Covering of a rectangle by squares

TL;DR

This paper characterizes when a 1 by b rectangle can be covered with equal squares on both sides in a single layer, using a specific mathematical condition involving rational parameters.

Contribution

It provides a necessary and sufficient condition for covering a rectangle with equal squares on both sides in one layer, expressed through a precise algebraic formula.

Findings

Derived a mathematical criterion involving rational parameters for covering rectangles with squares.

Established the equivalence condition for one-layer coverings with equal squares.

Contributed to geometric covering theory with a specific algebraic characterization.

Abstract

We prove that one can cover the $1 \times b$ rectangle by equal squares on both sides in one layer iff $b = p \pm \sqrt{p^2 - r^2} $, where $p \ge r \ge 0$ and $p,q \in \mathbb{Q}$.