N black balls and N white balls are placed in two urns so that each urn contains N balls. After each unit of time one ball is selected at random from each urn, and the two balls thus selected are interchanged. Let Xn be the number of white balls in the first urn after n units of time. Determine the transition matrix of the Markov chain X.

It looks as if this should depend on the number of white balls in the first urn in the first place, but apparently this is not true. How do we derive the transition matrix (and show it is the same regardless of the initial set up?)