I don't know if this is what you mean by a numerical example but here it goes. Suppose you rolled 4,2,6,1 then R looks like (for n=1,2,3,4) 0,2,0,1.

a) should be relatively simple (use the Markov property of the S_n) and for b), think about it in this way, suppose you are at 0 and want to go to 1. You have to roll either a 1, or a 5 (both give remainder 1). So from 0 to 1 has probability 1/3. Now try to figure the other ones out using the same method.

Hope this helps.