The Mondrian Puzzle: A Bound Concerning the $M(n) = 0$ Case

TL;DR

This paper establishes a lower bound on the count of integers less than a threshold for which the Mondrian Puzzle's minimal rectangle area difference is non-zero, contributing to understanding the puzzle's solutions.

Contribution

It provides a new lower bound on the number of integers where the minimal difference in rectangle areas in the Mondrian Puzzle is non-zero.

Findings

Lower bound on the count of integers with M(n) ≠ 0

Insights into the distribution of solutions to the Mondrian Puzzle

Connections to tileability and geometric constraints

Abstract

In response to the Numberphile video regarding the Mondrian Puzzle https://www.youtube.com/watch?v=49KvZrioFB0, we provide a lower bound on how many integers less than a given threshold $x$ satisfy $M(n) \neq 0$ where $M(n)$ is the quantity in which the Mondrian Puzzle is interested, i.e. the minimal difference in area between the largest and smallest rectangle in a set of incongruent, integer-sided rectangles which tile an $n$ by $n$ square.