This question is part of a larger problem I'm working on, but it seams to be causing me the most problems. Any help or tip would be appreciated. So here it goes. We have n objects which can have any of the m allowed values. If we find a pair of objects with equal values, we change their values to new random values (from 1 tom m). We repeat this until there are no more pairs. What is the distribution of the number of steps this game can last.? I did some numerical simulations but I would like to have an analytical solution. I only managed to get an analytical solution for the probability that the game ends in the first step. I also noticed that the probability has a peak close to the beginning (it's not monotonous as I first suspected).