SIMO Retiree Blog

(Choo Ray here.) I thought it may be a good idea to write about some of the more interesting problems that I have used in my recent sessions for students in the Singapore National Team.

 (Dylan here.) For this post (and perhaps my subsequent blog posts), I will share a math-adjacent slice of my experience, and then talk some (probably unrelated) math. I will also include upfront some problems related to the non-math/math I am talking about, in case you want to try them first.

(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 ...