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?

[Illustration]

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.

[Illustration]

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.

[Illustration]

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