THE FLY ON THE OCTAHEDRON.
(
Unicursal and Route Problems)
"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.
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