answer to this puzzle is shown in the illustration, where 
the numbers on the sixteen bottles all add up to 30 in the ten 




straight directions. The trick consists in the fact that, although 
the six bottles (3, 5, 6, 9, 10, and 15) in which the flowers have 
been placed are not removed, yet the sixteen need not occupy 
exactly the same position on the table as before. The square is. 
In fact, formed one step further to the left. 

23. — The Squire's Puzzle. 

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



SOLUTIONS 



177 



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 

(2.077) 12 



178 THE CANTERBURY PUZZLES 

look for some loophole in the«actual conditions as they were worded. 
If the Parson could get round -the source of the river, he could then 
cross every bridge once and once only on his way to church, as 
shown in the annexed illustration. That this was not prohibited 
we shall soon find. Though the plan showed all the bridges in 
his parish, it only showed " part of " the parish itself. It is not 
stated that the river did not take its rise in the parish, and since 
it leads to the only possible solution, we must assume that it did. 
The answer would be, therefore, as shown. It should be noted 
that we are clearly prevented from considering the possibility of 
getting round the mouth of the river, because we are told it " joined 
the sea some hundred miles to the south," while no parish ever 
extended a hundred miles I 

26. — The Haberdasher's Puzzle. 

The illustration will show how the triangular piece of cloth may 
be cut into four pieces that will fit together and form a perfect 




square. Bisect AB in D and BC in E ; produce the line AE 
to F making EF equal to EB ; bisect AF in G and describe the 



SOLUTIONS 



179 



arc AHF; produce EB to H, and EH is the length of the side 
of the required square ; from E with distance EH, describe the 
arc HJ, and make JK equal to BE ; now, from the points D 
and K drop perpendiculars on EJ at L and M. H you have 
done this accurately, you will now have the required directions for 
the cuts. 

I exhibited this problem before the Royal Society, at Burlington 
House, on 17th May 1905, and also at the Royal Institution in the 
following month, in the more general form : — *' A New Problem on 






Superposition : a demonstration that an equilateral triangle can 
be cut into four pieces that may be reassembled to form a square, 
with some examples of a general method for transforming all 
rectilinear triangles into squares by dissection." It was also issued 
as a challenge to the readers of the Daily Mail (see issues of ist 
and 8th February 1905), but though many hundreds of attempts 
were sent in there was not a single solver. Credit, however, is due 
to Mr. C. W. M'Elroy, who alone sent me the correct solution when 
I first published the problem in the Weekly Dispatch in 1902. 

I add an illustration showing the puzzle in a rather curious 



l8o THE CANTERBURY PUZZLES 

practical form, as it was made in polished mahogany with brass 
hinges for use by certain audiences. It will be seen that the four 
pieces form a sort of chain, and that when they are closed up in 
one direction they form the triangle, and when closed in the other 
direction they form the square. 

27. — The Dyer's Puzzle. 

The correct answer is 18,816 different ways. The general 
formula for six fieurs-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 
fleiu^-de-lys in the side of the square. Of course where n is even 
the remainders in rows and columns will be even, and where n is 
odd the remainders will be odd. 

For further solution, see No. 358 in A. in M. 

28. — The Great Dispute between the Friar and the Sompnour. 

In this little problem we attempted to show how, by sophistical 
reasoning, it may apparently be proved that the diagonal of a 
square is of precisely the same length as two of the sides. The 
puzzle was to discover the fallacy, because it is a very obvious 
fallacy if we admit that the shortest distance between two points 
is a straight line. But where does the error come in ? 

Well, it is perfectly true that so long as our zigzag path is 
formed of '* steps " parallel to the sides of the square that path must 
be of the same length as the two sides. It does not matter if you 
have to use the most powerful microscope obtainable ; the rule is 
always true if the path is made up of steps in that way. But 
the error lies in the assumption that such a zigzag path can ever 
become a straight line. You may go on increasing the number 
of steps infinitely — that is, there is no limit whatever theoretically 
to the number of steps that can be made — but you can never reach 
a straight line by such a method. In fact it is just as much 
a " jump " to a straight line if you have a billion steps as it is at 



SOLUTIONS i8i 

the very outset to pass from the two sides to the diagonal. It 
would be just as absurd to say we might go on dropping marbles into 
a basket until they become sovereigns as to say we can increase 
the number of our steps until they become a straight line. There 
is the whole thing in a nutshell. 



29. — Chaucer's Puzzle. 

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

30. — The Puzzle of the Canon's Yeoman. 

The number of different ways is 63,504. The general formula 
for such arrangements, when the number of letters in the sentence 
is 2M + I, and it is a palindrome without diagonal readings, is 

[4(2- - 1)]^. 

I think it will be well to give here a formula for the general 
solution of each of the four most common forms of the diamond- 
letter puzzle. By the word " line " I mean the complete diagonal. 
Thus in A, B, C, and D, the lines respectively contain 5, 5, 7, and 9 
letters. A has a non-palindrome line (the word being BOY), and 
the general solution for such cases, where the line contains 2n-\- 1 
letters, is 4(2" — i). Where the line is a single palindrome, with 
its middle letter in the centre, as in B, the general formula is 
[4(2" — i)Y. This is the form of the Rat-catcher's Puzzle, and 
therefore the expression that I have given above. In cases C and 
D we have double palindromes, but these two represent very 



iSa 



THE CANTERBURY PUZZLES 



different types. In C, where the line contains 4« — i letters, the 
general expression is 4(2*" — 2). But D is by far the most diffi- 
cult case of all. 

I had better here state that in the diamonds under consideration 
(i.) no diagonal readings are allowed — these have to be dealt with 
specially in cases where they are possible and admitted ; (ii.) 
readings may start anywhere ; (iii.) readings may go backwards 
and forwards, using letters more than once in a single reading, but 
not the same letter twice in immediate succession. This last con- 
dition will be understood if the reader glances at C, where it is 
impossible to go forwards and backwards in a reading without 
repeating the first O touched — a proceeding which I have said is 
not allowed. In the case D it is very different, and this is what 
accounts for its greater difficulty. The formula for D is this : 



(« + 



5) X 2-+* 4- (2"+' X ' ^ 3 X 5 ^ 



\z 



('»-') ' 



,*t + 4 



where the number of letters in the line is 4n + i. In the example 
given there are therefore 400 readings forn = 2. 
See also Nos. 256, 257, and 25^; in A. in M, 



Y 

YOY 

YOBOY 

YOY 

Y 



B 



L 

LEL 

LEVEL 

LEL 

L 



N 

NON 

NOOON 

NOONOON 

NOOON 

NON 

N 



L 

LEL 

LEVEL 

LEVEVEL 

LEVELEVEL 

LEVEVEL 

LEVEL 

LEL 

L 



SOLUTIONS 183 

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 a loaf. There- 
fore, as the Miller provided ^ and ate f , he contributed i to the 
Manciple's meal ; whereas the Weaver provided |, ate {, and con- 
tributed only I. Therefore, since they contributed to the Manciple 
in the proportion of 7 to i, they must divide the eight pieces of 
money in the same proportion. 



PUZZLING TIMES AT SOLVAMHALL CASTLE 

SIR HUGH EXPLAINS HIS PROBLEMS 

The friends of Sir Hugh de Fortibus were so perplexed over 
many of his strange puzzles that at a gathering of his kinsmen and 
retainers he undertook to explain his posers. 

" Of a truth," said he, " some of the riddles that I have put 




forth would greatly tax the wit of the unlettered knave to rede ; 
yet will I try to show the manner thereof in such way that^all may 
have understanding. For many there be who cannot of themselves 



IM 



SOLUTIONS 185 

do all these things, but will yet study them to their gain when they 
be given the answers, and will take pleasure therein." 



32. — The Game of Bandy-Ball. 

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

33. — Tilting at the Ring. 

"By my halidame ! " exclaimed Sir Hugh, '* if some of yon 
varlets had been put in chains, which for their sins they do truly 



deserve, then would they well know, mayhap, that the length of 
any chain having like rings is equal to the inner width of a ring 
multiplied by the number of rings and added to twice the thickness 
of the iron whereof it is made. It may be shown that the inner 
width of the rings used in the tilting was one inch and two-thirds 



1 86 THE CANTERBURY PUZZLES 

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 i| in. x 3 -f i in. « 6 in., 
and 1} in. X 9 + I in. *= 16 in. Thus De Goumay beat Malet 
by six rings. The drawing showing the rings may assist the reader 
in verifying tht answer and help him to see why the inner width of 
a link multiplied by the number of links and added to twice the 
thickness of the iron gives the exact length. It will be noticed that 
every link put on the chain loses a length equal to twice the thick- 
ness 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 I 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 



SOLUTIONS 



187 



the outside of the dungeon, so we must perforce start therefrom. 
Marry, then, my masters, the noble demoiselle must needs have 
been wasting in the other." 

The drawing will make this quite clear to the reader. The 
two ** odd cells " are indicated by the stars, and one of the many 
routes that will solve the puzzle is shown by the dotted line. It is 
perfectly certain that you must start at the lower star and end at 
the upper one ; therefore the cell with the star situated over the 
left eye must be the one sought. 

35. — The Archery Butt. 

" It hath been said that the proof of a pudding is ever in the 
eating thereof, and by the teeth of Saint George I know no better 




way of showing how this placing of the figures may be done than 
by the doing of it. Therefore have I in suchwise written the num- 



i88 



THE CANTERBURY PUZZLES 



bers that they do add up to twenty and three in all the twelve lines 
of three that are upon the butt." 

I tliink it well here to supplement the solution of De Fortibus 
with a few remarks of my own. The nineteen numbers may be 
so arranged that the lines will add up to any number we may 
choose to select from 22 to 38 inclusive, excepting 30. In some 
cases there are several different solutions, but in the case of 23 
there ^e only two. I give one of these. To obtain the second 
solution exchange respectively 7, 10, 5, 8, 9, in the illustration, 
with 13, 4, 17, 2, 15. Also exchange 18 with 12, and the other 
numbers may remain unmoved. In every instance there must be 
an even number in the central place, and any such number from 
2 to 18 may occur. Every solution has its complementary. Thus, 
if for every number in the accompanying drawing we substitute 
the difference between it and 20, we get the solution in the case of 
37. Similarly, from the arrangement in the original drawing, we 
may at once obtain a solution for the case of 38. 




36. — The Donjon Keep Window. 

In this case Sir Hugh had greatly perplexed his chief builder 
by demanding that he should make a window measuring one foot 
on every side and divided by bars into eight lights, having all 
their sides equal. The illustration will show how this was to be 



SOLUTIONS 



189 



done. It will be seen that if each side of the window measures 
one foot, then each of the eight triangular lights is six inches on 
every side. 

