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In a n x n chessboard a white knight sits on the top left corner, and a black knight on the bottom right corner. Starting with white, the two knights take turns to move at random, and with equal probability, to any of the (up to eight) available cells.

What is the expected total number of moves that will occur before one of the knights lands on the cell occupied by the other?

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  • $\begingroup$ @Mark S Yes, they can visit the same square indefinitely. $\endgroup$
    – Bernardo Recamán Santos
    Commented Mar 29, 2019 at 20:40
  • $\begingroup$ The top left and the bottom right are the same color. A knight move changes color. I think it can only be that the black knight steps onto the white knight. The white knight can't step on to the black knight I don't think. $\endgroup$
    – Mark S
    Commented Mar 29, 2019 at 20:40
  • $\begingroup$ From parity considerations, the black knight will be the capturing piece, so it will be an even number of plys (moves by both pieces) before such an encounter. I imagine (but don't know) that the expected number of moves is close to n^2. Gerhard "Hasn't Squared Off With This" Paseman, 2019.03.29. $\endgroup$
    – Gerhard Paseman
    Commented Mar 29, 2019 at 20:44
  • $\begingroup$ A suggestion for an approximation: starting with an infinite chessboard, a knight starts on one square; how many moves does it take to reach a specified other square on average? $\endgroup$
    – user44191
    Commented Mar 29, 2019 at 21:52

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