Covering a square by congruent squares

TL;DR

This paper investigates the maximum size of a square that can be covered by a given number of unit squares, exploring both interior and boundary coverage problems and providing solutions for specific cases.

Contribution

It establishes the equivalence of interior and boundary coverage for up to four squares and presents solutions for the case of five squares.

Findings

For n ≤ 4, interior and boundary coverage problems are equivalent.

Solutions for covering a square with 5 unit squares are provided.

The paper distinguishes the cases where n=5 for the two problems.

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 $n=5$.