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UNICURSAL
AND ROUTE PROBLEMS.
"I see them on their winding way."
REGINALD HEBER.
It is reasonable to suppose that from the earliest ages one man has asked
another such questions as these: "Which is the nearest way home?" "Which is the
easiest or pleasantest way?" "How can we find a way that will enable us to dodge
the mastodon and the plesiosaurus?" "How can we get there without ever crossing
the track of the enemy?" All these are elementary route problems, and they can
be turned into good puzzles by the introduction of some conditions that
complicate matters. A variety of such complications will be found in the
following examples. I have also included some enumerations of more or less
difficulty. These afford excellent practice for the reasoning faculties, and
enable one to generalize in the case of symmetrical forms in a manner that is
most instructive.
239.—A
JUVENILE PUZZLE.
For years I have been perpetually consulted by my juvenile friends about this
little puzzle. Most children seem to know it, and yet, curiously enough, they
are invariably unacquainted with the answer. The question they always ask is,
"Do, please, tell me whether it is really possible." I believe Houdin the
conjurer used to be very fond of giving it to his child friends, but I cannot
say whether he invented the little puzzle or not. No doubt a large number of my
readers will be glad to have the mystery of the solution cleared up, so I make
no apology for introducing this old "teaser."
The puzzle is to draw with three strokes of the pencil the diagram that the
little girl is exhibiting in the illustration. Of course, you must not remove
your pencil from the paper during a stroke or go over the same line a second
time. You will find that you can get in a good deal of the figure with one continuous stroke,
but it will always appear as if four strokes are necessary.
Another form of the puzzle is to draw the diagram on a slate and then rub it
out in three rubs.
Solution 240.—THE
UNION JACK.
The illustration is a rough sketch somewhat resembling the British flag, the
Union Jack. It is not possible to draw the whole of it without lifting the
pencil from the paper or going over the same line twice. The puzzle is to find
out just how much of the drawing it is possible to make without lifting
your pencil or going twice over the same line. Take your pencil and see what is
the best you can do.
Solution 241.—THE
DISSECTED CIRCLE.
How many continuous strokes, without lifting your pencil from the paper, do
you require to draw the design shown in our illustration? Directly you change
the direction of your pencil it begins a new stroke. You may go over the same
line more than once if you like. It requires just a little care, or you may find
yourself beaten by one stroke.
Solution 242.—THE
TUBE INSPECTOR'S PUZZLE.
The man in our illustration is in a little dilemma. He has just been
appointed inspector of a certain system of tube railways, and it is his duty to
inspect regularly, within a stated period, all the company's seventeen lines
connecting twelve stations, as shown on the big poster plan that he is
contemplating. Now he wants to arrange his route so that it shall take him over
all the lines with as little travelling as possible. He may begin where he likes
and end where he likes. What is his shortest route?
Could anything be simpler? But the reader will soon find that, however he
decides to proceed, the inspector must go over some of the lines more than once.
In other words, if we say that the stations are a mile apart, he will have to
travel more than seventeen miles to inspect every line. There is the little
difficulty. How far is he compelled to travel, and which route do you
recommend?
Solution 243.—VISITING
THE TOWNS.
A traveller, starting from town No. 1, wishes to visit every one of the towns
once, and once only, going only by roads indicated by straight lines. How many
different routes are there from which he can select? Of course, he must end his
journey at No. 1, from which he started, and must take no notice of cross roads,
but go straight from town to town. This is an absurdly easy puzzle, if you go
the right way to work.
Solution 244.—THE
FIFTEEN TURNINGS.
Here is another queer travelling puzzle, the solution of which calls for
ingenuity. In this case the traveller starts from the black town and wishes to
go as far as possible while making only fifteen turnings and never going along
the same road twice. The towns are supposed to be a mile apart. Supposing, for
example, that he went straight to A, then straight to B, then to C, D, E, and F,
you will then find that he has travelled thirty-seven miles in five turnings.
Now, how far can he go in fifteen turnings?
Solution 245.—THE
FLY ON THE OCTAHEDRON.
"Look here," said the professor to his colleague, "I have been watching that
fly on the octahedron, and it confines its walks entirely to the edges. What can
be its reason for avoiding the sides?"
"Perhaps it is trying to solve some route problem," suggested the other.
"Supposing it to start from the top point, how many different routes are there
by which it may walk over all the edges, without ever going twice along the same
edge in any route?"
The problem was a harder one than they expected, and after working at it
during leisure moments for several days their results did not agree—in fact,
they were both wrong. If the reader is surprised at their failure, let him
attempt the little puzzle himself. I will just explain that the octahedron is
one of the five regular, or Platonic, bodies, and is contained under eight equal
and equilateral triangles. If you cut out the two pieces of cardboard of the
shape shown in the margin of the illustration, cut half through along the dotted
lines and then bend them and put them together, you will have a perfect
octahedron. In any route over all the edges it will be found that the fly must
end at the point of departure at the top.
