# 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.

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.

• The solution is correct, but you can state it this way to make it simpler: Call the corners and the middles "A", call the edges "B". Now every closed path is alternating between A's and B's, so their total number should be the same. However, the number of A's is 14 and the number of B's is 12, which is a contradiction. – Puzzle Prime Sep 19 '15 at 22:40
• The three-dimensional checkerboard suggested by BenFrankel seems a good way of seeing this conceptually. The termite always moves from either a black to a white cubelet, or a white to a black cubelet. One of the two colours (shades?) is MC and the other is E. – Marconius Sep 19 '15 at 22:57

EDIT: Nevermind, misunderstood the question.

Color the cubelets so that the corners are black. There are 14 black and 13 white cubelets. At move 1, the termite is on a black cubelet, and the termite alternates between black and white for each move. Therefore at move 27, the termite will again be on a black cubelet, but the center is white. Thus the termite cannot complete such a journey.

• The termite can also start on an edge or face center of the larger cube. You general idea is good, though. – Marconius Sep 19 '15 at 21:22
• "center of a face" this misled me, I was thinking that the termite starts at the center of a face of the larger cube. – Ben Frankel Sep 19 '15 at 21:40
• @Zandar - If the 6 face centers are black, the interior center (body center) is white. – Marconius Sep 19 '15 at 21:45
• apologies for the confusion (there is a reference to "face of any of the outside cubes ..."). I think the term cubelet is good and would have made the question clearer - some of the wording is inherited. Your solution method just needs a minor tweak for the "other" case where the termite starts on an edge, i.e. white cubelet. +1 – Marconius Sep 19 '15 at 21:52