[math-fun] Nonexistence of cubed cube

I was scanning the cool Geometry Junkyard website (http://www.ics.uci.edu/~eppstein/junkyard/3d.html) and came across this link http://www.ics.uci.edu/~eppstein/junkyard/no-cubed-cube.html to a post by David Moews that explains why there can be no cubed cube (the same argument that appeared in Martin Gardner's Second Scientific American Book of Mathematical Puzzles and Diversions). But this reminded me of a question I've had for a long time: QUESTION: Can there be a cubed 3-torus? Start with a cubical 3-torus: an NxNxN cube with opposite faces identified. Can this 3-torus be tiled by unequal integer cubes (each of which is a union of some of the N^3 unit cubes that constitute the original cube) ? --Dan