Readers of _The Mill on the Floss_ will remember that whenever Mr.
Tulliver found himself confronted by any little difficulty he was
accustomed to make the trite remark, "It's a puzzling world." There can
be no denying the fact that we are surrounded on every hand by posers,
some of which the intellect of man has mastered, and many of which may be
said to be impossible of solution. Solomon himself, who may be supposed
to have been as sharp as most men at solving a puzzle, had to admit
"there be three things which are too wonderful for me; yea, four which I
know not: the way of an eagle in the air; the way of a serpent upon a
rock; the way of a ship in the midst of the sea; and the way of a man
with a maid."
Probing into the secrets of Nature is a passion with all men; only we
select different lines of research. Men have spent long lives in such
attempts as to turn the baser metals into gold, to discover perpetual
motion, to find a cure for certain malignant diseases, and to navigate
the air.
From morning to night we are being perpetually brought face to face with
puzzles. But there are puzzles and puzzles. Those that are usually
devised for recreation and pastime may be roughly divided into two
classes: Puzzles that are built up on some interesting or informing
little principle; and puzzles that conceal no principle whatever--such as
a picture cut at random into little bits to be put together again, or the
juvenile imbecility known as the "rebus," or "picture puzzle." The former
species may be said to be adapted to the amusement of the sane man or
woman; the latter can be confidently recommended to the feeble-minded.
The curious propensity for propounding puzzles is not peculiar to any
race or to any period of history. It is simply innate in every
intelligent man, woman, and child that has ever lived, though it is
always showing itself in different forms; whether the individual be a
Sphinx of Egypt, a Samson of Hebrew lore, an Indian fakir, a Chinese
philosopher, a mahatma of Tibet, or a European mathematician makes little
difference.
Theologian, scientist, and artisan are perpetually engaged in attempting
to solve puzzles, while every game, sport, and pastime is built up of
problems of greater or less difficulty. The spontaneous question asked by
the child of his parent, by one cyclist of another while taking a brief
rest on a stile, by a cricketer during the luncheon hour, or by a
yachtsman lazily scanning the horizon, is frequently a problem of
considerable difficulty. In short, we are all propounding puzzles to one
another every day of our lives--without always knowing it.
A good puzzle should demand the exercise of our best wit and ingenuity,
and although a knowledge of mathematics and a certain familiarity with
the methods of logic are often of great service in the solution of these
things, yet it sometimes happens that a kind of natural cunning and
sagacity is of considerable value. For many of the best problems cannot
be solved by any familiar scholastic methods, but must be attacked on
entirely original lines. This is why, after a long and wide experience,
one finds that particular puzzles will sometimes be solved more readily
by persons possessing only naturally alert faculties than by the better
educated. The best players of such puzzle games as chess and draughts are
not mathematicians, though it is just possible that often they may have
undeveloped mathematical minds.
It is extraordinary what fascination a good puzzle has for a great many
people. We know the thing to be of trivial importance, yet we are
impelled to master it; and when we have succeeded there is a pleasure and
a sense of satisfaction that are a quite sufficient reward for our
trouble, even when there is no prize to be won. What is this mysterious
charm that many find irresistible? Why do we like to be puzzled? The
curious thing is that directly the enigma is solved the interest
generally vanishes. We have done it, and that is enough. But why did we
ever attempt to do it?
The answer is simply that it gave us pleasure to seek the solution--that
the pleasure was all in the seeking and finding for their own sakes. A
good puzzle, like virtue, is its own reward. Man loves to be confronted
by a mystery, and he is not entirely happy until he has solved it. We
never like to feel our mental inferiority to those around us. The spirit
of rivalry is innate in man; it stimulates the smallest child, in play or
education, to keep level with his fellows, and in later life it turns men
into great discoverers, inventors, orators, heroes, artists, and (if they
have more material aims) perhaps millionaires.
