VARIOUS
GEOMETRICAL PUZZLES.
"So various are the tastes of men."
MARK AKENSIDE.
178.—THE
CARDBOARD BOX.
This puzzle is not difficult, but it will be found entertaining to discover
the simple rule for its solution. I have a rectangular cardboard box. The top
has an area of 120 square inches, the side 96 square inches, and the end 80
square inches. What are the exact dimensions of the box?
Solution
179.—STEALING
THE BELL-ROPES.
Two men broke into a church tower one night to steal the bell-ropes. The two
ropes passed through holes in the wooden ceiling high above them, and they lost
no time in climbing to the top. Then one man drew his knife and cut the rope
above his head, in consequence of which he fell to the floor and was badly
injured. His fellow-thief called out that it served him right for being such a
fool. He said that he should have done as he was doing, upon which he cut the
rope below the place at which he held on. Then, to his dismay, he found that he
was in no better plight, for, after hanging on as long as his strength lasted,
he was compelled to let go and fall beside his comrade. Here they were both
found the next morning with their limbs broken. How far did they fall? One of
the ropes when they found it was just touching the floor, and when you pulled
the end to the wall, keeping the rope taut, it touched a point just three inches
above the floor, and the wall was four feet from the rope when it hung at rest.
How long was the rope from floor to ceiling?
Solution 180.—THE
FOUR SONS.
Readers will recognize the diagram as a familiar friend of their youth. A man
possessed a square-shaped estate. He bequeathed to his widow the quarter of it
that is shaded off. The remainder was to be divided equitably amongst his four
sons, so that each should receive land of exactly the same area and exactly
similar in shape. We are shown how this was done. But the remainder of the story
is not so generally known. In the centre of the estate was a well, indicated by
the dark spot, and Benjamin, Charles, and David complained that the division was
not "equitable," since Alfred had access to this well, while they could not
reach it without trespassing on somebody else's land. The puzzle is to show how
the estate is to be apportioned so that each son shall have land of the same
shape and area, and each have access to the well without going off his own
land.
Solution 181.—THE
THREE RAILWAY STATIONS.
As I sat in a railway carriage I noticed at the other end of the compartment
a worthy squire, whom I knew by sight, engaged in conversation with another
passenger, who was evidently a friend of his.
"How far have you to drive to your place from the railway station?" asked the
stranger.
"Well," replied the squire, "if I get out at Appleford, it is just the same
distance as if I go to Bridgefield, another fifteen miles farther on; and if I
changed at Appleford and went thirteen miles from there to Carterton, it would
still be the same distance. You see, I am equidistant from the three stations,
so I get a good choice of trains."
Now I happened to know that Bridgefield is just fourteen miles from
Carterton, so I amused myself in working out the exact distance that the squire
had to drive home whichever station he got out at. What was the distance?
Solution 182.—THE
GARDEN PUZZLE.
Professor Rackbrain tells me that he was recently smoking a friendly pipe
under a tree in the garden of a country acquaintance. The garden was enclosed by
four straight walls, and his friend informed him that he had measured these and
found the lengths to be 80, 45, 100, and 63 yards respectively. "Then," said the
professor, "we can calculate the exact area of the garden." "Impossible," his
host replied, "because you can get an infinite number of different
shapes with those four sides." "But you forget," Rackbrane said, with a twinkle
in his eye, "that you told me once you had planted this tree equidistant from
all the four corners of the garden." Can you work out the garden's area?
Solution 183.—DRAWING
A SPIRAL.
If you hold the page horizontally and give it a quick rotary motion while
looking at the centre of the spiral, it will appear to revolve. Perhaps a good
many readers are acquainted with this little optical illusion. But the puzzle is
to show how I was able to draw this spiral with so much exactitude without using
anything but a pair of compasses and the sheet of paper on which the diagram was
made. How would you proceed in such circumstances?
Solution 184.—HOW
TO DRAW AN OVAL.
Can you draw a perfect oval on a sheet of paper with one sweep of the
compasses? It is one of the easiest things in the world when you know how.
Solution 185.—ST.
GEORGE'S BANNER.
At a celebration of the national festival of St. George's Day I was
contemplating the familiar banner of the patron saint of our country. We all
know the red cross on a white ground, shown in our illustration. This is the
banner of St. George. The banner of St. Andrew (Scotland) is a white "St.