** Of a truth, master builder," said De Fortibus slyly to the 
architect, ** I did not tell thee that the window must be square, as 
it is most certain it never could be." 



37. — The Crescent and the Cross. 

" By the toes of St. Moden," exclaimed Sir Hugh de Fortibus 
when this puzzle was brought up, " my poor wit hath never shaped 
a more cunning artifice or any more bewitching to look upon. It 
came to me as in a vision, and ofttimes have I marvelled at the 



1 



z 


3 


v^^^^fe^ 


^f 



thing, seeing its exceeding difficulty. My masters and kinsmen, 
it is done in this wise." 

The worthy knight then poirrted out that the crescent was of 
a particular and somewhat irregular form — the two distances atob 
and c to ^ being straight lines, and the arcs ac and bd being pre- 
cisely similar. He showed that if the cuts be made as in Figure I, 
the four pieces will fit together and form a perfect square, as shown 
in Figure 2, if we there only regard the three curved lines. By 
now making the straight cuts also shown in Figure 2, we get the 
ten pieces that fit together, as in Figure 3, and form a perfectly 
symmetrical Greek cross. The proportions of the crescent and 



190 THE CANTERBURY PUZZLES 

the cross in the original illustration were correct, and the solution 
con be demonstrated to be absolutely exact and not merely ap- 
proximate. 

I have a solution in considerably fewer pieces, but it is far 
more difl&cult to understand than the above method, in which the 
problem is simplified by introducing the intermediate square. 

ZS—The Amulet. 

The puzzle was to place your pencil on the A at the top of th« 
amulet and count in how many different ways you could trace out 
the word " Abracadabra " downwards, alwa3is passing from a 
letter to an adjoining one. 

B B 

R R R 

A A A A 

C C C C C 

A A A A A A 

D D D D D D D 

AAAAAAAA 

BBBBBBBBB 

RRRRRRRRRR 

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, 1024, is the answer 
thou dost seek." 

39. — The Snail on the Flagstaff. 

Though there was no need to take down and measure the staff, 
it is undoubtedly necessary to find its height before the answer 



SOLUTIONS 191 

can be given. It was well known among the friends and retainers 
of Sir Hugh de Fortibus that he was exactly six feet in height. 
It will be seen in the original picture that Sir Hugh's height is just 
twice the length of his shadow. Therefore we all know that the 
flagstaff will, at the same place and time of day, be also just twice 
as long as its shadow. The shadow of the staff is the same length 
as Sir Hugh's height ; therefore this shadow is six feet long, and 
the flagstaff must be twelve feet high. Now, the snail, by climbing 
up three feet in the daytime and slipping back two feet by night, 
really advances one foot in a day of twenty-four hours. At the 
end of nine days it is three feet from the top, so that it reaches its 
journey's end on the tenth day. 

The reader will doubtless here exclaim, " This is all very well ; 
but how were we to know the height of Sir Hugh ? It was never 
stated how tall he was I " No, it was not stated in so many words, 
but it was none the less clearly indicated to tl>e reader who is sharp 
in these matters. In the original illustration to the donjon keep 
window Sir Hugh is shown standing against a wall, the window in 
which is stated to be one foot square on the inside. Therefore, as 
his height will be found by measurement to be just six times the 
inside height of the window, he evidently stands just six feet in 
his boots I 

40. — Lady Isabel's Casket. 

The last puzzle was undoubtedly a hard nut, but perhaps diffi- 
culty 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 
j 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 
jonly possible solution, and it is a singular fact (though I cannot 
i here show the subtle method of working) that the number, sizes, 
and order of those squares are determined by the given dimensions 
of the strip of gold, and the casket can have no other dimensions 
than 20 inches square. The number in a square indicates the length 



192 



THE CANTERBURY PUZZLES 



in inches of the side of that square, so the accuracy of the answer 
can be checked almost at a glance. 

Sir Hugh de Fortibus made some general concluding remarks 
on the occasion that are not altogether uninteresting to-day. 



20 




ri 



m 



10 X i 



IZ 





20 



" Friends and retainers," he said, " if the strange offspring 
my poor wit about which we have held pleasant counsel to-nigl 
hath mayhap had some small interest for ye, let these mattei 
serve to call to mind the lesson that our fleeting life is rounded an^ 
beset with enigmas. Whence we came and whither we go be riddles,' 
and albeit such as these we may never bring within our under- 
standing, yet there be many others with which we and they that 



SOLUTIONS 193 

do come after us will ever strive for the answer. Whether success 
do attend or do not attend our labour, it is well that we make the 
attempt ; for 'tis truly good and honourable to train the mind, and 
the wit, and the fancy of man, for out of such doth issue all manner 
of good in ways unforeseen for them that do come after us." 



(2.077) 18 



THE MERRY MONKS OF RIDDLEWELL 

41. — The Riddle of the Fish-pond. 

Number the fish baskets in the illustration from i to 12 in the 
direction that Brother Jonathan is seen to be going. Starting 
from I, proceed as follows, where " x to 4 " means, take the fish 
from basket No. i and transfer it to basket No. 4 : — 

I to 4, 5 to 8, 9 to 12, 3 to 6, 7 to 10, II to 2, and complete 
the last revolution to i, making three revolutions in all. Or you 
can proceed this way : — 

4 to 7, 8 to II, 12 to 3, 2 to 5, 6 to 9, ID to I. 

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 tht 
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 remcnibcrcd, it was possible to accommodate under the 

104 



SOLUTIONS 



195 



conditions of the Abbot) must have been 27, and that, since three 
more than this number were actually provided with beds, the total 
number of pilgrims was 30. The accompanying diagram shows 




2 


T 
J- 


1 


T 


I 


1 


1 




^M 


1 


■B ■ 


T 






■ ■• 


1 


1 
J. 


I 


± 


1 



8 Rooms on Lower Floor 



dRoorns on Upper Floor 




6 Roomi oi\-L<ma Fiopr. 



how they might be arranged, and if in each instance we regard the 
upper floor as placed above the lower one, it will be seen that there 
are eleven persons on each side of the building, and twice as many 
above as below. 

43. — The Riddle of the Tiled Hearth. 

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



196 



THE CANTERBURY PUZZLES 




44. — The Riddle of the Sack of Wins, 

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-fiiths of 
a quarter of a pint of wine — that is, one-fifth of a pint — while wc 
transfer from the jug to the bottle an equal quantity (one-fifth of 
a pint) of water. 

45. — The Riddle of the Cellarer, 

T^iere 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 



SOLUTIONS 197 

99 pints ; after the second theft the wine in the cask would be 
Vtht pints (the square of 99 divided by 100) ; after the third theft 
there would remain -YATnf ("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 Dy 
the twenty-ninth power of lool 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 73yV^ pints of wine left in the cask. Consequently the 
cellarer stole nearly 26.03 pints. The monks doubtless omitted 
the answer for the reason that they had no tables of logarithms, 
and did not care to face the task of making that long and tedious 
calculation in order to get the quantity " to a nicety," as the wily 
cellarer had stipulated. 

By a simplified process of calculation, I have ascertained that 
the exact quantity of wine stolen would be 

26.029962661171957726998490768328505774732373764732355565- 

2999 

pints. A man who would involve the monastery in a fraction of 
fifty-eight decimals deserved severe punishment. 

46. — The Riddle of the Crusaders. 

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

47. — The Riddle of St. Edmondshury. 

The reader is aware that there are prime numbers and compo- 
site whole numbers. Now, i,iii,iii cannot be a prime number, 



198 THE CANTERBURY PUZZLES 

because if it were the only possible answers would be those proposed 
b}^ 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, i, 111,111 equals 239 x 4649 (both primes), and 
smce each cat killed more mice than there were cats, the answer 
must be 239 cats. See also the Introduction, p. 18. 

Treated generally, this problem consists in finding the factors, 

if any, of numbers of the form . 

Lucas, in his L' Arithmeiique Amusante, gives a number of 
curious tables which he obtained from an arithmetical treatise, 
called the Talkhys, by Ibn Albanna, an Arabian mathematician 
and astronomer of the first half of the thirteenth century. In the 
Paris National Library are several manuscripts dealing with the 
Talkhys, and a commentary by Alkala^adi, who died in i486. 
Among the tables given by Lucas is one giving all the factors of 
numbers of the above form up to m=i8. It seems almost incon- 
ceivable that Arabians of that date could find the factors where 
« = 17, as given in my Introduction. But I read Lucas as stating 
that they are given in Talkhys, though an eminent mathema- 
tician reads him differently, and suggests to me that they were 
discovered by Lucas himself. This can, of course, be settled by 
an examination of Talkhys, but this has not been possible during 
the war. 

The difficulty lies wholly with those cases where m is a prime 
number. If « = 2, we get the prime 11. The factors when « = 3, 5, 
II, and 13 are respectively (3 . 37), (41 . 271), (21,649 . 513,239), and 
(53 • 79 • 265371653). I have given in these pages the factors where 
n = 7 and 17. The factors when » = I9, 23, and 37 are unknown, if 
there are any.* When n = 29, the factors are (3,191 . 16,763 . 43,037 . 

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



SOLUTIONS 199 

62,003 . 77.843,839>397) '> when n = 31, one factor is 2,791 ; and when 
n = 4i, two factors are (83 . 1,231). 

As for the even values of n, the following curious series of 
factors will doubtless interest the reader. The numbers in brackets 
are primes. 

n= 2 = (11) 

n= 6 = (ii)xiii X91 

tJ = 10= (11) X II, III X (9,091) 

w = 14= (11) X 1, 111,111 X (909,091) 

«=i8=(ii) X 111,111,111 X 90,909,091 

Or we may put the factors this way : — 

n= 2 = (11) 

w= 6 = 111 X 1,001 

w = 10 = 11, III X 100,001 

ti = 14=1,111,111 X 10,000,001 

M = 18 = 111,111,111 X 1,000,000,001 

In the above two tables n is of the form 4W + 2. When n is of 
the form /[m the factors may be written down as follows : — 

n= 4=(ii)x(ioi) 

n= 8 = (11) X (loi) X 10,001 

M = 12 = (11) X (lOl) X 100,010,001 

n = i6 = (ii) X (loi) X 1,000,100,010,001. 

