A Bounded-Defect Construction for the Mondrian Art Problem

@misc{okuhn2026boundeddefect,
  title = {{A Bounded-Defect Construction for the Mondrian Art Problem}},
  author = {Cooper O'Kuhn and Freeman Slaughter},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19193},
  eprint = {2610.19193},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19193},
}

TY  - UNPB
TI  - A Bounded-Defect Construction for the Mondrian Art Problem
AU  - Cooper O'Kuhn
AU  - Freeman Slaughter
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.19193
UR  - https://arxiv.org/abs/2610.19193
AB  - 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 pmod630. 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 ℚ(√(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 ≡ 0 pmodk with defect bounded above by $560$. This constant bound offers a dramatic improvement on the previous quadratic bounds.
ER  -