Batman is chasing the joker. There are four places each could be, arranged in the corners of a square. Batman is moving from his corner to an adjacent corner at rate 1 (rate 1/2 to each). The Joker picks some rate v, and moves from his corner at rate v, in the same way. Find the expected time before they meet (are at the same corner). Hint: at each time they are either in opposite corners, nearby corners, or the same corner. Find transition probabilities between these.