Find a dissection of a 19x19x19 cube such that all the pieces are cubes with integer dimension and none of them are unit cubes.
(In reply to
re(2): first observations by Steve Herman)
OK I clearly didn't try hard enough. Although I don't know how to fill the cube here's two more ways to tile the 19x19 square:
12x12 and 7x7 at opposite corners. Fill the remaining 7x12 rectangles with stacks of four 3x3 and three 4x4.
7x7 dead center. eight 3x3s in opposite corners. The other two corners now have a 10x10 with a 4x4 corner missing: easily filled with 2x2s.
The trouble with filling a cube with any of these is once you use, say a 13x13x13 cube, you seem to need to tile a side of it.

Posted by Jer
on 20150613 21:02:11 