Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 1 2 2020 09 01 Counting closed billiard paths 171 177 3821 10.22060/ajmc.2020.17320.1026 EN Zahed Rahmati Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran 0000-0002-6942-3453 Sina Farahzad Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran Ali Rahmati Malek-Ashtar University of Technology, Tehran, Iran Journal Article 2019 11 04 Given a pool table enclosing a set of axis-aligned rectangles, with a total of n edges, this paper studies $\it{closed~billiard~paths}$. A closed billiard path is formed by following the ball shooting from a starting point into some direction, such that it doesn’t touch any corner of a rectangle, doesn’t visit any point on the table twice, and stops exactly at the starting position. The $\it{signature}$ of a billiard path is the sequence of the labels of edges in the order that are touched by the path, while repeated edge reflections like $abab$ are replaced by $ab$. We prove that the length of a signature is at most $4.5n−9$, and we show that there exists an arrangement of rectangles where the length of the signature is $1.25n+2$. We also prove that the number of distinct signatures for fixed shooting direction ($45^{\circ}$) is at most $1.5n−6$. https://ajmc.aut.ac.ir/article_3821_c04c19c2c2474dbf5f7ac4372c5b9af1.pdf