When n = 2, we have the prime number 11; when M = 3, the 
factors are 3 . 37 ; when w = 6, they are 11 . 3 . 37 . 7 . 13 ; when 
M = 9, they are 3^^ . 37 . 333,667. Therefore we know that factors of 
n = 18 are 11 . 3^ . 37 . 7 . 13 . 333,667, while the remaining factor is 
composite and can be split into 19 . 52579. This will show how the 
working may be simplified when n is not prime. 

^S.—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 



200 THE CANTERBURY PUZZLES 

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 posi- 
tion, as you never can have an alternative. The moves enclosed 
in brackets are to be played five times over : 6, 7, 8, 6, 5, 4, 7, 8, 
9. 10, 6, 5, 4, 3, 2, 7, 8, 9, 10, II (6, 5, 4, 3, 2, I), 6, 5, 4, 3, 2, 12, 
(7, 8, 9, 10, II, 12), 7. 8, 9, 10, II, I, 6, 5, 4, 3, 2, 12, 7, 8, 9, 10, II, 
^, 5, 4, 3, 2, 8, 9, 10, II, 4, 3, 2, 10, II, 2. We thus have made 118 
moves within the conditions, the black frogs have changed places 
with the white ones, and i and 12 are side by side in the positions 
stipulated. 

The general solution in the case of this puzzle is 3n' + 2w-2 
moves, where the number of frogs of each colour is n. The law 
governing the sequence of moves is easily discovered by an ex- 
amination of the simpler cases, where w = 2, 3, and 4. 

If, instead of 11 and 12 changing places, the 6 and 7 must 
interchange, the expression is n^ + ^n-h2 moves. If we give n the 
value 6, as in the example of the Frogs' Ring, the number of moves 
would be 62. 

For a general solution of the case where frogs of one colour 
reverse their order, leaving the blank space in the same position, 
and each frog is allowed to be moved in either direction (leaping, 
of course, over his own colour), see " The Grasshopper Puzzle " in 
A. in M ., p. 193. 



THE STRANGE ESCAPE OF THE 
KING^S JESTER 

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

49. — The Mysterious Rope, 

When the jester " divided his rope in half," it does not follow 
that he cut it into two parts, each half the original length of the 
rope. No doubt he simply untwisted the strands, and so divided 
it into two ropes, each of the original length, but one-half the thick- 
ness. He would thus be able to tie the two together and make a 
rope nearly twice the original length, with which it is quite con- 
ceivable 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 walk- 
ing without once removing his left hand (or right hand) from the 
wall. Starting from A, the dotted line will make the route clear 
when he goes to the left. If the reader tries the route to the right 
in the same way he will be equally successful ; in fact, the two 
routes unite and cover every part of the walls of the maze except 

those two detached parts on the left-hand side — one piece like a 

201 



202 



THE CANTERBURY PUZZLES 



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 en- 




closed by an isolated wall in the form of a split ring, the jester 
would simply have gone round and round this ring. 

See the article, " Mazes, and How to Thread Them,'* in ^4. in M. 

51. — The Secret Lock. 

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

52. — Crossing the Moat. 

No doubt some of my readers will smile at the statement that 
a m-an in a boat on smooth water can pull himself across with 
the tiller rope 1 But it is a fact. If the jester had fastened the 
end of his rope to the stem 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 MatJtematical Recreations, 



SOLUTIONS 



203 



53. — The Royal Gardens. 

This puzzle must have struck many readers as being absolutely 
impossible. The jester said : "I had, of a truth, entered every 
one of the sixteen gardens once, and never more than once." If 
we follow the route shown in the accompanying diagram, we find 
that there is no difficulty in once entering all the gardens but one 
before reaching the last garden containing the exit B. The diffi- 
culty is to get into the garden with a star, because if we leave the 
B garden we are compelled to enter it a second time before escaping, 
and no garden may be entered twice. The trick consists in the 



B 



U I. I., 

- +1+ + - 

-i+.+J+i- 



J L 



I. J 

A 



fact that you may enter that starred garden without necessarily 
leaving the other. If, when the jester got to the gateway where 
the dotted line makes a sharp bend, his intention had been to hide 
in the starred garden, but after he had put one foot through the 
doorway, upon the star, he discovered it was a false alarm and 
withdrew, he could truly say: ** I entered the starred garden, 
because I put my foot and part of my body in it ; and I did not 
enter the other garden twice, because, after once going in I never 
left it until I made my exit at B." This is the only answer possible, 
and it was doubtless that which the jester intended. 
See "The Languishing Maiden," in A. in M. 



204 



THE CANTERBURY PUZZLES 



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. 



I 



THE SQUIRE'S CHRISTMAS PUZZLE PARTY 

HOW THE VARIOUS TRICKS WERE DONE 

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



55. — The Three Teacups. 

Miss Charity Lockyer clearly must have had a trick up her 
sleeve, and I think it highly probable that it was conceived 
on the following lines. She proposed that ten lumps of sugar 
should be placed in three teacups, so that there should be an odd 




number of lumps in every cup. The illustration perhaps shows 
Miss Charity's answer, and the figures on the cups indicate the 
number of lumps that have been separately placed in them. By 
placing the cup that holds one lump inside the one that holds 
two lumps, it can be correctly stated that every cup contains an 
odd number of lumps. One cup holds seven lumps, another holds 
one lump, while the third cup holds three lumps. It is evident 



205 



2o6 THE CANTERBURY PUZZLES 

that if a cup contains another cup it also contains the contents 
of that second cup. 

There are in all fifteen different solutions to this puzzle. Here 
they are : — 



I 





9 


I 


4 


5 


9 







3 





7 


7 


o 


3 


7 


2 




I 


2 


7 


5 


2 


3 


5 


4 




5 





5 


3 


4 


3 


3 


6 




3 


2 


5 


I 


6 


3 


I 


8 





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

56. — The Eleven Pennies. 

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

^ 57. — The Christmas Geese. 

Farmer Rouse sent exactly loi geese to market. Jabez first sold 
Mr. Jasper Tyler half of the flock and half a goose over (that is, 
5oi+J, or 51 geese, leaving 50) ; he then sold Farmer Avent a 
third of what remained and a third of a goose over (that is, i6f -f- J, 
or 17 geese, leaving 33) ; he then sold Widow Foster a quarter of 
what remained and three-quarters of a goose over (that is, Sf-fi 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|-fT, or 5 geese, leaving 19). He then took these 19 back to his 
master. 

58. — The Chalked Numbers, 

Tliis little jest on the part of Major Trenchard is another trick 
puzzle, and the face of the roguish boy on the extreme right, with 



SOLUTIONS 207 

the figure 9 on his back, showed clearly that he was in the secret, 
whatever that secret might be. I have no doubt (bearing in mind 
the Major's hint as to the numbers being " properly regarded ") 
that his answer was that depicted in the illustration, where boy 
No. 9 stands on his head and so converts his number into 6. This 




makes the total 36 — an even number — and by making boys 3 and 
4 change places with 7 and 8, we get i 2 7 8 and 5346, the figures 
of which, in each case, add up to 18. There are just three other 
ways in which the boys may be grouped 1136 8—2 4 5 7, 1 4 6 7 
—2 3 5 8, and 2 3 6 7— I 4 5 

59. — Tasting the Plum Puddings. 

The diagram will show how this puzzle is to be solved. It is the 
only way within the conditions laid down. Starting at the pudding 
with holly at the top left-hand corner, we strike out all the puddings 
in twenty-one straight strokes, taste the steaming hot pudding at 
the end of the tenth stroke, and end at the second sprig of holly. 

Here we have an example of a chess rook's path that is not 
re-entrant, but between two squares that are at the greatest pos- 
sible distance from one another. For if it were desired to move, 
under the condition of visiting every square once and once only, 
from one corner square to the other corner square on the same 
diagonal, the feat is impossible. 

There are a good many different routes for passing from one 
sprig of holly to the other in the smallest possible number of moves 



208 



THE CANTERBURY PUZZLES 



— twenty-one — ^but I have not counted them. I have recorded 
fourteen of these, and possibly there are more. Any one of these 
would serve our purpose, except for the condition that the tenth 
stroke shall end at the steaming hot pudding. This was intro- 



0' a a- a a - q q -4 



a a a g^ o. cs (a ca 



a it a 



a ii a 
4 a a 



^r-a a 






■^— @ (II ^^ 

CI £1 




duced to stop a plurality of solutions — called by the maker of 
chess problems " cooks." I am not aware of more than one solu- 
tion to this puzzle ; but as I may not have recorded all the tours, 
I cannot make a positive statement on the point at the time of 
writing. 

6o. — Under the Mistletoe Bough. 

Everybody was found to have kissed everybody else once under 
the mistletoe, with the following additions and exceptions : No 
male kissed a male ; no man kissed a married woman except his 
own wife ; all the bachelors and boys kissed all the maidens and 
girls twice ; the widower did not kiss anybody, and the widows 
did not kiss each other. Every kiss was returned, and the double 
performance was to count as one kiss. In making a list of the 



SOLUTIONS 209 

company, we can leave out the widower altogether, because he 
took no part in the osculatory exercise. 

7 Married couples 14 

3 Widows 3 

12 Bachelors and Boys 12 

10 Maidens and Girls 10 

Total 39 Persons 

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

61. — The Silver Cubes. 

There is no limit to the number of different dimensions that will 
give two cubes whose sum shall be exactly seventeen cubic inches. 
Here is the answer in the smallest possible numbers. One of the 
silver cubes must measure 2f^||f inches along each edge, and the 
other must measure tJUt 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." 



(2,077) 14 



THE ADVENTURES OF THE PUZZLE CLUB 

62. — The Ambiguous Photograph. 

One by one the members of the Club succeeded in discovering 
the key to the mystery of the Ambiguous Photograph, except 
Churton, who was at length persuaded to " give it up." Herbert 
Baynes then pointed out to him that the coat that Lord Marksford 
was carrying over his arm was a lady's coat, because the buttons 
are on the left side, whereas a man's coat always has the buttons 
on the right-hand side. Lord Marksford would not be likely to 
walk about the streets of Paris with a lady's coat over his arm 
unless he was accompanying the owner. He was therefore walking 
with the lady. 

As they were talking a waiter brought a telegram to Baynes. 

** Here you are," he said, after reading the message. " A wire 
from Dovey : * Don't bother about photo. Find lady was the 
gentleman's sister, passing through Paris.' That settles it. You 
might notice that the lady was lightly clad, and therefore the coat 
might well be hers. But it is clear that the rain was only a sudden 
shower, and no doubt they were close to their destination, and she 
did not think it worth while to put the coat on." 

63. — The Cornish Cliff Mystery. 

