10.23638/DMTCS-22-1-21 https://dmtcs.episciences.org/6022 Rao, Hui Hui Rao Ren, Lei Lei Ren Wang, Yang Yang Wang Dissecting a square into congruent polygons We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number $\geq 3$, 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 $\geq 3$. 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 $q\geq 6$ or it is a right-angle trapezoid. Comment: 19 pages, 11 figure episciences.org Mathematics - Combinatorics 52B45, 05C45 arXiv.org - Non-exclusive license to distribute 2020-06-29 2020-06-29 2020-06-29 eng journal article arXiv:2001.03289 10.48550/arXiv.2001.03289 1365-8050 10.48550/arxiv.2001.03289 https://dmtcs.episciences.org/6022/pdf VoR application/pdf Discrete Mathematics & Theoretical Computer Science vol. 22 no. 1 Combinatorics Researchers Students