Solution 246.—THE
ICOSAHEDRON PUZZLE.
The icosahedron is another of the five regular, or Platonic, bodies having
all their sides, angles, and planes similar and equal. It is bounded by twenty
similar equilateral triangles. If you cut out a piece of cardboard of the form
shown in the smaller diagram, and cut half through along the dotted lines, it
will fold up and form a perfect icosahedron.
Now, a Platonic body does not mean a heavenly body; but it will suit the purpose of our
puzzle if we suppose there to be a habitable planet of this shape. We will also
suppose that, owing to a superfluity of water, the only dry land is along the
edges, and that the inhabitants have no knowledge of navigation. If every one of
those edges is 10,000 miles long and a solitary traveller is placed at the North
Pole (the highest point shown), how far will he have to travel before he will
have visited every habitable part of the planet—that is, have traversed every
one of the edges?
Solution 247.—INSPECTING
A MINE.
The diagram is supposed to represent the passages or galleries in a mine. We
will assume that every passage, A to B, B to C, C to H, H to I, and so on, is
one furlong in length. It will be seen that there are thirty-one of these
passages. Now, an official has to inspect all of them, and he descends by the
shaft to the point A. How far must he travel, and what route do you recommend?
The reader may at first say, "As there are thirty-one passages, each a furlong
in length, he will have to travel just thirty-one furlongs." But this is
assuming that he need never go along a passage more than once, which is not the
case. Take your pencil and try to find the shortest route. You will soon
discover that there is room for considerable judgment. In fact, it is a
perplexing puzzle.
Solution 248.—THE
CYCLISTS' TOUR.
Two cyclists were consulting a road map in preparation for a little tour
together. The circles represent towns, and all the good roads are represented by
lines. They are starting from the town with a star, and must complete their tour
at E. But before arriving there they want to visit every other town once, and
only once. That is the difficulty. Mr. Spicer said, "I am certain we can find a
way of doing it;" but Mr. Maggs replied, "No way, I'm sure." Now, which of them
was correct? Take your pencil and see if you can find any way of doing it. Of
course you must keep to the roads indicated.
Solution 249.—THE
SAILOR'S PUZZLE.
The sailor depicted in the illustration stated that he had since his boyhood
been engaged in trading with a small vessel among some twenty little islands in
the Pacific. He supplied the rough chart of which I have given a copy, and
explained that the lines from island to island represented the only routes that
he ever adopted. He always started from island A at the beginning of the season,
and then visited every island once, and once only, finishing up his tour at the
starting-point A. But he always put off his visit to C as long as possible, for
trade reasons that I need not enter into. The puzzle is to discover his exact
route, and this can be done with certainty. Take your pencil and, starting at A,
try to trace it out. If you write down the islands in the order in which you
visit them—thus, for example, A, I, O, L, G, etc.—you can at once see if you
have visited an island twice or omitted any. Of course, the crossings of the
lines must be ignored—that is, you must continue your route direct, and you are
not allowed to switch off at a crossing and proceed in another direction. There
is no trick of this kind in the puzzle. The sailor knew the best route. Can you
find it?
Solution 250.—THE
GRAND TOUR.
One of the everyday puzzles of life is the working out of routes. If you are
taking a holiday on your bicycle, or a motor tour, there always arises the
question of how you are to make the best of your time and other resources. You
have determined to get as far as some particular place, to include visits to
such-and-such a town, to try to see something of special interest elsewhere, and
perhaps to try to look up an old friend at a spot that will not take you much
out of your way. Then you have to plan your route so as to avoid bad roads,
uninteresting country, and, if possible, the necessity of a return by the same
way that you went. With a map before you, the interesting puzzle is attacked and
solved. I will present a little poser based on these lines.
I give a rough map of a country—it is not necessary to say what particular
country—the circles representing towns and the dotted lines the railways
connecting them. Now there lived in the town marked A a man who was born there,
and during the whole of his life had never once left his native place. From his
youth upwards he had been very industrious, sticking incessantly to his trade,
and had no desire whatever to roam abroad. However, on attaining his fiftieth
birthday he decided to see something of his country, and especially to pay a
visit to a very old friend living at the town marked Z. What he proposed was this: that he would
start from his home, enter every town once and only once, and finish his journey
at Z. As he made up his mind to perform this grand tour by rail only, he found
it rather a puzzle to work out his route, but he at length succeeded in doing
so. How did he manage it? Do not forget that every town has to be visited once,
and not more than once.
Solution 251.—WATER,
GAS, AND ELECTRICITY.
