Dissecting a square into congruent polygons

Abstract

We study the dissection of a square into congruent convex polygons. Yuan et al. [Dissecting the square into five congruent parts, Discrete Math. 339 (2016) 288-298] asked whether, if the number of tiles is a prime number ≥ 3, it is true that the tile must be a rectangle. We conjecture that the same conclusion still holds even if the number of tiles is an odd number ≥ 3. Our conjecture has been confirmed for triangles in earlier works. We prove that the conjecture holds if either the tile is a convex q-gon with q ≥ 6 or it is a right-angle trapezoid.