Melville's explanation of the Cornish Cliff Mystery was very 
simple when he gave it. Yet it was an ingenious trick that the 
two criminals adopted, and it would have completely succeeded 

had not our friends from the Puzzle Club accidentally appeared on 

210 



SOLUTIONS 211 

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 back\irards to the stile, 
carrying his own boots. 

This Uttle manoeuvre accounts for the smaller footprints show- 
ing a deeper impression at the heel, and the larger prints a deeper 
impression at the toe ; for a man will walk more heavily on his heels 
when going forward, but will make a deeper impression with the 
toes in walking backwards. It will also account for the fact that 
the large footprints were sometimes impressed over the smaller 
ones, but never the reverse ; also for the circumstance that the 
larger footprints showed a shorter stride, for a man will necessarily 
take a smaller stride when walking backwards. The pocket-book 
was intentionally dropped, to lead the police to discover the foot- 
prints, and so be put on the wrong scent. 

64. — The Runaway Motor-Car. 

Russell found that there are just twelve five-figure numbers 
that have the peculiarity that the first two figures multiplied by 
the last three — all the figures being different, and there being no 
—will produce a number with exactly the same five figures, in a 
iifferent order. But only one of these twelve begins with a i — 
fiamely, 14926. Now, if we multiply 14 by 926, the result is 12964, 
which contains the same five figures. The number of the motor- 
car was therefore 14926. 

Here are the other eleven numbers : — 24651, 42678, 51246, 
57834' 75231. 78624, 87435, 72936, 65281, 65983, and 86251. 

Compare with the problems in " Digital Puzzles," section of 
A, in M., and with Nos. 93 and loi in these pages. 

65. — The Mystery of Ravensdene Park. 

The diagrams show that there are two different ways in which 
the routes of the various persons involved in the Ravensdene 



212 



THE CANTERBURY PUZZLES 



Mystery may be traced, without any path ever crossing another. 
It depends whether the butler, E, went to the north or the south 
of the gamekeeper's cottage, and the gamekeeper. A, went to the 
south or the north of the hall. But it will be found that the only 
persons who could have approached Mr. Cyril Hastings without 



1 ^^ 



Z B 




BB 




crossing a path were the butler, E, and the man, C. It was, how- 
ever, a fact that the butler retired to bed five minutes before mid- 
night, whereas Mr. Hastings did not leave his friend's house until 
midnight. Therefore the criminal must have been the man who 
entered the park at C. 



66. — The Buried Treasure. 

The field must have contained between 179 and 180 acres — to 
be more exact, 179.37254 acres. Had the measurements been 3, 
2, and 4 furlongs respectively from successive corners, then the 
field would have been 209.70537 acres in area. 

One method of solving this problem is as follows. Find the area 
of triangle APB in terms of x, the side of the square. Double 
the result =A;jy. Divide by x and then square, and we have the 
value of y'^ in terms of x. Similarly find value of z^ in terms 
of X ; then solve the equation 3^^+ 2^=32, which will come out in 
the form ^— 20^;^=— 37. Therefore x'^=io-{- 763= 17.937254 
square furlongs, very nearly, and as there are ten acres in one 
square furlong, this equals 179.37254 acres. If we take the nega- 
tive root of the equation, we get the area of the field as 20.62746 
acres, in which case the treasure would have been buried outside 



I 



SOLUTIONS 



213 



the field, as in Diagram 2. But this solution is excluded by the 
condition that the treasure was buried in the field. The words 





^ -D 

were, " The document . . . states clearly that the field is square, 
and that the treasure is buried in it/' 



THE PROFESSOR'S PUZZLES 

67. — The Coinage Puzzle. 

The point of this puzzle turns on the fact that if the magic 
square were to be composed of whole numbers adding up 15 in 
all ways, the two must be placed in one of the comers. Otherwise 
fractions must be used, and these are supplied in the puzzle by the 



45. 
64 


45. 

4s 


2s6i 


Is 


5s. 


<5k 

2s, 


5s. 
gs6d 


2s 


5i. 
6d. 



employment of sixpences and half-crowns. I give the arrange- 
ment requiring the fewest possible current English coins — fifteen. 
It will be seen that the amount in each comer is a fractional one, 
the sum required in the total being a whole number of shillings. 



68. — The Postage Stamps Puzzles, 

The first of these puzzles is based on a similar principle, though 
it is really much easier, because the condition that nine of the 



214 



SOLUTIONS 



215 



stamps must be of different values makes their selection a simple 
matter, though how they are to be placed requires a little thought 





^ 






U 




, Fl 










y 




2i 






m 









Sf 






Sd 




y 



or trial until one knows the rule respecting putting the fractions 
in the comers. I give the solution. 

I also show the solution to the second stamp puzzle. All the 



^ 










i-i 




Mi 






L] 


9i 






lioj] 


[13 


|2a| 


ill 





y 



columns, rows, and diagonals add up is. 6d. There is no stamp 
on one square, and the conditions did not forbid this omission. The 



21 6 THE CANTERBURY PUZZLES 

stamps at present in circulation are these : — Id., id., i\d., 2d., 2jd., 
3^., 4^., 5^., dd., gd., lod., is., 2s. 6d., 5s., 10s., £1, and £5. 

In the first solution the numbers are in arithmetical progression 
— I, ij, 2, 2j, 3, 3j, 4, 4j, 5. But any nine numbers will form a 
magic square if we can write them thus : — 

123 

789 
13 14 15 

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

012 

567 
10 II 12 

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

2 2i 3 

4i 5 5i 
7 7i^ 

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

— 2 —I o 

567 
12 13 14 

69. — The Frogs and Tumblers. 

It is perfectly true, as the Professor said, that there is only one 
solution (not counting a reversal) to this puzzle. The frogs that 
jump are George in the third horizontal row; Chang, the artful- 
looking batrachian at the end of the fourth row ; and Wilhelmina. 



SOLUTIONS 2,7 

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 

oo®ooooo 
oooo®ooo 

0,00000®; 
O O; O ® O O O O' 



® C'i O o 

o c! o o 

o ®" o ©' 



0000 
o o ® o 
0000 



ooooo®oo 

her youth and sex, performs the very creditable saltatory feat of 
leaping to the fourth tumbler in the fourth row. In their new 
positions, as shown in the accompanying diagram, it will be found 
that of the eight frogs no two are in line vertically, horizontally, 
or diagonally. 

70. — Romeo and Juliet. 

This is rather a difficult puzzle, though, as the Professor re- 
marked when Hawkhurst hit on the solution, it is " just one of 
those puzzles that a person might solve at a glance " by pure luck. 
Yet when the solution, with its pretty, symmetrical arrangement, 
is seen, it looks ridiculously simple. 

It will be found that Romeo reaches Juliet's balcony after 
visiting every house once and only once, and making fourteen 
turnings, not counting the turn he makes at starting. These are 



2l8 



THE CANTERBURY PUZZLES 




the fewest turnings possible, and the problem can only be solved 
by the route shown or its reversal. 

71. — Romeo's Second Journey. 

In order to take his trip through all the white squares only 
with the fewest possible turnings, Romeo would do well to adopt 




the route I have shown, by means of which only sixteen turnings 
are required to perform the feat. The Professor informs me that 



SOLUTIONS 219 

the Helix Aspersa, or common or garden snail, has a peculiar aver- 
sion to making turnings— so much so that one specimen with which 
he made experiments went off in a straight line one night and has 



never come back since. 



72. — The Frogs who would a-wooing go. 

This is one of those puzzles in which a plurality of solutions is 
practically unavoidable. There are two or three positions into 
which four frogs may jump so as to form five rows with four 



^. 



$^ ^ "^ ^'-^ ^ 



in each row, but the case I have given is the most satisfactory 
arrangement. 

The frogs that have jumped have left their astral bodies behind, 
in order to show the reader the positions which they originally 
occupied. Chang, the frog in the middle of the upper row, suffer- 
ing 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. respec- 
tively ; while the frog in the middle of the lower row, whose name 
the Professor forgot to state, goes direct S. 



MISCELLANEOUS PUZZLES 

73. — The Game of Kayles. 

To win at this game you must, sooner or later, leave your op- 
ponent an even number of similar groups. Then whatever he 
does in one group you repeat in a similar group. Suppose, for 
example, that you leave him these groups : o . o . 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 o . 00000000000, 
the first player can alwajrs win, but only by knocking down the 
sixth or tenth kayle (counting the one already fallen as the second), 
and this leaves in either case o . 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. Let us suppose that he knocks down the single one ; then 
we play to leave him 00 . 0000000. Now, whatever he does we 
can afterwards leave him either 000 . 000 or o . 00 . 000. We know 
why the former wins, and the latter wins also ; because, however 
he may play, we can always leave him either o . o, or o . o . . o, 
or 00 . 00, as the case may be. The complete analysis I can now 
leave for the amusement of the reader. 

74. — The Broken Chessboard. 

The illustration will show how the thirteen pieces can be put 
together so as to construct the perfect board, and the reverse prob- 

290 



SOLUTIONS 



221 




lem of cutting these particular pieces out will be found equally 
entertainine^. 



entertaining. 

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

75- — The Spider and the Fly, 

Though this problem was much discussed in the Daily Mail 
from i8th January to 7th February 1905, when it appeared to 
create great public interest, it was actually first propounded by 
me in the Weekly Dispatch of 14th June 1903. 

Imagine the room to be a cardboard box. Then the box may 
be cut in various different ways, so that the cardboard may be laid 
flat on the table. I show four of these ways, and indicate in every 
case the relative positions of the spider and the fly, and the straight- 
ened course which the spider must take without going off the 
cardboard. These are the four most favourable cases, and it will 
be found that the shortest route is in No. 4, for it is only 40 feet in 
length (add the square of 32 to the square of 24 and extract the 
square root). It will be seen that the spider actually passes along 
five of the six sides of the room ! Having marked the route, fold 
the box up (removing the side the spider does not use), and the 
appearance of the shortest course is rather surprising. If the 



222 



THE CANTERBURY PUZZLES 



spider had taken what most persons will consider obviously the 
shortest route (that shown in No. i), he would have gone 42 feet I 
Route No. 2 is 43.174 feet in length, and Route No. 3 is 40.718 feet. 



12 i^ 



loTT 



42 U 

FLOOR 



2 



} 



,v:^.^- 



FLOOR 









3 


..:t>-''- 


- A 


B 


'' FLOOR 













y 


A 

• 


4 


o*'"'-' 

*?-- 




B 


'' FLOOR 











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



76. — The Perplexed Cellarman. 

