Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24491220201001Counting closed billiard paths171177382110.22060/ajmc.2020.17320.1026ENZahedRahmatiDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)SinaFarahzadDepartment of Mathematics and Computer Science, Amirkabir University of TechnologyAliRahmatiMalek-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 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 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◦ ) is at most 1.5n − 6.https://ajmc.aut.ac.ir/article_3821_4ed954ba6c2fb01e9af05e47ec0a785e.pdf