Generalized knightʹs tours on rectangular chessboards Original Research Article

In [Math. Mag. 64 (1991) 325–332], Schwenk has completely determined the set of all integers image and image for which the image chessboard admits a closed knightʹs tour. In this paper, (i) we consider the corresponding problem with the knightʹs move generalized to image-knightʹs move (defined in the paper, Section 1). (ii) We then generalize a beautiful coloring argument of Pósa and Golomb to show that various image chessboards do not admit closed generalized knightʹs tour (Section 3). (iii) By focusing on the image-knightʹs move, we show that the image chessboard does not have a closed generalized knightʹs tour if image and image and determine almost completely which image chessboards have a closed generalized knightʹs tour (Section 4). In addition, (iv) we present a solution to the (standard) open knightʹs tour problem (Section 2).