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



SOLUTIONS 



222 



yy. — Making a Flag. 

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




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

78. — Catching the Hogs. 

A very short examination of this puzzle game should convince 
the reader that Hendrick can never catch the black hog, and that 
the white hog can never be caught by Katriin. 

Each hog merely runs in and out of one of the nearest comers, 
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 Katriin 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 



224 THE CANTERBURY PUZZLES 

pigs : in their simplicity and ignorance of the peculiarities of 
Dutch hogs, each went after the wrong animal. 

The little principle involved in this puzzle is that known to 
chess-players as '* getting the opposition." The rule, in the case 
of my puzzle (where the moves resemble rook moves in chess, with 
the added condition that the rook may only move to an adjoining 
square), is simply this. Where the number of squares on the same 
row, between the man or woman and the hog, is odd, the hog can 
never be captured ; where the number of squares is even, a capture 
is possible. The number of squares between Hendrick and the 
black hog, and between Katriin and the white hog, is i (an odd 
number), therefore these individuals cannot catch the animals 
they are facing. But the number between Hendrick and the white 
hog, and between Katriin and the black one, is 6 (an even number), 
therefore they may easily capture those behind them. 

79. — The Thirty-one Game. 

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

80. — The Chinese Railways. 

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



SOLUTIONS 



225 



The reader may wish to know how many different solutions 
there are to the puzzle. To this I should answer that the number 
is indeterminate, and I will explain why. If we simply consider 
the case of line A alone, then one route would be Diagram 2, another 
3, another 4, and another 5. If 3 is different from 2, as it un- 
doubtedly is, then we must regard 5 as different from 4. But a 




glance at the four diagrams, 2, 3, 4, 5, in succession will show that 
we may continue this " winding up " process for ever ; and as there 
will always be an unobstructed way (however long and circuitous) 
from stations B and E to their respective main lines, it is evident 
that the number of routes for line A alone is infinite. Therefore 
the number of complete solutions must also be infinite, if railway 
lines, like other lines, have no breadth ; and indeterminate, unless 

(2,077) 15 



226 THE CANTERBURY PUZZLES 

we are told the greatest number of parallel lines that it is possible 
to construct in certain places. If some clear condition, restricting 
these '* windings up," were given, there would be no great difficulty 
in giving the number of solutions. With any reasonable limitation 
of the kind, the number would, I calculate, be little short of two 
thousand, surprising though it may appear. 

8i. — 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 con- 
ditions. In the case of our puzzle there would be no difficulty 
in making the magic square with 9 missing ; but with i 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 
.0. Now, since the recurring decimal .^ is equal to |, and there- 
fore to I, it is evident that, although the clown who bears the figure 
I 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 i. The 
troupe can consequently be grouped in the following manner : — 

7 5 

246 
3 8 .d 

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 multip 



SOLUTIONS 227 

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 16393442622950819672131 147540983- 
60655737704918032787. If you want to multiply this by 71, all 
you have to do is to place another i 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 multi- 
pliers, but the number to be multiplied may always be increased. 
Thus, if you prefix to 41096 the number 41095890, repeated any 
number of times, the result may always be multiplied by 83 in the 
wizard's peculiar manner. 

If we add the figures of any number together and then, if neces- 
sary, again add, we at last get a single-figure number. This I call 
the *' digital root." Thus, the digital root of 521 is 8, and of 697 
it is 4. This digital analysis is extensively dealt with in A. in M. 
Now, it is evident that the digital roots of the two numbers 
required by the puzzle must produce the same root in sum and 
product. This can only happen when the roots of the two numbers 
are 2 and 2, or 9 and 9, or 3 and 6, or 5 and 8. Therefore the two- 
figure multiplier must have a digital root of 2, 3, 5, 6, 8, or 9. There 
are ten such numbers in each case. I write out all the sixty, then 
I strike out all those numbers where the second figure is higher 
than the first, and where the two figures are alike (thirty-six numbers 
in all) ; also all remaining numbers where the first figure is odd 
and the second figure even (seven numbers) ; also all multiples 
of 5 (three more numbers). The numbers 21 and 62 I reject on 
inspection, for reasons that I will not enter into. I then have left, 
out of the original sixty, only the following twelve numbers: 
83, 63, 81, 84, 93, 42, 51, 87, 41, 86, 53, and 71. These are the only 
possible multipliers that I have really to examine. 

My process is now as curious as it is simple in working. First 
trying 83, I deduct 10 and call it 73. Adding o's to the second 
figure, I say if 30000, etc., ever has a remainder 43 when divided 
by 73, the dividend will be the required multiplier for 83. I get 



228 THE CANTERBURY PUZZLES 

the 43 in this way. The only multiplier of 3 that produces an 8 in 
the digits place is 6. I therefore multiply 73 by 6 and get 438, or 
43 after rejecting the 8. Now, 300,000 divided by 73 leaves the 
remainder 43, and the dividend is 4,109. To this I add the 6 men- 
tioned above and get 41,096 x 83, the example given on page 129. 

In trying the even numbers there are two cases to be con- 
sidered. Thus, taking 86, we may say that if 60000, etc., when 
divided by 76 leaves either 22 or 60 (because 3x6 and 8x6 both 
produce 8), we get a solution. But I reject the former on inspec- 
tion, and see that 60 divided by 76 is o, leaving a remainder 60. 
Therefore 8 x 86 = 688, the other example. It will be found in 
the case of 71 that looooo, etc., divided by 61 gives a remainder 
42, (7 X 61 = 427) after producing the long dividend at the beginning 
of this article, with the 7 added. 

The other multipliers fail to produce a solution, so 83, 86, and 
71 are the only three possible multipliers. Those who are familiar 
with the principle of recurring decimals (as somewhat explained in 
my next note on No. 83, " The Ribbon Problem ") will understand 
the conditions under which the remainders repeat themselves after 
certain periods, and will only find it necessary in two or three cases 
to make any lengthy divisions. It clearly follows that there is 
an unlimited number of multiplicands for each multiplier. 



83. — The Ribbon Problem. 

The solution is as follows : Place this rather lengthy number 
on the ribbon, 021276595744680851063829787234042553191439- 
3617. It may be multiplied by any number up to 46 inclusive 
to give the same order of figures in the ring. The number pre- 
viously given can be multiplied by any number up to 16. I made 
the limit 9 in order to put readers off the scent. The fact is these 
two numbers are simply the recurring decimals that equal yV 
and -ij respectively. Multiply the one by seventeen and the oth( 
by forty-seven, and you will get all nines in each case. 

In transforming a vulgar fraction, say ^V- to a decii 



SOLUTIONS 229 

fraction, we proceed as below, adding as many noughts to the 
dividend as we like until there is no remainder, or until we get 
a recurring series of figures, or until we have carried it as far as 
we require, since every additional figure in a never-ending decimal 
carries us nearer and nearer to exactitude. 

17) 100 (.058823 
85 

150 
136 

140 
136 

40 
34 

60 
51 



Now, since all powers of 10 can only contain factors of the 
powers of 2 and 5, it clearly follows that your decimal never will 
come to an end if any other factor than these occurs in the de- 
nominator of your vulgar fraction. Thus, J, J, and J give us the 
exact decimals, .5, .25, and .125 ; i and -^ give us .2 and .04 ; 
xV and ttV give us .1 and .05 : because the denominators are all 
composed of 2 and 5 factors. But if you wish to convert J, J, 
or I, your division sum will never end, but you will get these 
decimals, .33333, etc., .166666, etc., and .142857142857142857, 
etc., where, in the first case, the 3 keeps on repeating for ever 
and ever ; in the second case the 6 is the repeater, and in the 
last case we get the recurring period of 142857. In the case of 
A (in " The Ribbon Problem ") we find the circulating period 
to be .0588235294117647. 

Now, in the division sum above, the successive remainders are 



230 



THE CANTERBURY PUZZLES 



1, 10, 15, 14, 4, 6, 9, etc., and these numbers I have inserted around 
the inner ring of the diagram. It will be seen that every number 
from I to 16 occurs once, and that if we multiply our ribbon number 
by any one of the numbers in the inner ring its position indicates 
exactly the point at which the product will begin. Thus, if we 
multiply by 4, the product will be 235, etc. ; if we multiply by 6, 




352, etc. We can therefore multiply by any number from i to 
16 and get the desired result. 

The kernel of the puzzle is this : Any. prime number, with the 
exception of 2 and 5, which are the factors of 10, will exactly 
divide without remainder a number consisting of as many nines as 
the number itself, less one. Thus 999999 (six 9's) is divisible by 7, 
sixteen 9's are divisible by 17, eighteen 9's by 19, and so on. This 
is always the case, though frequently fewer 9's will suffice ; for one 
9 is divisible by 3, two by 11, six by 13, when our ribbon rule for^ 
consecutive multipliers breaks down and another law comes in* 
Therefore, since the o and 7 at the ends of the ribbon may nol 



SOLUTIONS 



231 



be removed, we must seek a fraction with a prime denominator 
ending in 7 that gives a full period circulator. We try 37. and 
find that it gives a short period decimal, .027, because 37 exactly 
divides 999]; it, therefore, will not do. We next examine 47, and 
find that it gives us the full period circulator, in 46 figures, at the 
beginning of this article. 

If you cut any of these full period circulators in half and place 
one half under the other, you will find that they will add up all 
9's ; so you need only work out one half and then write down the 
complements. Thus, in the ribbon above, if you add 05882352 to 
941 17647 the result is 99999999, and so with our long solution 
number. Note also in the diagram above that not only are the 
opposite numbers on the outer ring complementary, always making 
9 when added, but that opposite numbers in the inner ring, our 
remainders, are also complementary, adding to 17 in every case. 
I ought perhaps to point out that in limiting our multipliers to the 
first nine numbers it seems just possible that a short period cir- 
culator might give a solution in fewer figures, but there are reasons 
for thinking it improbable. 



A 

A 



84. — The Japanese Ladies and the Carpet, 

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



2 


1 












*v^^ 4 


3 


5 


6^ 


2::. 



232 



THE CANTERBURY PUZZLES 



If with the three equal squares we form the rectangle IDBA, 
then the mean proportional of the two sides of the rectangle will 
be the side of a square of equal area. Produce AB to C, making 



I 


/■ 


H 




F 

D \ 




1 5 


3 
6 


2 


1 


^ 


V 




5 E B t 



EC equal to BD. Then place the point of the compasses at E 
(midway between A and C) and describe the arc AC. I am show- 
ing the quite general method for converting rectangles to squares, 
but in this particular case we may, of course, at once place our 
compasses at E, which requires no finding. Produce the line BD, 
cutting the arc in F, and BF will be the required side of the square. 
Now mark off AG and DH, each equal to BF, and make the 
cut IG, and also the cut HK from H, perpendicular to ID. The 
six pieces produced are numbered as in the diagram on last page. 

