Tiling Rectangles and the Plane using Squares of Integral Sides

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.