Split the Cube

Can a cube be split into finely many smaller cubes, all with different sizes?

It is impossible. Assume the opposite and consider the smallest cube on the bottom, C1. Since all cubes surrounding it are larger, they must “wall in” its top face. We repeat the same argument for the smallest cube which lies on the top face of C1, call it C2. If all surrounding cubes are larger, they must wall in its top face. Thus, we can create an infinite sequence of cubes with decreasing sizes lying on top of each other: C1, C2, C3, etc. Since the initial cube is split into a finite number of smaller cubes, we get a contradiction.

Unknown Author

We do not know where this puzzle originated from. If you have any information, please let us know via email.