It will be seen that I have here given the reverse method first : 

r4 to cut the three small squares into six 

pieces to form a large square. In the case 

of our puzzle we can proceed as follows : — 

Make LM equal to half the diagonal 

ON. Draw the line NM and drop from 

L a perpendicular on NM. Then LP 

will be the side of all the three squares 

of combined area equal to the large 

square QNLO. The reader can now 

cut out without difficulty the six pieces, 

as shown in the numbered square on the last page. 




SOLUTIONS 233 

85. — Captain Longbow and the Bears. 
It might have struck the reader that the story of the bear 
impaled on the North Pole had no connection with the problem 
that followed. As a matter of fact it is essential to a solution. 
Eleven bears cannot possibly be arranged to form of themselves 
seven rows of bears with four bears in every row. But it is 
a different matter when Captain Longbow informs us that " they 

? s 
I 




had so placed themselves that there were " seven rows of four 
bears. For if they were grouped as shown in the diagram, so that 
three of the bears, as indicated, were in line with the North Pole, 
that impaled animal would complete the seventh row of four, 
which cannot be obtained in any other way. It obviously does not 
affect the problem whether this seventh row is a hundred miles 
long or a hundred feet, so long as they were really in a straight 
line — a point that might perhaps be settled by the captain's pocket 
compass. 

86. — The English Tour, 

It was required to show how a resident at the town marked A 
might visit every one of the towns once, and only once, and finish 



234 THE CANTERBURY PUZZLES 

up his tour at Z. This puzzle conceals a little trick. After the 
solver has demonstrated to his satisfaction that it cannot be done 
in accordance with the conditions as he at first understood them, 
he should carefully examine the wording in order to find some 
flaw. It was said, " This would be easy enough if he were able to 
cut across country by road, as well as by rail, but he is not." 




Now, although he is prohibited from cutting across country by 
road, nothing is said about his going by sea ! F If, therefore, we 
carefully look again at the map, we shall find that two towns, and 
two only, lie on the sea coast. When he reaches one of these 
towns he takes his departure on board a coasting vessel and sails 
to the other port. The annexed illustration shows, by a dark 
line, the complete route. 

This problem should be compared with No. 250, *' The Grand 
Tour," in A. in M. It can be simplified in practically an 



SOLUTIONS 235 

identical manner, but as there is here no choice on the first stage 
from A, the solutions are necessarily quite different. See also 
solution to No. 94. 

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

E5678 
^ ^ = 10 moves. 



= 2 moves. 

— 5 moves. 

o ,^ = amoves. 

87654321 ^ 

Twenty-six moves in all. 

^S. — The Eccentric Market-woman. 

The smallest possible number of eggs that Mrs. Covey coiild 
have taken to market is 719. After selling half the number and 
giving half an egg over she would have 359 left ; after the second 
transaction she would have 239 left ; after the third deal, 179 ; 
and after the fourth, 143. This last number she could divide 
equally among her thirteen friends, giving each 1 1, 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, 



1234 
E56 


123 %7 
56 


4 


E312 87 
E 


4 



236 THE CANTERBURY PUZZLES 

jump from 3 to i, and write it down on i ; and so on. The second 
word can be inserted in the same order. The solution depends on 
finding those words in which the second and eighth letters are the 
same, and also the fourth and sixth the same, because these letters 
interchange without destroying the words. MARITIMA (or sea- 
pink) would also solve the puzzle if it were an English word. 
Compare with No. 226 in A . in M. 

go,— 'The Round Table, 
Here is the way of arranging the seven men : — • 



A 


B 


C 


D 


E 


F 


G 


A 


C 


D 


B 


G 


E 


F 


A 


D 


B 


C 


F 


G 


E 


A 


G 


B 


F 


E 


C 


D 


A 


F 


C 


E 


G 


D 


B 


A 


E 


D 


G 


F 


B 


C 


A 


C 


E 


B 


G 


F 


D 


A 


D 


G 


C 


F 


E 


B 


A 


B 


F 


D 


E 


G 


C 


A 


E 


F 


D 


C 


G 


B 


A 


G 


E 


B 


D 


F 


C 


A 


F 


G 


C 


B 


E 


D 


A 


E 


B 


F 


C 


D 


G 


A 


G 


C 


E 


D 


B 


F 


A 


F 


D 


G 


B 


C 


E 



Of course, at a circular table, A will be next to the man at the 
end of the line. 

I first gave this problem for six persons on ten days, in the 
Daily Mail for the 13th and i6th 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 \^ -i) [n- 2 ) ^^yg jj^g comparatively easy method 



SOLUTIONS 237 

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

The solutions for all cases up to 12 inclusive are given in A. 
in M,, pp. 205, 206. 

91. — The Five Tea Tins. 

There are twelve ways of arranging the boxes without 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, 
including the example that I gave, not counting the reversals and 
reflections of these arrangements as different. Jaenisch, in his 
Analyse Mathematique au jeu des £checs (1862), quotes the 
statement that there are just twenty-one solutions to the little 
problem on which this puzzle is based. As I had myself only 
recorded seventeen, I examined the matter again, and found that 



'238 THE CANTERBURY PUZZLES 

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 

E. 160025 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 
jrield by reversal and reflection eight arrangements each. There- 
fore the pigs may be placed in (2 x 2) + (4 x i) + (8 x 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 withj 
one another ; but there is only one way of doing it (if we do notj 
count reversals as different), as follows : 105030. 

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. 



SOLUTIONS 



239 



94. — Foxes and Geese. 

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



10-5 


II— 6 


12-7 


5- 

8- 


-12 

-3 
7—2 


6—1 
9—10 
8-3 


7-6 


1-8 
12—7 


2—9 
1—8 


3—4 
6—1 


4-9 



3—4 10—5 9—10 4— II 



-12 



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

In A. in M., p. 230, I have explained fully my "buttons 
and string " method of solving puzzles on chequered boards. In 
Diagram A is shown the puzzle in the form in which it may be pre- 



£1, 


K 


H, 


4 


S 


6 


y 


s 


9 


«. 


A 


^ 




sented on a portion of the chessboard with six knights. A com- 
parison with the illustration on page 141 will show that I have 
there dispensed with the necessity of explaining the knight's move 
to the uninstructed reader by lines that indicate those moves. The 



240 



THE CANTERBURY PUZZLES 



two puzzles are the same thing in different dress. Now compare 
page 141 with Diagram B, and it will be seen that by disentangling 
the strings I have obtained a simplified diagram without altering 
the essential relations between the buttons or discs. The reader will 
now satisfy himself without any difficulty that the puzzle requires 
eleven moves for the foxes and eleven for the geese. He will see 
that a goose on i or 3 must go to 8, to avoid being one move from 
a fox and to enable the fox on 11 to come on to the ring. If we 
play I — 8, then it is clearly best to play 10 — 5 and not 12 — 5 for 
the foxes. When they are all on the circle, then they simply 
promenade round it in a clockwise direction, taking care to reserve 
8 — 3 and 5 — 12 for the final moves. It is thus rendered ridicu- 
lously easy by this method. See also notes on solutions to Nos. 13 
and 85. 

95. — Robinson Crusoe's Table. 

The diagram shows how the piece of wood should be cut in two 
pieces to form the square table-top. A, B, C, D are the corners of 




the table. The way in which the piece E fits into the piece F will 
be obvious to the eye of the reader. The shaded part is the wood 
that is discarded. 



SOLUTIONS 241 

96. — The Fifteen Orchards, 

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

97. — The Perplexed Plumber. 

The rectangular closed cistern that shall hold a given quantity 
of water and yet have the smallest possible surface of metal must 
be a perfect cube — ^that is, a cistern every side of which is a square. 
For 1,000 cubic feet of water the internal dimensions will be 
10 ft. X 10 ft. X 10 ft., and the zinc required will be 600 square feet. 
In the case of a cistern without a top the proportions will be ex- 
actly half a cube. These are the ** exact proportions " asked for 
in the t econd 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 out- 
side, 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-an[;led triangle, of which the length and 
width oi the paper are the other two sides. 
In the puzzle given, the lengths of the two sides of the triangle 
are 40 ft. (one-fifth of 200 ft.) and 16 ft. 8 in. Therefore the 

(2,077) 16 



/ 




m 


M . 




' i 


/"M 




< 


'M 




> 


M 




i 


/ 




« 


A 




T 



242 THE CANTERBURY PUZZLES 

hypotenuse is 43 ft. 4 in. The length of the garland is therefore 
five times as long — 216 ft. 8 in. A curious feature of the puzzle is 
the fact that with the dimensions given the result is exactly the 
sum of the height and the circumference. 



99. — The Two Errand Boys, 

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

100. — On the Ramsgaie Sands. 
Just six different rings may be formed without breaking the 



conditions. 


Here is one 


way of effecting the arrangements. 


A 


BCD 


EFGHIJKLM 


A 


C E G 


I KMBDFHJ L 


A 


D G J 


MCFI LBEHK 


A 


E I M 


DHLCGKBFJ 


A 


F K C 


HMEJBGLDI 


A 


G M F 


LEKDJCI BH 



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

loi. — The Three Motor-Cars. 

The only set of three numbers, of two, three, and five figures 
respectively, that will fulfil the required conditions is 27x594 = 
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 



SOLUTIONS 



243 



the numbers might contain one, four, and five figures respectively, 
there would be many correct answers, such as 3x5,694 = 17,082; 
but it is a curious fact that there is only one answer to the problem 
as propounded, though it is no easy matter to prove that this is 
the case. 

102. — A Reversible Magic Square. 

It will be seen that in the arrangement given every number is 
different, and all the columns, all the rows, and each of the two 



II 


XX 


.1 

62 


29 


69 


11 


\X 


X\ 


IX 


61 


XS 


12 


U 


19 


21 


61 



179 

diagonals, add up 179, whether you turn the page upside down or 
not. The reader will notice that I have not used the figures 3, 4, 
5. 8, or 0. 

103. — The Tube Railway. 