Andrew's Cross" on a blue ground. That of St. Patrick (Ireland) is a similar
cross in red on a white ground. These three are united in one to form our Union
Jack.
Now on looking at St. George's banner it occurred to me that the following
question would make a simple but pretty little puzzle. Supposing the flag
measures four feet by three feet, how wide must the arm of the cross be if it is
required that there shall be used just the same quantity of red and of white
bunting?
Solution 186.—THE
CLOTHES LINE PUZZLE.
A boy tied a clothes line from the top of each of two poles to the base of
the other. He then proposed to his father the following question. As one pole
was exactly seven feet above the ground and the other exactly five feet, what
was the height from the ground where the two cords crossed one another?
Solution 187.—THE
MILKMAID PUZZLE.
Here is a little pastoral puzzle that the reader may, at first sight, be led
into supposing is very profound, involving deep calculations. He may even say
that it is quite impossible to give any answer unless we are told something
definite as to the distances. And yet it is really quite "childlike and
bland."
In the corner
of a field is seen a milkmaid milking a cow, and on the other side of the field
is the dairy where the extract has to be deposited. But it has been noticed that
the young woman always goes down to the river with her pail before returning to
the dairy. Here the suspicious reader will perhaps ask why she pays these visits
to the river. I can only reply that it is no business of ours. The alleged milk
is entirely for local consumption.
"Where are you going to, my pretty
maid?"
"Down to the river, sir," she
said.
"I'll not choose your
dairy, my pretty maid."
"Nobody axed
you, sir," she said.
If one had any curiosity in the matter, such an independent spirit would
entirely disarm one. So we will pass from the point of commercial morality to
the subject of the puzzle.
Draw a line from the milking-stool down to the river and thence to the door
of the dairy, which shall indicate the shortest possible route for the milkmaid.
That is all. It is quite easy to indicate the exact spot on the bank of the
river to which she should direct her steps if she wants as short a walk as
possible. Can you find that spot?
Solution 188.—THE
BALL PROBLEM.
A stonemason was engaged the other day in cutting out a round ball for the
purpose of some architectural decoration, when a smart schoolboy came upon the
scene.
"Look here," said the mason, "you seem to be a sharp youngster, can you tell
me this? If I placed this ball on the level ground, how many other balls of the
same size could I lay around it (also on the ground) so that every ball should
touch this one?"
The boy at once gave the correct answer, and then put this little question to
the mason:—
"If the surface of that ball contained just as many square feet as its volume
contained cubic feet, what would be the length of its diameter?"
The stonemason could not give an answer. Could you have replied correctly to
the mason's and the boy's questions?
Solution 189.—THE
YORKSHIRE ESTATES.
I was on a visit to one of the large towns of Yorkshire. While walking to the
railway station on the day of my departure a man thrust a hand-bill upon me, and
I took this into the railway carriage and read it at my leisure. It informed me
that three Yorkshire neighbouring estates were to be offered for sale. Each
estate was square in shape, and they joined one another at their corners, just
as shown in the diagram. Estate A contains exactly 370 acres, B contains 116
acres, and C 74 acres.
Now, the little triangular bit of land enclosed by the three square estates
was not offered for sale, and, for no reason in particular, I became curious as
to the area of that piece. How many acres did it contain?
Solution 190.—FARMER
WURZEL'S ESTATE.
I will now present another land problem. The demonstration of the answer that
I shall give will, I think, be found both interesting and easy of
comprehension.
Farmer Wurzel owned the three square fields shown in the annexed plan,
containing respectively 18, 20, and 26 acres. In order to get a ring-fence round
his property he bought the four intervening triangular fields. The puzzle is to
discover what was then the whole area of his estate.
Solution 191.—THE
CRESCENT PUZZLE.
Here is an easy geometrical puzzle. The crescent is formed by two circles,
and C is the centre of the larger circle. The width of the crescent between B
and D is 9 inches, and between E and F 5 inches. What are the diameters of the
two circles?
Solution 192.—THE
PUZZLE WALL.