In starting on a tour through the wide realm of Puzzledom we do well to
remember that we shall meet with points of interest of a very varied
character. I shall take advantage of this variety. People often make the
mistake of confining themselves to one little corner of the realm, and
thereby miss opportunities of new pleasures that lie within their reach
around them. One person will keep to acrostics and other word puzzles,
another to mathematical brain-rackers, another to chess problems (which
are merely puzzles on the chess-board, and have little practical relation
to the game of chess), and so on. This is a mistake, because it restricts
one's pleasures, and neglects that variety which is so good for the
brain.
And there is really a practical utility in puzzle-solving. Regular
exercise is supposed to be as necessary for the brain as for the body,
and in both cases it is not so much what we do as the doing of it from
which we derive benefit. The daily walk recommended by the doctor for the
good of the body, or the daily exercise for the brain, may in itself
appear to be so much waste of time; but it is the truest economy in the
end. Albert Smith, in one of his amusing novels, describes a woman who
was convinced that she suffered from "cobwigs on the brain." This may be
a very rare complaint, but in a more metaphorical sense many of us are
very apt to suffer from mental cobwebs, and there is nothing equal to the
solving of puzzles and problems for sweeping them away. They keep the
brain alert, stimulate the imagination, and develop the reasoning
faculties. And not only are they useful in this indirect way, but they
often directly help us by teaching us some little tricks and "wrinkles"
that can be applied in the affairs of life at the most unexpected times
and in the most unexpected ways.
There is an interesting passage in praise of puzzles in the quaint
letters of Fitzosborne. Here is an extract: "The ingenious study of
making and solving puzzles is a science undoubtedly of most necessary
acquirement, and deserves to make a part in the meditation of both sexes.
It is an art, indeed, that I would recommend to the encouragement of both
the Universities, as it affords the easiest and shortest method of
conveying some of the most useful principles of logic. It was the maxim
of a very wise prince that 'he who knows not how to dissemble knows not
how to reign'; and I desire you to receive it as mine, that 'he who knows
not how to riddle knows not how to live.'"
How are good puzzles invented? I am not referring to acrostics, anagrams,
charades, and that sort of thing, but to puzzles that contain an original
idea. Well, you cannot invent a good puzzle to order, any more than you
can invent anything else in that manner. Notions for puzzles come at
strange times and in strange ways. They are suggested by something we see
or hear, and are led up to by other puzzles that come under our notice.
It is useless to say, "I will sit down and invent an original puzzle,"
because there is no way of creating an idea; you can only make use of it
when it comes. You may think this is wrong, because an expert in these
things will make scores of puzzles while another person, equally clever,
cannot invent one "to save his life," as we say. The explanation is very
simple. The expert knows an idea when he sees one, and is able by long
experience to judge of its value. Fertility, like facility, comes by
practice.
Sometimes a new and most interesting idea is suggested by the blunder of
somebody over another puzzle. A boy was given a puzzle to solve by a
friend, but he misunderstood what he had to do, and set about attempting
what most likely everybody would have told him was impossible. But he was
a boy with a will, and he stuck at it for six months, off and on, until
he actually succeeded. When his friend saw the solution, he said, "This
is not the puzzle I intended--you misunderstood me--but you have found
out something much greater!" And the puzzle which that boy accidentally
discovered is now in all the old puzzle books.
Puzzles can be made out of almost anything, in the hands of the ingenious
person with an idea. Coins, matches, cards, counters, bits of wire or
string, all come in useful. An immense number of puzzles have been made
out of the letters of the alphabet, and from those nine little digits and
cipher, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.
It should always be remembered that a very simple person may propound a
problem that can only be solved by clever heads--if at all. A child
asked, "Can God do everything?" On receiving an affirmative reply, she at
once said: "Then can He make a stone so heavy that He can't lift it?"
Many wide-awake grown-up people do not at once see a satisfactory answer.
