Closed Monochromatic Bishops' Tours
In chess, the bishop is unique as it is locked to a single color on the black and white board. This makes a closed tour in which the bishop visits every square on the board exactly once and returns to its starting position impossible. When can two bishops, one black and one white, legally visit every square (of their respective colors) exactly once and return to their starting positions? Such a tour will be called a closed monochromatic bishops' tour. In this article, necessary and sufficient conditions for the existence of a monochromatic bishop's tour for the rectangular m x n board are proven. Furthermore, a monochromatic knight's move is defined for the three dimensional chessboard and a closed monochromatic knight's tour is provided for the cube of side 6
DeMaio, J. (2006). Closed monochromatic bishops' tours. Journal of Recreational Mathematics, 34(3), 196-203.