There was a small lake, around which four poor men built their cottages. Four
rich men afterwards built their mansions, as shown in the illustration, and they
wished to have the lake to themselves, so they instructed a builder to put up
the shortest possible wall that would exclude the cottagers, but give themselves
free access to the lake. How was the wall to be built?
Solution 193.—THE
SHEEP-FOLD.
It is a curious fact that the answers always given to some of the best-known
puzzles that appear in every little book of fireside recreations that has been
published for the last fifty or a hundred years are either quite unsatisfactory
or clearly wrong. Yet nobody ever seems to detect their faults. Here is an
example:—A farmer had a pen made of fifty hurdles, capable of holding a hundred
sheep only. Supposing he wanted to make it sufficiently large to hold double
that number, how many additional hurdles must he have?
Solution 194.—THE
GARDEN WALLS.
A speculative country builder has a circular field, on which he has erected
four cottages, as shown in the illustration. The field is surrounded by a brick
wall, and the owner undertook to put up three other brick walls, so that the
neighbours should not be overlooked by each other, but the four tenants insist
that there shall be no favouritism, and that each shall have exactly the same
length of wall space for his wall fruit trees. The puzzle is to show how the
three walls may be built so that each tenant shall have the same area of ground,
and precisely the same length of wall.
Of course, each garden must be entirely enclosed by its walls, and it must be
possible to prove that each garden has exactly the same length of wall. If the
puzzle is properly solved no figures are necessary.
Solution 195.—LADY
BELINDA'S GARDEN.
Lady Belinda is an enthusiastic gardener. In the illustration she is depicted
in the act of worrying out a pleasant little problem which I will relate. One of
her gardens is oblong in shape, enclosed by a high holly hedge, and she is
turning it into a rosary for the cultivation of some of her choicest roses. She wants to
devote exactly half of the area of the garden to the flowers, in one large bed,
and the other half to be a path going all round it of equal breadth throughout.
Such a garden is shown in the diagram at the foot of the picture. How is she to
mark out the garden under these simple conditions? She has only a tape, the
length of the garden, to do it with, and, as the holly hedge is so thick and
dense, she must make all her measurements inside. Lady Belinda did not know the
exact dimensions of the garden, and, as it was not necessary for her to know, I
also give no dimensions. It is quite a simple task no matter what the size or
proportions of the garden may be. Yet how many lady gardeners would know just
how to proceed? The tape may be quite plain—that is, it need not be a graduated
measure.
Solution 196.—THE
TETHERED GOAT.
Here is a little problem that everybody should know how to solve. The goat is
placed in a half-acre meadow, that is in shape an equilateral triangle. It is
tethered to a post at one corner of the field. What should be the length of the
tether (to the nearest inch) in order that the goat shall be able to eat just
half the grass in the field? It is assumed that the goat can feed to the end of
the tether.
Solution 197.—THE
COMPASSES PUZZLE.
It is curious how an added condition or restriction will sometimes convert an
absurdly easy puzzle into an interesting and perhaps difficult one. I remember
buying in the street many years ago a little mechanical puzzle that had a
tremendous sale at the time. It consisted of a medal with holes in it, and the
puzzle was to work a ring with a gap in it from hole to hole until it was
finally detached. As I was walking along the street I very soon acquired the
trick of taking off the ring with one hand while holding the puzzle in my
pocket. A friend to whom I showed the little feat set about accomplishing it
himself, and when I met him some days afterwards he exhibited his proficiency in
the art. But he was a little taken aback when I then took the puzzle from him
and, while simply holding the medal between the finger and thumb of one hand, by
a series of little shakes and jerks caused the ring, without my even touching
it, to fall off upon the floor. The following little poser will probably prove a
rather tough nut for a great many readers, simply on account of the restricted
conditions:—
Show how to find exactly the middle of any straight line by means of the
compasses only. You are not allowed to use any ruler, pencil, or other
article—only the compasses; and no trick or dodge, such as folding the paper,
will be permitted. You must simply use the compasses in the ordinary legitimate
way.
Solution 198.—THE
EIGHT STICKS.
I have eight sticks, four of them being exactly half the length of the
others. I lay every one of these on the table, so that they enclose three
squares, all of the same size. How do I do it? There must be no loose ends
hanging over.
Solution 199.—PAPA'S
PUZZLE.