Yet the difficulty lies merely in the absurd, though cunning, form of the
question, which really amounts to asking, "Can the Almighty destroy His
own omnipotence?" It is somewhat similar to the other question, "What
would happen if an irresistible moving body came in contact with an
immovable body?" Here we have simply a contradiction in terms, for if
there existed such a thing as an immovable body, there could not at the
same time exist a moving body that nothing could resist.
Professor Tyndall used to invite children to ask him puzzling questions,
and some of them were very hard nuts to crack. One child asked him why
that part of a towel that was dipped in water was of a darker colour than
the dry part. How many readers could give the correct reply? Many people
are satisfied with the most ridiculous answers to puzzling questions. If
you ask, "Why can we see through glass?" nine people out of ten will
reply, "Because it is transparent;" which is, of course, simply another
way of saying, "Because we can see through it."
Puzzles have such an infinite variety that it is sometimes very difficult
to divide them into distinct classes. They often so merge in character
that the best we can do is to sort them into a few broad types. Let us
take three or four examples in illustration of what I mean.
First there is the ancient Riddle, that draws upon the imagination and
play of fancy. Readers will remember the riddle of the Sphinx, the
monster of 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 Oedipus, who pointed out that
man walked on his hands and feet in the morning of life, at the noon of
life he walked erect, and in the evening of his days he supported his
infirmities with a stick. When the Sphinx heard this explanation, she
dashed her head against a rock and immediately expired. This shows that
puzzle solvers may be really useful on occasion.
Then there is the riddle propounded by Samson. It is perhaps the first
prize competition in this line on record, the prize being thirty sheets
and thirty changes of garments for a correct solution. The riddle was
this: "Out of the eater came forth meat, and out of the strong came forth
sweetness." The answer was, "A honey-comb in the body of a dead lion."
To-day this sort of riddle survives in such a form as, "Why does a
chicken cross the road?" to which most people give the answer, "To get to
the other side;" though the correct reply is, "To worry the chauffeur."
It has degenerated into the conundrum, which is usually based on a mere
pun. For example, we have been asked from our infancy, "When is a door
not a door?" and here again the answer usually furnished ("When it is
a-jar") is not the correct one. It should be, "When it is a negress (an
egress)."
There is the large class of Letter Puzzles, which are based on the
little peculiarities of the language in which they are written--such as
anagrams, acrostics, word-squares, and charades. In this class we also
find palindromes, or words and sentences that read backwards and forwards
alike. These must be very ancient indeed, if it be true that Adam
introduced himself to Eve (in the English language, be it noted) with the
palindromic words, "Madam, I'm Adam," to which his consort replied with
the modest palindrome "Eve."
Then we have Arithmetical Puzzles, an immense class, full of diversity.
These range from the puzzle that the algebraist finds to be nothing but a
"simple equation," quite easy of direct solution, up to the profoundest
problems in the elegant domain of the theory of numbers.
Next we have the Geometrical Puzzle, a favourite and very ancient branch
of which is the puzzle in dissection, requiring some plane figure to be
cut into a certain number of pieces that will fit together and form
another figure. Most of the wire puzzles sold in the streets and
toy-shops are concerned with the geometry of position.
But these classes do not nearly embrace all kinds of puzzles even when we
allow for those that belong at once to several of the classes. There are
many ingenious mechanical puzzles that you cannot classify, as they stand
quite alone: there are puzzles in logic, in chess, in draughts, in cards,
and in dominoes, while every conjuring trick is nothing but a puzzle, the
solution to which the performer tries to keep to himself.
There are puzzles that look easy and are easy, puzzles that look easy and
are difficult, puzzles that look difficult and are difficult, and puzzles
that look difficult and are easy, and in each class we may of course have
degrees of easiness and difficulty. But it does not follow that a puzzle
that has conditions that are easily understood by the merest child is in
itself easy. Such a puzzle might, however, look simple to the uninformed,
and only prove to be a very hard nut to him after he had actually tackled
it.
