Minimizing the Number of Tiles in a Tiled Rectangle

TL;DR

This paper presents a strategy to reduce the number of tiles in a rectangle tiling where each tile has at least one integer side, offering a new approach to the rectangle tiling theorem.

Contribution

It introduces a novel strategy for minimizing the number of tiles in rectangle tilings with at least one integer side, providing a new solution to the rectangle tiling theorem.

Findings

A strategy exists to reduce the number of tiles without violating border conditions.

The approach offers an alternative proof to the rectangle tiling theorem.

The method applies to tilings with rectangles having at least one integer side.

Abstract

In this paper, we prove that if a finite number of rectangles, every of which has at least one integer side, perfectly tile a big rectangle then there exists a strategy which reduces the number of these tiles (rectangles) without violating the condition on the borders of the tiles. Consequently this strategy leads to yet another solution to the famous rectangle tiling theorem.