ow, how far can he go in
fifteen turnings?

[Illustration]

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?"

[Illustration]

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.

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