SIMO Retiree Blog

(Etienne here.) The date was 24 October 2021. It was late at night, and I was just about to tuck myself in. After all, the next day was the A-Level Physics practical! I needed to be well-rested (spoiler alert - I actually didn't). I was lying comfortably and about to bid a temporary farewell to the waking world, when suddenly, out of nowhere, I was attacked by a sudden thought: For what $r$ could I tile a square with rectangles whose sides had ratio $r$?

(Dylan here.) Today, I'll be solving a (pseudorandomly chosen) combinatorics problem. But instead of just explaining the solution, I will detail my messy thought process as I work through it in real time.  After solving the problem, I will do a brief meta-commentary, and try to abstract the problem. Finally, for the more mathematically mature, I will talk about what possibly lies beyond it: various extensions and related areas. Let's begin. 1     The Problem (APMO 2023 P1) Let $n\geq 5$ be an integer. Consider $n$ squares with side lengths $1,2,\dots,n$ respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly $2$ other squares.  1.1     First Thoughts In no particular order: This is a  construction  problem.  The orientation of the ...