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 — 6×6×6 — and can also be expressed as a sum of three smaller cubes:

6^3 = 5^3 + 4^3 + 3^3

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 corners^{1}: i.e.,

6^3 = 4^3 + 6\cdot 4^2 + 12\cdot 4^1 + 8 \cdot 4^0

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
*n*-dimensional hypercube of size*x*≥2 breaks into*k*-dimensional polytope boundary chunks as*x*^{n}^{n−k}*n*,*k*)*x*−2)^{k}, where C(*n*,*k*) is the binomial coefficient. ↩︎