A solution to a great brain bender problem presented in this video by 3Blue1Brown.
The number of collisions between a mass
m1 = 1kg and mass m2 = m1 * 10^n will result in exactly pi * 10^(n-1) collisions, due to the fact that the two governing equations (conservation of momentum and conservation of kinetic energy) represent a system of equations of a circle and a line with variable y-intercept. The intersections of these two functions represents the solution as shown in the figure.