On the covering of $n+\epsilon$ square with $n^2+1$ unit squares for $n\ge 4$

Abstract

In 2006, Alexander Soifer conjectured that one cannot fully cover a square of side length $>n$ with $n^2+O(1)$ unit squares. For small $n\in \{2,3\}$, in 2009, Janusz Januszewski proved that it is impossible to fully cover a square of side length $>n$ with exactly $n^2+1$ unit squares. Recently in 2023, Baek and Lee proved that it is impossible to fully cover an equilateral triangle of side length $>n$ with exactly $n^2+1$ unit equilateral triangles whose sides are parallel to it. There were also some progress on a related problem: what is the largest square with side length $S(k)$ that can be fully covered by $k$ unit squares. However, there have been no improvement on the original conjecture. In this work, new tools have been developed that led to the proof of this conjecture for $n=4$, with potential applications to related problems.

arXiv comment9 pages, 6 figures, intended for journal publication