There are some half-dozen puzzles, as old as the hills, that are perpetually
cropping up, and there is hardly a month in the year that does not bring
inquiries as to their solution. Occasionally one of these, that one had thought
was an extinct volcano, bursts into eruption in a surprising manner. I have
received an extraordinary number of letters respecting the ancient puzzle that I
have called "Water, Gas, and Electricity." It is much older than electric
lighting, or even gas, but the new dress brings it up to date. The puzzle is to
lay on water, gas, and electricity, from W, G, and E, to each of the three
houses, A, B, and C, without any pipe crossing another. Take your pencil and
draw lines showing how this should be done. You will soon find yourself landed
in difficulties.
Solution 252.—A
PUZZLE FOR MOTORISTS.
Eight motorists drove to church one morning. Their respective houses and
churches, together with the only roads available (the dotted lines), are shown.
One went from his house A to his church A, another from his house B to his
church B, another from C to C, and so on, but it was afterwards found that no
driver ever crossed the track of another car. Take your pencil and try to trace
out their various routes.
Solution 253.—A
BANK HOLIDAY PUZZLE.
Two friends were spending their bank holiday on a cycling trip. Stopping for
a rest at a village inn, they consulted a route map, which is represented in our
illustration in an exceedingly simplified form, for the puzzle is interesting
enough without all the original complexities. They started from the town in the
top left-hand corner marked A. It will be seen that there are one hundred and
twenty such towns, all connected by straight roads. Now they discovered that
there are exactly 1,365 different routes by which they may reach their
destination,
always travelling either due south or due east. The puzzle is to discover which
town is their destination.
Of course, if you find that there are more than 1,365 different routes to a
town it cannot be the right one.
Solution 254.—THE
MOTOR-CAR TOUR.
In the above diagram the circles represent towns and the lines good roads. In
just how many different ways can a motorist, starting from London (marked with
an L), make a tour of all these towns, visiting every town once, and only once,
on a tour, and always coming back to London on the last ride? The exact reverse
of any route is not counted as different.
Solution 255.—THE
LEVEL PUZZLE.
This is a simple counting puzzle. In how many different ways can you spell
out the word LEVEL by placing the point of your pencil on an L and then passing
along the lines from letter to letter. You may go in any direction, backwards or
forwards. Of course you are not allowed to miss letters—that is to say, if you
come to a letter you must use it.
Solution 256.—THE
DIAMOND PUZZLE.
IN how many different ways may the word DIAMOND be read in the arrangement
shown? You may start wherever you like at a D and go up or down, backwards or
forwards, in and out, in any direction you like, so long as you always pass from
one letter to another that adjoins it. How many ways are there?
Solution 257.—THE
DEIFIED PUZZLE.
In how many different ways may the word DEIFIED be read in this arrangement
under the same
conditions as in the last puzzle, with the addition that you can use any letters
twice in the same reading?
Solution 258.—THE
VOTERS' PUZZLE.
Here we have, perhaps, the most interesting form of the puzzle. In how many
different ways can you read the political injunction, "RISE TO VOTE, SIR," under
the same conditions as before? In this case every reading of the palindrome
requires the use of the central V as the middle letter.
Solution 259.—HANNAH'S
PUZZLE.
A man was in love with a young lady whose Christian name was Hannah. When he
asked her to be his wife she wrote down the letters of her name in this
manner:—
and promised that she would be his if he could tell her correctly in how many
different ways it was possible to spell out her name, always passing from one
letter to another that was adjacent. Diagonal steps are here allowed. Whether
she did this merely to tease him or to test his cleverness is not recorded, but
it is satisfactory to know that he succeeded. Would you have been equally
successful? Take your pencil and try. You may start from any of the H's and go
backwards or forwards and in any direction, so long as all the letters in a
spelling are adjoining one another. How many ways are there, no two exactly
alike?
Solution 260.—THE
HONEYCOMB PUZZLE.
Here is a little puzzle with the simplest possible conditions. Place the
point of your pencil on a letter in one of the cells of the honeycomb, and trace
out a very familiar proverb by passing always from a cell to one that is
contiguous to it. If you take the right route you will have visited every cell
once, and only once. The puzzle is much easier than it looks.
Solution 261.—THE
MONK AND THE BRIDGES.
In this case I give a rough plan of a river with an island and five bridges.
On one side of the river is a monastery, and on the other side is seen a monk in
the foreground. Now, the monk has decided that he will cross every bridge once,
and only once, on his return to the monastery. This is, of course, quite easy to
do, but on the way he thought to himself, "I wonder how many different routes
there are from which I might have selected." Could you have told him? That is
the puzzle. Take your pencil and trace out a route that will take you once over all the five
bridges. Then trace out a second route, then a third, and see if you can count
all the variations. You will find that the difficulty is twofold: you have to
avoid dropping routes on the one hand and counting the same routes more than
once on the other.
Solution
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