Consider a wooden 5x5x5 block. A termite wishes to eat the block in the following way

1) It starts with an external, central 1x1x1 cube of any face.
2) The termite eats it, and the heads towards a neighboring 1x1x1 cube.
3) It repeats step 2 until it can't go any further.

Determine:

a) A possible path for the termite to follow, in order to eat every single 1x1x1 cube.

b) If it's possible for the termite to eat all 1x1x1 cubes, knowing it ate the central internal cube last.

(In reply to Reminds me of Konigsberg Puzzle by brianjn)

In the classic bridge problem, the point was to use all of the bridges once and only once. Euler proved that it was impossible because there were more than two vertices (islands and banks)that were connected to an odd number of paths (bridges).

What you propose is to consider the cube as a graph with 125 (not 27 -- it's 5³, not 3³) vertices and 300 paths. It can be done with graph theory, but it won't be as simple as the Konigsberg problem. Insted of all paths being traced, there will only be 124 of the 300 paths traced.

Posted by TomM on 2003-05-12 18:13:07