Cubing a Cube Proof
Back in June I wrote an article called Mrs Perkin's Quilt in which the object was to divide a square into smaller squares, but it is only one formation of a whole genre of square splitting problems. One of the common types is to attempt to split a square into smaller squares of distinct sizes.
This is only possible for squares of certain sizes and the theory goes quite deep (and has the unique distinction of a major breakthrough being made by one of the leading mathematicians sharing the problem with his mum and her finding a brand new division which lead to a general case). However in this article I want to bring everything up a dimension and talk about dividing a cube into cubes of distinct sizes.
It turns out it is impossible and we can prove it by contradiction. Let's assume we can divide a cube into a finite number of smaller cubes of distinct sizes. Consider the smallest cube which can be found on the face of the bottom cube. This cube can't be in the corner, because trying to fit the cubes on each of the three surrounding faces to fit around it will result in them bumping into each other. Similarly, putting the smallest cube of the bottom face on the edge leads to similar problems. It must be in the centre of the face.
Around this smallest cube of the face we will have to surround it and each of the cubes on all four sides will tower above it, at least slightly. This creates a small alcove space above it. In this space there must be a smallest square somewhere in the middle and so exactly the same logic applies. This continues forever creating an infinite regress. This contradicts our assumption that we could divide it into a finite number of distinct cubes and so it is impossible. This proof ports itself straight into all higher dimensions as well, but my visualisation of the proof quickly gives out above 3D.