Knight’s Tours on 3 x n Chessboards with a Single Square Removed

DOI: 10.4236/ojdm.2013.31012   PDF   HTML   XML   5,656 Downloads   7,758 Views   Citations

Abstract

The following theorem is proved: A knights 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.

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

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

  
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