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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?

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243.—­VISITING THE TOWNS.

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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.

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?

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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?”

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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.