Amirkabir University of TechnologyAUT Journal of Mathematics and Computing1220201001Counting closed billiard paths171177382110.22060/ajmc.2020.17320.1026ENSinaFarahzadDepartment of Mathematics and Computer Science, Amirkabir University of TechnologyAliRahmatiMalek-Ashtar University of Technology, Tehran, IranZahedRahmatiDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)Journal 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.<br />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 $ab$.<br /><br />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 (ang{45}) is at most $1.5n-6$.https://ajmc.aut.ac.ir/article_3821_c20d01188914b161943269bccd3f81c1.pdf