A Bounded-Defect Construction for the Mondrian Art Problem

Abstract

We construct an infinite family of solutions to the Mondrian art problem with absolute defect bounded above by $280$, of side lengths $n_0=315$, $n_1=106785$, and $n_{j+2}=340n_{j+1}-n_j$, where $n_j$ is congruent to $315(\mathrm{mod}630)$. This absolute bound shows that the minimum defect does not necessarily increase with $n$, providing a counterexample to a conjecture which has been open since the problem's original proposal. Our design is derived from a well-known seven-rectangle dissection over the real quadratic field $\mathbb{Q}(\sqrt{19})$, with non-integer side lengths; we tailor the side lengths to be integers via an infinite Pell family. Using the same method, we devise a second family with defect $560$ that contains infinitely many side lengths which are divisible by any given positive integer, implying that for any $k$ there are infinitely many side lengths $n\equiv 0(\mathrm{mod}k)$ with defect bounded above by $560$. This constant bound offers a dramatic improvement on the previous quadratic bounds.

arXiv comment9 pages, email comments are welcome!