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© Copyright 2000, Jim Loy
Let me repeat the problem. In a room that it 12x12x30, there is a
spider near the top of one of the smaller walls. She is one unit (foot or
whatever) from the ceiling, and centered between the side walls. She sees a fly
on the opposite wall, one unit from the floor and centered between the side
walls. What is the shortest path that she can travel (on walls, ceiling, an/or
floor) to get to the fly (which remains stationary the entire time)?
Solution: The simplest way to solve this is to flatten out
the walls, ceiling, and floor to make a map of the room. This particular
spider, after all, lives in a one dimensional world (for the purposes of this
puzzle). There are many possible paths. On the right, we see three likely
paths. We can find the lengths of two of these paths using the Pythagorean
Theorem. We have right triangles with sides 24x32x40 and 19x37x41.59. The
direct route by the ceiling (or floor which is more dangerous for a spider) is
42. The shortest path is 40. A fourth path, just using walls (and no floor or
ceiling), is 43.17. In three-space, the shortest path looks like this:
You will be happy to know that the story had a happy ending. The
fly was delicious. Return to original puzzle.
Addendum:
The legendary Sam Lloyd published a different version of this puzzle (after Dudeney published his), with a different solution. Instead of a spider and a fly, he was stringing a wire from a door bell button to an electrical bell on the opposite wall. Lloyd's puzzle is different in that the two points that we want to connect are not one foot from the edge of a wall, but three feet. The same paths are possible, but the lengths are different. You may want to try to solve this version. The correct path length is 41.79. I won't tell you which one of the three paths is the correct one, in the second diagram above.