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.
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.
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