geometry - Termite eating through a large cube composed of 27 smaller cubes while not moving diagonally

The is a large cube formed by gluing together 27 smaller cubes of uniform size (see figure). A termite starts at the center of a face of any of the outside cubes and bores a path that takes him once through each 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 complete his journey by entering the central cube for the first time?
If possible, show how; if not, prove it.

This puzzle is from Martin Garner's The Colossal Book of Short Puzzles and Problems.

I hope the problem is not too boring (ahem) for members of this forum.

Answer

The termite is doomed to fail.

There are 8 corner cubes, 12 edge cubes, 6 middle cubes, and 1 center cube. Since the last cube must be the center cube, the second to last cube must be a middle cube.

Middle cubes and Corner cubes behave roughly the same way. They can't link to each other, but they can both link to an Edge cubes. Because of this, we will refer to Middle cubes and Corner cubes with MC and Edge cubes with E.

Any path including MCs and Es will need to alternate between the two because Es and MCs can't link to themselves. We have 14 MCs and 12 Es, however, so no matter how you figure it there aren't enough Es to link together all the MCs. You would need at least 13. Sorry, termite.

Thanks to Artur Kirkoryan for coming up with a way to simplify my proof and make it much more understandable.