Tiling Rectangles and the Plane using Squares of Integral Sides

TL;DR

This paper investigates the problem of tiling rectangles with distinct integral squares, providing solutions for specific cases and discussing open problems related to tiling the plane and rectangles with such squares.

Contribution

It offers a solution for tiling rectangles with a set of squares containing a specific number of odd sides and explores open problems in tiling with integral squares.

Findings

Tiling the plane with a set of odd numbers is impossible.

Tiling with an infinite sequence with exactly two odd numbers is impossible.

New solutions are provided for certain cases of tiling rectangles with integral squares.

Abstract

We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not. Previously, it has been proved that tiling the plane is not feasible using a set of odd numbers or an infinite sequence of natural numbers including exactly two odd numbers. The problem is open for different situations in which the number of odd numbers is arbitrary. In addition to providing a solution to this special case, we discuss some open problems to tile the plane and rectangles in this paper.