For example, if we write down nineteen ones to form the number
1,111,111,111,111,111,111, and then ask for a number (other than 1 or
itself) that will divide it without remainder, the conditions are
perfectly simple, but the task is terribly difficult. Nobody in the world
knows yet whether that number has a divisor or not. If you can find one,
you will have succeeded in doing something that nobody else has ever
done.[A]
The number composed of seventeen ones, 11,111,111,111,111,111, has only
these two divisors, 2,071,723 and 5,363,222,357, and their discovery is
an exceedingly heavy task. The only number composed only of ones that we
know with certainty to have no divisor is 11. Such a number is, of
course, called a prime number.
The maxim that there are always a right way and a wrong way of doing
anything applies in a very marked degree to the solving of puzzles. Here
the wrong way consists in making aimless trials without method, hoping to
hit on the answer by accident--a process that generally results in our
getting hopelessly entangled in the trap that has been artfully laid for
us.
Occasionally, however, a problem is of such a character that, though it
may be solved immediately by trial, it is very difficult to do by a
process of pure reason. But in most cases the latter method is the only
one that gives any real pleasure.
When we sit down to solve a puzzle, the first thing to do is to make
sure, as far as we can, that we understand the conditions. For if we do
not understand what it is we have to do, we are not very likely to
succeed in doing it. We all know the story of the man who was asked the
question, "If a herring and a half cost three-halfpence, how much will a
dozen herrings cost?" After several unsuccessful attempts he gave it up,
when the propounder explained to him that a dozen herrings would cost a
shilling. "Herrings!" exclaimed the other apologetically; "I was working
it out in haddocks!"
[A] See footnote on page 198.
It sometimes requires more care than the reader might suppose so to word
the conditions of a new puzzle that they are at once clear and exact and
not so prolix as to destroy all interest in the thing. I remember once
propounding a problem that required something to be done in the "fewest
possible straight lines," and a person who was either very clever or very
foolish (I have never quite determined which) claimed to have solved it
in only one straight line, because, as she said, "I have taken care to
make all the others crooked!" Who could have anticipated such a quibble?
Then if you give a "crossing the river" puzzle, in which people have to
be got over in a boat that will only hold a certain number or combination
of persons, directly the would-be solver fails to master the difficulty
he boldly introduces a rope to pull the boat across. You say that a rope
is forbidden; and he then falls back on the use of a current in the
stream. I once thought I had carefully excluded all such tricks in a
particular puzzle of this class. But a sapient reader made all the people
swim across without using the boat at all! Of course, some few puzzles
are intended to be solved by some trick of this kind; and if there
happens to be no solution without the trick it is perfectly legitimate.
We have to use our best judgment as to whether a puzzle contains a catch
or not; but we should never hastily assume it. To quibble over the
conditions is the last resort of the defeated would-be solver.
Sometimes people will attempt to bewilder you by curious little twists in
the meaning of words. A man recently propounded to me the old familiar
problem, "A boy walks round a pole on which is a monkey, but as the boy
walks the monkey turns on the pole so as to be always facing him on the
opposite side. Does the boy go around the monkey?" I replied that if he
would first give me his definition of "to go around" I would supply him
with the answer. Of course, he demurred, so that he might catch me either
way. I therefore said that, taking the words in their ordinary and
correct meaning, most certainly the boy went around the monkey. As was
expected, he retorted that it was not so, because he understood by "going
around" a thing that you went in such a way as to see all sides of it. To
this I made the obvious reply that consequently a blind man could not go
around anything.
He then amended his definition by saying that the actual seeing all sides
was not essential, but you went in such a way that, given sight, you
could see all sides. Upon which it was suggested that consequently you
could not walk around a man who had been shut up in a box! And so on. The
whole thing is amusingly stupid, and if at the start you, very properly,
decline to admit any but a simple and correct definition of "to go
around," there is no puzzle left, and you prevent an idle, and often
heated, argument.
