# Is it possible to assemble copies of this shape into a cube?

A couple of friends of mine were discussing a problem concerning this shape:

Is it possible to assemble enough of these to form a cube?

I have discovered a lot of impossible positions but was not successful in creating something useful. We have managed to build a 12x12x4 tower with leftover blocks at the top, however.

Maybe someone here has any ideas on how to tackle this problem? My next steps would be to try and extend our 12x12x4 tower, and if that doesn't work, to write a program to search for solutions.

• We can't see whether the rear bottom corner is a hole or a cubie. Oct 22, 2023 at 18:11
• This should have been a great question in Puzzling SE Oct 23, 2023 at 2:59
• If you manage creating a block of size $a \cdot b \cdot c$, you can always turn this into a cube with size $(a \cdot b \cdot c) \cdot (a \cdot b \cdot c) \cdot (a \cdot b \cdot c)$. Oct 23, 2023 at 13:47
• Side note: $8$ copies of the version with a hole at the back, as commented by @shoover, can be assembled into a $4 \times 4 \times 4$ cube. Oct 26, 2023 at 8:03

• @Henry A brute force search finds no solution for $6\times6\times6$. Search on $9\times9\times9$ is currently running on my home PC while I'm at work. Oct 23, 2023 at 14:33
• Search finished as expected: no solutions for $9\times9\times9$. There is no way to cover a $9\times9$ face while staying within the bounds of the cube. Oct 23, 2023 at 22:20