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Amusements in Mathematics

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.
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?
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.
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?
"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?"