Erdős Problem #106 — Packing $k^2+1$ squares in a square

Erdős asked whether the largest possible total side length of $k^2+1$ squares packed in a unit square is exactly $k$, with arbitrary orientations; the case $k=4$ is $f(17)=4$.

Known results

• Erdős proved $f(2)=1$.
• Newman proved $f(5)=2$.
• Cauchy–Schwarz gives $f(k^2)=k$.
• Halász and Praton obtained reductions and bounds for the broader family $f(k^2+2c+1)$; Baek, Koizumi, and Ueoro proved the axis-parallel analogue $g(k^2+1)=k$ in 2024.

July 29, 2026 candidate counterexample

A repository gives exact rational coordinates for seventeen squares with total side length $4.000124312403920968>4$, claiming $f(17)>4$ and, via known monotonicity, failure for every $k\ge4$. It explicitly says the construction has not been checked by a human referee; AlphaEvolve's separate search did not produce it.

Current status (as of September 2026): The axis-parallel case is proved, while the unrestricted conjecture remains unsettled and the proposed $k=4$ counterexample is unverified.