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 twentysix 2's. The inside can easily be filled with twentythree 3's.
Each side of the cube can be tiles this way using a total of four 5's, seventysix 2's, and onehundredthirtyeight 3's.
The total volume used is 4834.
19^34834 = 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 20150613 22:09:34 