Here is a puzzle by Pappus, who lived at Alexandria about the end of the
third century. It is the fifth proposition in the eighth book of his
Mathematical Collections. I give it in the form that I presented it some
years ago under the title "Papa's Puzzle," just to see how many readers would
discover that it was by Pappus himself. "The little maid's papa has taken two
different-sized rectangular pieces of cardboard, and has clipped off a
triangular piece from one of them, so that when it is suspended by a thread from
the point A it hangs with the long side perfectly horizontal, as shown in the
illustration. He has perplexed the child by asking her to find the point A on
the other card, so as to produce a similar result when cut and suspended by a
thread." Of course, the point must not be found by trial clippings. A curious and pretty point is
involved in this setting of the puzzle. Can the reader discover it?
Solution 200.—A
KITE-FLYING PUZZLE.
While accompanying my friend Professor Highflite during a scientific
kite-flying competition on the South Downs of Sussex I was led into a little
calculation that ought to interest my readers. The Professor was paying out the
wire to which his kite was attached from a winch on which it had been rolled
into a perfectly spherical form. This ball of wire was just two feet in
diameter, and the wire had a diameter of one-hundredth of an inch. What was the
length of the wire?
Now, a simple little question like this that everybody can perfectly
understand will puzzle many people to answer in any way. Let us see whether,
without going into any profound mathematical calculations, we can get the answer
roughly—say, within a mile of what is correct! We will assume that when the wire
is all wound up the ball is perfectly solid throughout, and that no allowance
has to be made for the axle that passes through it. With that simplification, I
wonder how many readers can state within even a mile of the correct answer the
length of that wire.
Solution 201.—HOW
TO MAKE CISTERNS.
Our friend in the illustration has a large sheet of zinc, measuring (before
cutting) eight feet by three feet, and he has cut out square pieces (all of the
same size) from the four corners and now proposes to fold up the sides, solder
the edges, and make a cistern. But the point that puzzles him is this: Has he
cut out those square pieces of the correct size in order that the cistern may
hold the greatest possible quantity of water? You see, if you cut them very
small you get a very shallow cistern; if you cut them large you get a
tall and slender one. It is all a question of finding a way of cutting put these
four square pieces exactly the right size. How are we to avoid making them too
small or too large?
Solution 202.—THE
CONE PUZZLE.
I have a wooden cone, as shown in Fig. 1. How am I to cut out of it the
greatest possible cylinder? It will be seen that I can cut out one that is long
and slender, like Fig. 2, or short and thick, like Fig. 3. But neither is the
largest possible. A child could tell you where to cut, if he knew the rule. Can
you find this simple rule?
Solution 203.—CONCERNING
WHEELS.
There are some curious facts concerning the movements of wheels that are apt
to perplex the novice. For example: when a railway train is travelling from
London to Crewe certain parts of the train at any given moment are actually
moving from Crewe towards London. Can you indicate those parts? It seems absurd
that parts of the same train can at any time travel in opposite directions, but
such is the case.
In the accompanying illustration we have two wheels. The lower one is
supposed to be fixed and the upper one running round it in the direction of the
arrows. Now, how many times does the upper wheel turn on its own axis in making
a complete revolution of the other wheel? Do not be in a hurry with your answer,
or you are almost certain to be wrong. Experiment with two pennies on the table
and the correct answer will surprise you, when you succeed in seeing it.
Solution 204.—A
NEW MATCH PUZZLE.
In the illustration eighteen matches are shown arranged so that they enclose
two spaces, one just twice as large as the other. Can you rearrange them (1) so
as to enclose two four-sided spaces, one exactly three times as large as the
other, and (2) so as to enclose two five-sided spaces, one exactly three times
as large as the other? All the eighteen matches must be fairly used in each
case; the two spaces must be quite detached, and there must be no loose ends or
duplicated matches.
Solution 205.—THE
SIX SHEEP-PENS.
Here is a new little puzzle with matches. It will be seen in the illustration
that thirteen matches, representing a farmer's hurdles, have been so placed that
they enclose six sheep-pens all of the same size. Now, one of these hurdles was
stolen, and the farmer wanted still to enclose six pens of equal size with the
remaining twelve. How was he to do it? All the twelve matches must be fairly
used, and there must be no duplicated matches or loose ends.
Solution
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