Investigating the Mondrian Puzzle

Last semester, I researched a new-ish mathematical problem with a partner. This work was part of a class I took, MATH 496: Introduction to Mathematical Research, taught by Bruce Reznick. The class was relaxed and seminar-style, but we still got a lot of work done, eventually delivering it as a presentation and paper at the end of the semester. Though we didn't quite prove what we set out to show, we got pretty close. If you happen to know a bit about Diophantine equations, you might be able to finish what we've started!

Our problem has to do with the Mondrian Puzzle, named after the works of modern artist Piet Mondrian. It asks: is it possible to fill a square with multiple rectangles, leaving no gaps, such that:

While irrational solutions to this puzzle have been found (dropping the requirement that the rectangles be integer-sided), no true solutions have been constructed. The mathematical community believes that this puzzle is impossible to solve, but nobody's proved it yet. We took a stab at it for a certain class of possible constructions. I'm not going to recount the whole story in this post, mostly because I haven't added LaTeX support on this blog yet, but you can read our paper here.