Prove that cube cannot be partitioned into $n>1$ cubes, such that each of them has different side length.
I believe tallhis is not hard problem, but I just do not have an idea how to start. I tried to prove it by contradiction. I assumed that large cube is tiled with a $n>1$ cubes such that at all of small cubes have different side length. Then I tried to use induction.
For $n=2$ it is trivial that it is impossible because in that case one side of a small cube must be twice of other side. For $n=3$ it is impossible because wherever we place first small cube, the remaining area cannot be a cuboid so we cannot partition the remaining area into $2$ cubes. For $n=4$ I do not see an easy way to prove it.
I do not believe that induction is the best method, but any solution will be appreciated.