What type of Markov Chain is a random walk of a Knight on a chessboard? 
Assume we have the following chessboard and we have a knight that starts at the top left corner of the board. On every move the Knight chooses reachable square (i.e. a valid chess move a Knight can make to that square so moving in the shape of an L for the Knight.) Consider a Markov Chain that represents the random walk of the Knight. What kind of Markov Chain is this? Irreducible, reducible, aperiodic, periodic?
My educated guess is that'd it be irreducible since given at state the Knight has the guaranteed possibility to get back to a state it started at by reversing it's actions. Then it'd have to be aperiodic because it's irreducible. Is this correct?

A quick simulation suggests that the chain in periodic. The knight will never return to start after an odd number of iterations.
A knight starts at position (X, Y). After a significantly large number of iterations (N), the knight is guaranteed to be on one of 50 tiles. These 50 tiles depend on whether (X+Y+N) is even or odd.
This makes intuitive sense since the vector a knight moves always has an L1 norm equal to 3 (an odd number). On the other hand, the vector a bishop moves always has an even L1 norm. Likewise a bishop always stays on either white or black tiles since the color of any tile is a function of (X+Y)%2.


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*Red "X" shows the starting position.

*Faint "0" shows a position with zero probability the knight may be there on the next iteration.

*Look closely and you'll notice that the probability after 100 iterations depends on the starting position.

