(Knight)\u3csup\u3e3\u3c/sup\u3e: A Graphical Perspective of the Knight\u27s Tour on a Multi-Layered Chess Board

The Knight’s Tour is an interesting question related to the game of chess. In chess, the Knight must move two squares in one direction (forward, backward, left, right) followed by one square in a perpendicular direction. The question of the Knight’s Tour follows: Does there exist a tour for the Knight that encompasses every single square on the chess board without revisiting any squares? The existence of Knight’s Tours has been proven for the standard 8x8 chess board. Furthermore, the Knight’s Tour can also exist on boards with different sizes and shapes. There has been a lot of research into tours on two-dimensional boards. In this project, we explore the question of the Knight’s Tour on multi-layered chess boards. In other words, would it still be possible for a Knight’s Tour to exist on a chess board if there was a third dimension of movement that the Knight could take? This thesis will look at the Knight’s Tour on a two-dimensional board, both standard and rectangular, and will then examine the existence of Knight’s Tours on a multi-layered chess board. Finally, the Knight’s Tour will also be explored on a three dimensional cube on which the Knight can only move on the face of the cube. Throughout the thesis, we will use concepts of Graph Theory to explore tours on the different types of boards