This was asked to me on Dec 1 in campus interviews of one of the companies.
We have a cube. An ant is on one of the corners. It moves randomly with equal probability in all the three directions. What is the expected number of steps to reach the opposite corner.
Note that the probability of the ant reaching the opposite corner in 2 steps is 0, in 3 steps is 6/27, in 4 steps is 0 and so on..
(I was able to solve it :D)
Update (11/12/09): Solution posted by Anshum (Jaadu, CSE, IITB) in comments!!