Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24491220200901Counting closed billiard paths171177382110.22060/ajmc.2020.17320.1026ENZahedRahmatiDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran0000-0002-6942-3453SinaFarahzadDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, IranAliRahmatiMalek-Ashtar University of Technology, Tehran, IranJournal Article20191104Given 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_9489e0477a249fc2d912a25b0c2814a4.pdf