Arithmetic billiards, or: How can a bouncing ball compute stuff?

Here's a math surprise: if you bounce a billiard ball around a table of length L and width W, then its path can "compute" the greatest common divisor (gcd) and least common multiple (lcm) of those two numbers L and W. But how does this happen? This activity lets you play around with bouncing a ball around different sized tables, to explore what happens. The table has four pockets, one at each corner. What leads to the ball falling into a particular pocket? How does that explain the ball somehow computing a common factor of the lengths of the table? Note that the greatest common divisor (gcd) sometimes gets referred to by other names, including greatest common factor (gcf) and highest common factor (hcf). Those are all just different names for the same thing. For more info about this topic, check out: https://en.wikipedia.org/wiki/Arithmetic_billiards and https://plus.maths.org/content/arithmetic-billiards-0 For some very interesting explorations and conjectures to explore in this billiards game, see pages 11-13 of "Discovering the Art of Mathematics: Patterns" by Julian F. Fleron and Philip K. Hotchkiss with Volker Ecke and Christine von Renesse: https://www.artofmathematics.org/sites/default/files/books/patternsbook-2018-06-01-www.pdf

By Raj Raizada