Description

The following is a classical result of Max Dehn from 1903:
A rectangle can be tiled with finitely many squares (of possibly different sizes) if and only if the ratio of the sides of the rectangle is rational.

The theorem has several beutiful proofs, none of them are easy to find. Five years ago at a Conjecture and Proof course at the BSM with two students, Changshuo Liu and Stephen Lacina we discovered a new proof, which uses simultaneous diophantine approximation. Our (unpublished) proof also gives an almost sharp upper estimate for the enumerator and denominator of the ratio of the sides of the rectangle as a function of the number of squares used in the tiling. The sharp estimate is also known but, as far as I saw, the proof uses the above theorem of Max Dehn, while our argument gives both.

By solving the problems of the Prelimary Assignment you can reproduce all of our arguments and results.

One can suspect that this approach might be also used for other variants of this problem and might lead to interesting new results. The aim of this project is to explore these possibilities.