(Random walk on a clock). Consider the numbers 1; 2; : : : 12 written around a ring as they usually are on a clock. Consider a Markov chain that at any point jumps with equal probability to the two adjacent numbers.

(a) What is the expected number of steps that Xn will take to return to its starting position?

(b) What is the probability Xn will visit all the other states before returning to its starting position?
The next three examples continue Example 1.34. Again we represent our chessboard as f.i; j / W 1 i; j 8g. How do you think that the pieces bishop, knight, king, queen, and rook rank in their answers to (b)?