Oh, What A Knight!
A knight moves on a chessboard according the standard rules of chess, but in a totally random manner. Thus, at each move, the knight is equally likely to move to any of the squares it can reach. For example, a knight on a corner square will move to either of the two squares it can reach with probability \(\tfrac12\), while a knight in the centre of the board can move to any one of eight possible squares, each with probability \(\tfrac18\).
If the knight starts its journey at a corner square, what is the expected number of moves that it takes to return to its starting square?