Repainting balls until a single colour remains

A box contains $n$ balls, each painted a different colour. At each step you draw two balls at random, repaint the second ball in the colour of the first, and put both back. What is the expected number of steps until every ball has the same colour?

$(n-1)^2$. Fix the colour that eventually wins and condition on it: with $k$ balls of that colour the count moves up or down by one with equal probability, and solving the conditioned birth-death recursion gives an expected absorption time of $(n-1)^2$ (simulation agrees, e.g. $9$ for $n=4$).