Square packing in a square

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a real $x\geq1$, let $N(x)$ be the maximum number of closed unit squares contained in $[0,x]^2$ with pairwise disjoint interiors. Each small square may be translated and rotated independently. Define the wasted area by $W(x)=x^2-N(x)$. In particular, $W(m)=0$ for every positive integer $m$, since an integer-sided square admits the usual grid tiling.

Statement recorded in the supplied source.

Determine the asymptotic behavior of $W(x)$ as the real side length $x\to\infty$, including its dependence on side lengths between integers. Any asymptotic description must account for $W(m)=0$ at integer $m$. [S2]

Short English statement

How does the area left unused by the best packing of unit squares in a large square grow as the side length increases, including the fluctuations between exact integer tilings?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200031 (AMR-071-0031). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hong Duc Bui, Square Packing with Asymptotically Smallest Waste Only Needs Good Squares (2025). Section 2, Definitions 2, 4 and 5, pp. 1–2. https://arxiv.org/pdf/2504.09489.