When you have grasped your conditions, always see if you cannot simplify
them, for a lot of confusion is got rid of in this way. Many people are
puzzled over the old question of the man who, while pointing at a
portrait, says, "Brothers and sisters have I none, but that man's father
is my father's son." What relation did the man in the picture bear to the
speaker? Here you simplify by saying that "my father's son" must be
either "myself" or "my brother." But, since the speaker has no brother,
it is clearly "myself." The statement simplified is thus nothing more
than, "That man's father is myself," and it was obviously his son's
portrait. Yet people fight over this question by the hour!
There are mysteries that have never been solved in many branches of
Puzzledom. Let us consider a few in the world of numbers--little things
the conditions of which a child can understand, though the greatest minds
cannot master. Everybody has heard the remark, "It is as hard as squaring
a circle," though many people have a very hazy notion of what it means.
If you have a circle of given diameter and wish to find the side of a
square that shall contain exactly the same area, you are confronted with
the problem of squaring the circle. Well, it cannot be done with
exactitude (though we can get an answer near enough for all practical
purposes), because it is not possible to say in exact numbers what is the
ratio of the diameter to the circumference. But it is only in recent
times that it has been proved to be impossible, for it is one thing not
to be able to perform a certain feat, but quite another to prove that it
cannot be done. Only uninstructed cranks now waste their time in trying
to square the circle.
Again, we can never measure exactly in numbers the diagonal of a square.
If you have a window pane exactly a foot on every side, there is the
distance from corner to corner staring you in the face, yet you can never
say in exact numbers what is the length of that diagonal. The simple
person will at once suggest that we might take our diagonal first, say an
exact foot, and then construct our square. Yes, you can do this, but then
you can never say exactly what is the length of the side. You can have it
which way you like, but you cannot have it both ways.
All my readers know what a magic square is. The numbers 1 to 9 can be
arranged in a square of nine cells, so that all the columns and rows and
each of the diagonals will add up 15. It is quite easy; and there is only
one way of doing it, for we do not count as different the arrangements
obtained by merely turning round the square and reflecting it in a
mirror. Now if we wish to make a magic square of the 16 numbers, 1 to 16,
there are just 880 different ways of doing it, again not counting
reversals and reflections. This has been finally proved of recent years.
But how many magic squares may be formed with the 25 numbers, 1 to 25,
nobody knows, and we shall have to extend our knowledge in certain
directions before we can hope to solve the puzzle. But it is surprising
to find that exactly 174,240 such squares may be formed of one particular
restricted kind only--the bordered square, in which the inner square of
nine cells is itself magic. And I have shown how this number may be at
once doubled by merely converting every bordered square--by a simple
rule--into a non-bordered one.
Then vain attempts have been made to construct a magic square by what is
called a "knight's tour" over the chess-board, numbering each square that
the knight visits in succession, 1, 2, 3, 4, etc.; and it has been done,
with the exception of the two diagonals, which so far have baffled all
efforts. But it is not certain that it cannot be done.
Though the contents of the present volume are in the main entirely
original, some very few old friends will be found; but these will not, I
trust, prove unwelcome in the new dress that they have received. The
puzzles are of every degree of difficulty, and so varied in character
that perhaps it is not too much to hope that every true puzzle lover will
find ample material to interest--and possibly instruct. In some cases I
have dealt with the methods of solution at considerable length, but at
other times I have reluctantly felt obliged to restrict myself to giving
the bare answers. Had the full solutions and proofs been given in the
case of every puzzle, either half the problems would have had to be
omitted, or the size of the book greatly increased. And the plan that I
have adopted has its advantages, for it leaves scope for the mathematical
enthusiast to work out his own analysis. Even in those cases where I have
given a general formula for the solution of a puzzle, he will find great
interest in verifying it for himself.