Each morning, an individual leaves his house and goes on a run. He is equally likely to leave from the front door or back door. Upon leaving the house, he chooses a pair of running shoes (or goes barefoot if there are no shoes near the door from which the runner departs). Upon his return, independently of the door from which he left, he is equally likely to enter the house through either door. He leaves his shoes near the door through which he enters. Suppose that the runner owns 4 pairs of running shoes. Let Xn be the number of pairs of shoes at the front door each morning before he goes on the run. Thenis a Markov chain.