Covering a square by congruent squares

TL;DR

The paper determines the maximal square edge length $S(n)$ that can be covered by $n$ unit squares and the related boundary-covering value $S_{bd}(n)$. Using geometric coverings, $L$-shape embeddings, and a vertex/side/center analysis, it obtains exact small-n results, including $S(2)=1$, $S(3)=\sqrt{(1+\sqrt{5})/2}$, $S(4)=2$, and $S(5)=2$, with a parallel boundary-analysis giving $S_{bd}(4)=2$ and $S_{bd}(5)=\frac{1}{2}\sqrt{2}\sqrt{\sqrt{13-8\sqrt{2}}+1}+1\approx 2.072$, plus the recurrence $S_{bd}(n+4)=S_{bd}(n)+\sqrt{2}$ for $n\ge 4$ and a conjecture about $S(6)$. The work clarifies when boundary coverage mirrors interior coverage (for $n\le 4$) and provides precise structural descriptions of optimal configurations, including a unique boundary configuration for $n=5$. Overall, it contributes exact small-n characterizations and methodological insight into covering problems by congruent unit squares.

Abstract

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate also the variant in which the goal is to cover only the boundary of a square. We show that these two problems are equivalent for $n \leq 4$, but not for $n=5$. For both problems, we also present the solutions for $n=5$.

Figures (6)

• Figure 1: Three unit squares $S_1, S_2, S_3$ covering the boundary of a square (and also the whole square) $S$ of edge length $\sqrt{\varphi}$. The square $S$ is denoted as a region with dashed boundary; $S_3$ is a homothetic copy of $S$ with a vertex of $S$ as homothety center. The squares $S_1$ and $S_2$ are rotated copies of $S_3$ by angles $\pm \alpha$, where $\alpha = \arccos \frac{1}{\sqrt{\varphi}}$.
• Figure 2: Five unit squares covering the boundary of a square $S$ of edge length $\frac{1}{2}\sqrt{2}\sqrt{\sqrt{13-8\sqrt{2}}+1}+1\approx 2.072$. The boundary of $S$ not belonging to the edges of the covering unit squares is denoted by a dashed line.
• Figure 3: Notation for the proof.
• Figure 4: Left-hand side panel: the graph of $f(\alpha)$. Right-hand side panel: The graph of the numerator of $f'(\alpha)$; the denominator of $f'(\alpha)$ is $\sqrt{ \left( \sqrt{\varphi} -A_1(\alpha) \right)^2-1 }$, which is positive on the investigated interval. Computation shows that the function can be extended to a continuous function on the given interval.
• Figure 5: Illustrations for the proof of Theorem \ref{['thm:biggern']} in Cases 2 and 3. Panel (a): Case 2, Panel (b): Case 3.

Theorems & Definitions (6)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof