Pi from Billiards
A small block slides toward a larger block near a wall. Count every elastic collision — block-on-block and block-on-wall. When the mass ratio is 100N, the total number of collisions gives the first N+1 digits of π. The phase-space diagram reveals why: collisions trace an arc of a circle, and counting bounces is the same as counting how many times a line intersects that arc.
Higher mass ratios need more collisions. Increase speed for 104 and 106.
In phase space, the velocity state (v1, v2) bounces between two lines constrained by energy conservation (a circle) and momentum. The number of bounces equals the angle of the arc divided by a unit angle — which is exactly π when masses are powers of 100.