There are 640 different routes. A general formula for puzzles of 
this kind is not practicable. We have obviously only to consider the 
variations of route between B and E. Here there are nine sections 
or " lines," but it is impossible for a train, under the conditions, 
to traverse more than seven of these lines in any route. In the 
following table by '* directions " is meant the order of stations 



244 



THE CANTERBURY PUZZLES 



irrespective of " routes." Thus, the *' direction " BCDE gives 
nine " routes," because there are three ways of getting from B to 
C, and three ways of getting from D to E. But the " direction " 
BDCE admits of no variation ; therefore yields only one route. 



2 two-line directions of 3 routes 
I three-line 



I „ „ . 


> 9 


2 four-line „ 


, 6 


2 „ „ , 


, 18 


6 five-line „ 


. 6 


2 „ „ , 


, 18 


2 six-line 


, 36 


12 seven-line „ 


, 36 



Total 



6 

I 

9 
12 

36 
36 
36 
72 
432 

640 



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



104. — The Skipper and the Sea-Serpent. 

Each of the three pieces was clearly three cables long. But 
Simon persisted in assuming that the cuts were made transversely, 
or across, and that therefore the complete length was nine cables. 
The skipper, however, explained (and the point is quite as veracious 
as the rest of his yarn) that his cuts were made longitudinally — 
straight from the tip of the nose to the tip of the tail ! The com- 
plete 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. 



SOLUTIONS 



245 



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 all the ladies immodestly kissed the curate, although they 
were not (except the sisters) kissed by him in return ? No ; be- 
cause, in that case, it would be found that there must have been 
twelve girls, not one of whom was a sister, which is contrary to the 
conditions. If, again, it should be held that the sisters might not, 
according to the wording, have kissed their brother, although he 
kissed them, I reply that in that case there must have been twelve 
girls, all of whom must have been his sisters. And the reference 
to the ladies who might have worked exclusively of the sisters shuts 
out the possibility of this. 

106. — The Adventurous Snail. 

At the end of seventeen days the snail will have climbed 17 ft., 
and at the end of its eighteenth day-time task it will be at the top. 
It instantly begins slipping while sleeping, and will be 2 ft. down 
the other side at the end of the eighteenth day of twenty-four hours. 
How long will it take over the remaining 18 ft. ? If it slips 2 ft. 
at night it clearly overcomes the tendency to slip 2 ft. during the 
daytime, in climbing up. In rowing up a river we have the stream 
against us, but in coming down it is with us and helps us. If the 
snail can climb 3 ft. and overcome the tendency to slip 2 ft. in 
twelve hours' ascent, it could with the same exertion crawl 5 ft. a 



246 THE CANTERBURY PUZZLES 

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 
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 2mn will be the three sides of a 
rational right-angled triangle. Here m and n are cdled generating 
numbers. To form three such triangles of equal area, we use the 
following simple formula, where m is the greater number : — 

mn + fyi?-\-n'^ = a 

m^-n^^b 

2mn ■\-n^ = c 

Now, if we form three triangles from the following pairs of 
generators, a and b, a and c, a and i + 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 CoUingwood, p. 343), 
" Sat up last night till 4 a.m., over a tempting problem, sent me 
from New York, ' to find three equal rational-sided right-angled 
triangles.' I found two . . . but could not find three I " 

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 



SOLUTIONS 247 

all we have to do is to form a R.A.T. from the generators z^ and Ofl,, 
and give each side the denominator 2z[l>^-W), and we get the 
required answer in fractions. If we multiply all three sides of the 
original triangle by the denominator, we shall get at once a solution 
in whole numbers. 

The answer to our puzzle in smallest possible numbers is as 
follows :- 



First Prince . . 


. 518 


1320 


1418 


Second Prince . . 


. 280 


2442 


2458 


Third Prince . . 


. 231 


2960 


2969 


Fourth Prince . . 


. Ill 


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 
shown, from the numbers 3 and 4, which give the generators 
37' 7 J 37' 33 > 37* 40- These three pairs of numbers solve the 
indeterminate equation, c^h - h^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, II ; 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. — Tlato and the Nines. 
The following is the simple solution of the three nines puzzle : — 

9 + 9 
•9 

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



248 THE CANTERBURY PUZZLES 

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 comer 
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 pit- 
falls. But Nought may safely lead anything and secure a draw, 
and he can only win through Cross's blunders. 

no. — 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 varia- 
tion of the game is to bar the centre for the first move of the first 
player. In that case the second player should take the centre at 
once. This should always end in a draw, but to ensure it the first 
player must play to two adjoining corners (such as i and 3) on his 
first and second moves. The game then requires great care on 
both sides. 

III. — 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 lo-acre 
field, and 6 bullocks eat the grass on 10 acres in 16 weeks. There- 
fore, if 6 bullocks keep down the growth on 10 acres, 24 will keep 
down the growth on 40 acres. 



SOLUTIONS 249 

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, 

4^ »» »» 40 >. 8 „ 

192 „ „ 40 » 2 „ 

64 „ „ 40 „ 6 „ 

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

112. — The Great Grangemoor Mystery. 

We were told that the bullet that killed Mr. Stanton Mowbray 
struck the very centre of the clock face and instantly welded to- 
gether the hour, minute, and second hands, so that all revolved 
in one piece. The puzzle was to tell from the fixed relative posi- 
tions of the three hands the exact time when the pistol was fired. 

We were also told, and the illustration of the clock face bore 
out the statement, that the hour and minute hands were exactly 
twenty divisions apart, ** the third of the circumference of the dial." 
Now, there are eleven times in twelve hours when the hour hand 
is exactly twenty divisions ahead of the minute hand, and eleven 
times when the minute hand is exactly twenty divisions ahead of 
the hour hand. The illustration showed that we had only to con- 
sider the former case. If we start at four o'clock, and keep on 
adding i h. 5 m. 2y^Y sec, we shall get all these eleven times, the 
last being 2 h. 54 min. 32yt sec. Another addition brings us back 
to four o'clock. If we now examine the clock face, we shall find 
that the seconds hand is nearly twenty-two divisions behind the 
minute hand, and if we look at all our eleven times we shall find 
that only in the last case given above is the seconds hand at this 
distance. Therefore the shot must have been fired at 2 h. 54 min. 
32y\ sec. exactly, or, put the other way, at 5 min. 27y\ sec. to 
three o'clock. This is the correct and only possible answer to the 
puzzle. 



250 THE CANTERBURY PUZZLES 

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 102 1, from 
which it is evident that they were of that miniature description 
that finds favour in the nursery. The general solution is that for 
n men the number must be w(m"+^) - (« - i), 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 w = i, the final distribution 
only benefits the dog. Of course, in every case each man steals an 
«th of the number of biscuits, after giving the odd one to the dog. 



INDEX 



Abracadabra," 64. 

Age and Kinship Puzzles, 20, 28. 

Albanna, Ibn, 198. 

Ale, Measuring the, 29. 

Algebraical Puzzles. See Arithmetical 
Puzzles. 

Alkala^adi, 198. 

Amulet, The, 64, 190. 

Archery Butt, The, 60, 187. 

Arithmetical Puzzles, 18, 26, 34, 36, 
45, 46, 51, 56, 69, 61, 64, 65, 73, 74, 
75, 88, 89, 91, 92, 103, 107, 122, 125, 
128, 129, 130, 136, 137, 139, 143, 
147, 148, 160, 151, 162, 163, 164, 167, 
168, 161. 

Arrows, The Nine, 32. 

Astronomical Problem, 55. 

Bags, Four Money, 46. 

Ball, W. W. Rouse, 202. 

Bandy-Ball, The Game of, 58, 185. 

Bears, Capt. Longbow and the, 132, 233. 

Bergholt, Ernest, 237. 

Bewley, E. D., 237. 

Biscuits, The Tramps and the, 160, 250. 

Block, Cutting a Wood, 160, 250. 

Blocks, The Number, 139, 238. 

Bottles, Sharing the, 122. 

Bottles, The Sixteen, 45. 

Bridges, The Eight, 48. 

Bridging the Ditch, 83, 204. 

Brooch, Cutting the, 41. 

Buried Treasure, 107, 212. 

Buttons and String Method, 171, 239. 



Canon's Yeoman, Puzzle of the, 65, 

181. 
Canterbury Pilgrim's Puzzle, 33. 
Canterbury Puzzles, 23, 163. 
Card Puzzle, 125. 
Carpenter's Puzzle, The, 31, 168. 
Carpet, Japanese Ladies and, 131, 231. 
Carroll, Lewis, 246. 
Casket, Lady Isabel's, 67, 191. 
Cats and Mice, 75. 
Cellarer, The Riddle of the, 73, 196. 
Cellarman, The Perplexed, 122, 222. 
Chalked Numbers, The, 89, 206. 
Chaucer's Puzzle, 54, 181. 
Cheeses on Stools, 24. 
Chessboard Problems, 21, 25, 32, 51, 72, 

82, 90, 11.3, 114,116, 119, 124, 138, 141. 
Chessboard, The Broken, 119, 220. 
Chifu-Chemulpo Puzzle, 134, 235. 
Chinese RaHways, The, 127, 224. 
Christmas Puzzle Party, The Squire's, 

86, 205. 
Cisterns, Making, 146. 
Clerk of Oxenford's Puzzle, The, 29, 167. 
Cliff Mystery, The Cornish, 99, 210. 
Clock Puzzle, 158. 
Cloth, Cutting the, 50. 
Clowns, The Eight, 128, 226. 
Club, Adventures of the Puzzle, 94, 210. 
Coinage Puzzle, The, 111, 214. 
Coin Magic Square, 111. 
Column, The Nelson, 146, 241. 
Combination and Group Problems, 38, 

39, 61, 70, 122, 137, 147. 



251 



252 



INDEX 



Cook's Puzzle, The, 36, 171. 
Cbmish Cliff Mystery, Tho, 90, 210. 
Counter Problems, Moving, 24, 35, 69, 

77, 124, 135, 136, 141. 
Counting out Puzzle, 39. 

Crescent and the Cross, The, 63, 189. 
Crossing River Problems, 82, 83. 
Crusaders, The Riddle of the, 74, 197. 
Crusoe s Table, Robinson, 142, 240. 
Cubes, Sums of Two, 174, 209. 
The Silver, 92, 209. 

DaUy Mail, 179, 221, 236. 
Decimals, Recurring, 228, 229. 
Demoiselle, The Noble, 59, 186. 
Diamond I-iotter Puzzles, 181. 
Digital Analysis, 228. 
Puzzles, 18, 26, 90, 103, 129, 139, 

148, 154. 
Dispute between Friar and Sompnour, 

51, 180. 
Dissection Puzzles, 29, 30, 35, 41, 49, 

63, 123, 131, 142. 
Ditch, Bridging the, 83, 204. 
Divisors of Numbers, To Find, 170. 
Doctor of Physic, The Puzzle of the, 42, 

174. 
Donjon Keep Window, The, 62, 188. 
Dorcas Society, The, 151, 245. 
Dormitory Puzzle, The, 70. 
Dungeon, The Death's-head, 60. 
Dungeons, The Nine, 35. 
Dyer's Puzzle, The, 50, 180. 

