A while ago I arbitrarily decided that I needed a favourite three-digit number (don’t ask) and ended up choosing 216. It’s a nice cube number — — and can also be expressed as a sum of three smaller cubes:
The Wikipedia article for the number has a diagram showing one way to reassemble a 6×6×6 cube into three smaller cubes, but I’ve been playing around looking for other, more aesthetically pleasing methods. Here’s one I found.
First, we break the 6×6×6 cube into its 4×4×4 interior, six 4×4×1 faces, twelve 4×1×1 edges, and eight 1×1×1 corners1: i.e.,
The 4×4×1 faces can be combined with seven of the edges and one of the corners to build a 5×5×5 cube. The remaining five edges can be split into ten 2×1×1 chunks and arranged with the remaining seven corners to form a 3×3×3 cube.
There are many more ways to construct three cubes from the pieces of a 6×6×6 cube. What’s your favourite?
As an aside, this decomposition can be done with any size cube and even in any number of dimensions. An -dimensional hypercube of size breaks into -dimensional polytope boundary chunks as ↩