Posted by: yanzhang | January 2, 2011

## Balls and Urns for the New Year

This puzzle, brought to my attention by Kartik, has been bugging me:

For some n, put $n$ balls into an urn, all different colors. Each move, pick two balls uniformly randomly, and recolor the second ball the color of the first. What is the expected number of moves for the balls to all become the same color?

Numerical evidence suggests $(n-1)^2$, meaning if there is no nice proof then there is no justice in the world. However, I have no idea how to get this. Warning: the general problem (having an arbitrary number of each color) seems to be messy, so I think a “nice” solution must avoid the general problem.

Happy new year,

-Y