The Termite Problem

From: Martin Gardner. My Best Mathematical and Logic Puzzles. 1994 Dover Publications

Imagine a large cube formed by gluing together 27 smaller wooden cubes of uniform size as shown.

A termite starts at the center of the face of any one of the outside cubes and bores a path that takes him once throught every cube. His movement is always parallel to a side of the large cube, never diagonal.

Is it possible for the termite to bore through each of the 26 outside cubes once and only once, then finish his trip by entering the central cube for the first time? If possible, show how it can be done; if impossible, prove it...

It is assumed that the termite, once it has bored into a small cube, follows a path entirely within the large cube. Otherwise, it could crawl out on the surface of the large cube and move along the surface to a new spot of entry. If this were permitted, there would, of course, be no problem.