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

It can't be done.

Since 19 is prime, each axis must contain either an odd number of prime blocks, (e.g. 11+5+3) or an odd number of prime blocks plus 2s (e.g. 11+4*2). Other cases (e.g. 13+3+3) reduce to these. [1]

We call a row of odd prime blocks along the side of a face of the cube a 'spine'. We call a tiling of a face of the cube a 'plating'. The upper vertices of the cube are U1,U2,U3,U4, and the corresponding lower vertices are L1,L2,L3, L4.

Assume we spine each of six sides U1U2,U2U3,U3L3,L3L4,U1L1,L1L4 with, say, 5 size 3 cubes, leaving voids of size 4 in the corners U2, L1, L3, We can easily fill those voids with size 2s and start to plate all six sides with size 2s.

But there is a problem at corner L2. Because of the voids at U2, L1, L3, there are still lengths of 15 to plate, which cannot be covered by size 2s alone. If we spine that corner with a size 3, the cube is too small, and there are still lengths of 15 to plate. But if we spine that corner with, say, a size 5, then there are some lengths of 11 to plate, see [1].

Hence if the sides of the cube are prime, there will always be one corner that cannot be filled under dissection by either an odd or an even prime cube.

It follows at once that the dissection is impossible.

 

Edited on June 14, 2015, 1:16 am

Posted by broll on 2015-06-14 00:26:06