Questions about Markov Chain: Fix an integer k. States are the hours 1, 2, . . . , 12 which we think of as arranged around a clock face. A k-random walk adds or subtracts k hours on the clock, with probability 1/2 each (so for example, a 3-walk starting at state 2 will step to either 5 or 11).

For k = 1, 2, 3, 4, 5, 6 do the following:

(a) Decide if the k-walk is irreducible or not.

(b) Find the equivalence classes for the <-> relation.

(c) Decide if the walk is aperiodic.

(d) Decide the period of state 12 (or of any other state).