This is a short note based on the paper
Playing pool with π (the number π from a billiard point of view) by G. Galperin in 2003.
Let’s start with two hard balls, denoted by and , of masses on the positive real axis with position , and a rigid wall at the origin. Without loss of generality we assume . Then push the ball towards , and count the total number of collisions (ball-ball and ball-wall) till the escapes to faster than .
Case. : first collision at , then rests, and move towards the wall; second collision at , then gains the opposite velocity and moves back to ; third collision at , then rests, and move towards .
Total counts , which happens to be first integral part of . Well, this must be coincidence, one might wonder.
However, Galperin proved that, if we set , then gives the integral part of . For example, ; and .