Another Knight On The Tiles

A knight moves on a chessboard according the standard rules of chess, but in a random manner. At each move, then, the knight is equally likely to move to any of the squares it can reach. Thus a knight at a corner square of the board will move to either of the two squares it can reach with probability \(\tfrac12\), while a knight in the centre of the board will move to any one of eight possible squares, each with probability \(\tfrac18\).

This chessboard has a "safe zone," consisting of the four central squares. The knight starts its journey on one of the squares in the safe zone. The expected number of moves it must make until it returns to any square in the safe zone can be written as \(\tfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

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