How do I solve this question:

2. At the right is a section of an infinite

grid--enough has been diagrammed to

make the pattern clear. Some nodes are

white and some are grey. Two nodes are

"neighbours" if they are connected by an

edge. In each time unit, a counter

occupying a node moves to a neighbouring

node choosing each of its three neighbours,

with equal probability 1/3.

(a) Suppose a counter is on a grey node.

Calculate, by tracking the different

possible paths of a counter, the

probability that it is on a grey node after

3 moves.

(b) Find a way of keeping track of the change in the number of counters on white and grey nodes

over time. We are looking here for a dynamical system.

(c) Find a steady state distribution of counters on white and grey nodesthat is, proportions that

remain constant over time.

(d) Use the eigenvalues and eigenvectors of the system of (b) to get equations for the state at any

time n and then set n = 3 and check your answer to (a).