Dissecting the square into seven or nine congruent parts

TL;DR

This paper presents a computer-assisted proof that dividing a square into seven or nine congruent convex polygons necessarily results in all tiles being rectangles.

Contribution

It provides the first computer-based proof confirming that such partitions into seven or nine congruent convex polygons must be composed of rectangles.

Findings

All seven or nine congruent convex polygons in a square are rectangles.

Computer-based proof confirms the geometric restriction for these partitions.

No non-rectangular convex polygons can form such partitions into seven or nine parts.

Abstract

We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles.