Knight’s Tours on 3 x n Chessboards with a Single Square Removed
ABSTRACT
The following theorem is proved: A knight’s tour exists on all 3 x n chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the center square is removed, n = 5, n = 7 when any square other than square (2, 2) or (2, 6) is removed, n = 9 when square (1, 3), (3, 3), (1, 7), (3, 7), (2, 4), (2, 6), (2, 2), or (2, 8) is removed, or when square (1, 3), (2, 4), (3, 3), (1, n – 2), (2, n – 3), or (3, n – 2) is removed.
The following theorem is proved: A knight’s tour exists on all 3 x n chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the center square is removed, n = 5, n = 7 when any square other than square (2, 2) or (2, 6) is removed, n = 9 when square (1, 3), (3, 3), (1, 7), (3, 7), (2, 4), (2, 6), (2, 2), or (2, 8) is removed, or when square (1, 3), (2, 4), (3, 3), (1, n – 2), (2, n – 3), or (3, n – 2) is removed.
Cite this paper
A. Miller and D. Farnsworth, "Knight’s Tours on 3 x n Chessboards with a Single Square Removed," Open Journal of Discrete Mathematics, Vol. 3 No. 1, 2013, pp. 56-59. doi: 10.4236/ojdm.2013.31012.
A. Miller and D. Farnsworth, "Knight’s Tours on 3 x n Chessboards with a Single Square Removed," Open Journal of Discrete Mathematics, Vol. 3 No. 1, 2013, pp. 56-59. doi: 10.4236/ojdm.2013.31012.
References
[1] J. J. Watkins, “Across the Board: The Mathematics of Chessboard Problems,” Princeton University Press, Princeton, 2004. [2] A. J. Schwenk, “Which Rectangular Chessboards Have a Knight’s Tour?” Mathematics Magazine, Vol. 64, No. 5, 1991, pp. 325-332. doi:10.2307/2690649 [3] J. DeMaio and T. Hippchen, “Closed Knight’s Tours with Minimal Square Removal for All Rectangular Boards,” Mathematics Magazine, Vol. 82, No. 3, 2009, pp. 219-225. doi:10.4169/193009809X468869
[1] J. J. Watkins, “Across the Board: The Mathematics of Chessboard Problems,” Princeton University Press, Princeton, 2004. [2] A. J. Schwenk, “Which Rectangular Chessboards Have a Knight’s Tour?” Mathematics Magazine, Vol. 64, No. 5, 1991, pp. 325-332. doi:10.2307/2690649 [3] J. DeMaio and T. Hippchen, “Closed Knight’s Tours with Minimal Square Removal for All Rectangular Boards,” Mathematics Magazine, Vol. 82, No. 3, 2009, pp. 219-225. doi:10.4169/193009809X468869