In combinatorics and recreational mathematics, cubing the cube refers to the analogue in three dimensions of squaring the square: that is, given a cube C, to divide it into finitely many smaller cubes, not all congruent.

Unlike the case of squaring the square, a hard but soluble question, cubing the cube is impossible. This can be shown by a relatively simple argument. In fact if you take a squared square, and consider the smallest square S in it, and stand cubes of the same side on each of the squares, they 'tower over' the cube standing at S. If there are cubes standing on that cube, in a cubed cube, their lower faces must square the upper square above S. This gives rise to an infinite descent, unless at some stage we hit a square squared by equal squares. This can be made the basis for an argument that cubing the cube is not possible in unobvious ways.