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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.</span

Knight’s Tours on 3 x &lt;i&gt;n&lt;/i&gt; Chessboards with a Single Square Removed

Knight’s Tours on 3 x <i>n</i> Chessboards with a Single Square Removed