Find a dissection of a 19x19x19 cube such that all the pieces are cubes with integer dimension and none of them are unit cubes.

Tile one side as follows.  Put a 5 at two opposite corners.   Tile the rest of edges with twenty-six 2's.  The inside can easily be filled with twenty-three 3's.

Each side of the cube can be tiles this way using a total of four 5's, seventy-six 2's, and one-hundred-thirty-eight 3's.

The total volume used is 4834.
19^3-4834 = 2025 remaining.
2025/27=125

If the region on the inside were cube shaped we could fill it with 3's.  But it isn't.  I wish I had a good way to model this. 

-------------------------

Scratch this...
The 3s near the edges of adjacent sides would overlap each other.  I thought I was close too...

Edited on June 13, 2015, 10:29 pm

Posted by Jer on 2015-06-13 22:09:34