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

78, 201. 
Executioner, The, 78. 

Fallacy of Square's Diagonal, 62, 
Farmer's Oxen, The, 167, 248. 



Fermat, P. de, 174. 

Fish-pond, The Riddle of the, 69, 194. 

Flag, Making a, 123, 223, 

Fleurs-de-Lys, Sixty-four, 50. 

Flour, The Nine Sacks of, 26. 

Fly, The Spider and the, 121, 221. 

Footprints Puzzles, 101, 105. 

Four Princes, The, 153, 246. 

Foxes and Geese, 140, 239. 

Franklin's Puzzle, The, 44, 176. 

Friar and Sompnour's Dispute, 51, 180. 

Friar's Puzzle, The, 46, 177. 

Frogs and Tumblers, The, 113, 216. 

who would a-wooing go, 116, 219. 

Frogs' Ring, The Riddle of the, 76, 199. 

Games, Puzzle, 118, 125, 156, 157. 
Gardens, The Royal, 82, 203. 
Geese, The Christmas, 88, 206. 
Geometrical Problems, 52, 62, 67, 121, 

131, 144, 146. 
Grangemoor Mystery, The, 158, 249. 
Group Problems, Combination and. See 

Combination and Group Problems. 

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

Isabel's Casket, Lady, 67, 191. 

Jaenisch, 237. 

Japanese Ladies and the Carpet, 131, 

231. 
Jester, Strange Escape of the King's, 

78, 201. 

Kayles, The Game of, 118, 220. 

Kennels, The Nine, 39. 

King's Jester, Strange Escape of the, 

78, 201. 
Knight's Puzzle, The, 26, 165. 



I 



INDEX 



253 



Lady Isabel's Casket, 67, 191. 
V Arithmetique Amusante, 198. 
Legendre, 175. 
Letter Puzzles, 16. 
Lock, The Secret, 80, 202. 
Locomotive and Speed Puzzle, 147. 
Longbow and the Bears, Capt., 132, 233. 
Lucas, Edouard, 175, 198, 242. 

M'Elroy, C. W., 179. 

Magdalen, Chart of the, 41. 

Magic Square, A Reversible, 149, 243. 

Square Problems, 21, 29, 44, 111, 

112, 128, 149. 
Manciple's Puzzle, The, 56, 183. 
Man of Law's Puzzle, The, 34, 170. 
Market Woman, The Eccentric, 135, 

235. 
Marksford and the Lady, Lord, 96. 
Maze, The Underground, 79, 201. 
Measuring, Weighing, and Packing 

Puzzles, 29, 31, 55, 72, 73, 160. 
Merchant's Puzzle, The, 33, 170. 
Merry Monks of Riddlewell, 68, 194. 
Miller's Puzzle, The, 26, 164. 
Miscellaneous Puzzles, 118, 220. 
Mistletoe Bough, Under the, 91, 208. 
Moat, Crossing the, 81, 202. 
Money, Dividing the, 57. 
Monks of Riddlewell, The Merry, 68, 

194. 
Monk's Puzzle, The, 39, 172. 
Montucla, 246. 

Motor-Car, The Runaway, 103, 211. 
Motor-Cars, The Three, 147, 242. 
Moving Counter Problems. See Counter 

Problems, Moving. 

Nash, W., 237. 

Nelson Column, The, 146, 241. 
Newton, Sir Isaac, 248. 
Nines, Plato and the, 154, 247. 
Noble DemoiseUe, The, 59, 186. 
Noughts and Crosses, 156, 248. 



Number Blocks, The, 139, 238. 
Numbers on Motor-Cars, 103, 148. 

Partition of. 46. 

The Chalked, 89, 206. 

Nun's Puzzle, The, 32, 169. 

Ones, Numbers composed only of, 

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

Packing Puzzles, Measuring, Weighing, 

and. See Measuring. 
Palindromes, 17. 
Pardoner's Puzzle, The, 25, 164. 
Parental Command, A, 28. 
Park, Mystery of Ravensdene, 105, 211. 
Parson's Puzzle, The, 47, 177. 
Party, The Squire's Christmas Puzzle, 

86, 205. 
Pellian Equation, 197. 
Pennies, The Eleven, 88, 206. 
Phials, The Two, 42. 
Photograph, The Ambiguous, 94, 210. 
Pie and the Pasty, The, 36. 
Pilgrimages, The Fifteen, 25. 
Pilgrims' Manner of Riding, 34. 

The Riddle of the, 70, 194. 

Pillar, The Carved Wooden, 31. 
Plato and the Nines, 154, 247. 
Ploughman's Puzzle, The, 43, 175. 
Plumber, The Perplexed, 144, 241. 
Plum Puddings, Tasting the, 90, 207. 
Points and Lines Problems, 43, 116, 

133. 
Porkers, The Four, 138, 237. 
Postage Stamps Puzzle, The, 112, 214. 
Primrose Puzzle, The, 136, 235. 
Princes, The Four, 153, 246. 
Prioress, The Puzzle of the, 41, 173. 
Professor's Puzzles. The. 110. 214. 



254 



INDEX 



Puzzle Club, Adventures of the, 94 210. 
Puzzles, How to solve, 18. 

How they are made, 14. 

Sophistical, 15. 

The exact conditions of, 18. 

The mysterious charm of, 12. 

The nature of, 11. 

The utility of, 13. 

The variety of, 13, 16. 

Unsolved, 20. 

Puzzling Times at Solvamhall Castle, 58, 

184. 
Pyramids, Triangular, 163. 

Railway Puzzle, 134. 

The Tube, 149, 243. 

Railways, The Chinese, 127, 224- 
Ramsgate Sands, On the, 147, 242, 
Rat-catcher's Riddle, The, 56. 
Ravensdene Park, Mystery of, 105, 211. 
Reve's Puzzle, The, 24, 163. 
Ribbon Problem, The, 130, 228. 
Riddles, old, 16. 
Riddlewell, The Merry Monks of, 68, 

194. 
River Crossing Problems, 82, 83. 
Robinson Crusoe's Table, 142, 240. 
Romeo and Juliet, 114, 217. 
Romeo's Second Journey, 116, 218. 
Rook's Path, The, 207. 
Rope, The Mysterious, 79, 201. 
Round Table, The, 137, 236. 
Route Problems, Unicursal and. See 

Unicursal. 

Sack Wine, The Riddle of the, 72, 196. 
St. Edmond8bury,The Riddle of, 75, 197. 
Sands, On the Ramsgate, 147, 242. 
Sea-Serpent, The Skipper and the, 150, 

244. 
Shield, Squares on a, 27. 
Shipman's Puzzle, The, 40, 173. 
Skipper and the Sea-Serpent, The, 150, 

244. 



Snail on the Flagstaff, The, 65, 190. 

The Adventurous, 152, 246. 

Snails, The Two, 115, 217. 
Solvamhall Castle, Puzzling Times at, 

68, 184. 
Sompnour's and Friar's Dispute, 51, 

180. 

Puzzle, The, 38, 172. 

Spherical Surface of Water, 181. 

Spider and the Fly, The, 121, 221. 

Square and Triangle, The, 49. 

Square Field, The, 107. 

Squares, Problem of, 74. 

Square, Three Squares from One, 131, 

231. 
Squire's Christmas Puzzle Party, The, 

86, 205. 

Puzzle, The, 45, 176. 

Yeoman, The Puzzle of the, 31, 

168. 
Stamps, Counting Postage, 137. 

Magic Squares of, 112. 

Puzzle, The Postage, 112, 214. 

Superposition, Problem on, 179. 
Sylvester, 176. 

Table, Robinson Crusoe's, 142, 240. 

The Round, 137, 236. 

Talkhya, 198. 
Tapestry, Cutting the, 30. 
Tapiser's Puzzle, The, 30, 167. 
Teacups, The Three, 87, 205. 
Tea Tins, The Five, 137, 237. 
Thirty-one Game, The, 125, 224. 
Tiled Hearth, The Riddle of the 

195. 
Tilting at the Ring, 59, 185. 
Tour, The English, 134, 233. 
Towns, Visiting the, 134. 
Tramps and the Biscuits, The, 160, 250 
Treasure, The Buried, 107, 212. 
Trees, The Sixteen Oak, 44. 
Triangle and Square, 49. 
Triangles of Equal Area, 153, 246 



I 



INDEX 



2SS 



riangular numbers, 163. 
ub© Railwav, The, 149, 243. 



Tnicurs&l and Route Problems, 40, 45, 
48, 66, 60, 83, 90, 106, 127, 134, 149. 

i^earer'a Puzzle, The, 35, 171. 
/celdy Dispatch, 179, 221. 



Weighing, and Packing Puzzles, Measur- 
ing. See Measuring. 
Wife of Bath's Riddles, The, 27, 166. 
Window, The Donjon Keep, 62, 188. 
Wine, Stealing the, 73. 
Wizard's Arithmetic, The, 129, 226. 
Wood Block, Cutting a, 160, 250. 
Wreath on Column, 146, 241. 



THE END. 



PRINTED IN GREAT BRITAIN AT 
THE PRESS OF THE PUBLISHERS. 



PRESS OPINIONS ON 
"THE CANTERBURY PUZZLES." 

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*• A capital book of posers." — Daily News. 

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well for the sanity of his readers that the author gives a list of solutions at 
the end of the book." — Observer, 

" A book that will provide much entertainment for Christmas gatherings 
. . . ingenious puzzles and problems invented by * Sphinx,' the Puzzle 
King." — The Captain. 

" Mr. Dudeney, whose reputation is world-wide as the puzzle and problem 
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appearance as its contents are fascinating." — English Mechanic and World of 
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" An exceedingly ingenious constructor and solver of fascinating puzzles, 
mathematical and otherwise.'' — School Guardian. 

"A book which ought to be highly popular ... it is all mighty in- 
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" It is matter for delight that Mr. Henry E. Dudeney has collected into 
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. . . contains quite a number of ingenious new mental problems ... a 
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"Took up more of the reviewer's time than he could well afford to give 
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"A well-known master puzzler . . . provides an abundance of seasonable 
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" Extremely ingenious book, which abounds in problems that will 
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