Suppose there are $n$ boxes, each containing $n$ balls of $n$ different colors. Each time, I randomly select a box, and randomly select a pair of balls from that box, repaint the first to match the second, and put the pair back into the chosen box. What is the expected number of steps until for all the boxes, the balls in each box are of the same color? (the color of different boxes need not be the same).
This problem is inspired by the version where there is only one box. And using the Markov chain one can derive that the expected number, in that case, is $(n-1)^2$. The way to do it is to condition on the case of the first color, derive the transition probability for each state (# of first color ball in the box), and work out the recursion to determine the formula. However, when there are multiple boxes, I'm stumped on how to apply Markov chain in this scenario. Simply writing out all the states seem a little bit tedious. Is there any more efficient way to solve the problem?