Another fun and easy way to create Knight Tours is M-O-D-U-L-A-R. Break down an 8x8 square chess board into smaller rectangles such as two 3x4 and two 4x5 rectangles. To create open and closed Knight Tours, only five simple patterns should be learnt. When using rotation or reflection, only two patterns are actually needed to complete an open Knight's Tour.
Check out KT2x3x4.htm to see a simple 2x3x4 Closed Knight's Tour using the 3x4 module B previously listed. The following Closed Knight's Tour, created in Bryce 5.0, is encapsulated within 64 glass cubes. It shows the same 8x8 modular closed tour as above.
I just noticed (April 29, 2007) in the 1917 publication of Amusements in Mathematics by Henry Ernest Dudeney, that Henry shows a similar, but slightly different, modular Knight's Tour as the closed tour shown above. On page 103, Henry presents the modular knight tour puzzle (#338 - The Board in Compartments) with the following solution shown on page 229.
The same M-O-D-U-L-A-R design for creating 2d 8x8 Knight Tours, can be used to create 3d 8x8x8 cubic Knight Tours. The following graphic shows all eight 8x8 levels of an 8x8x8 cube. The Knight's Tour starts on the first level with the highlighted green square. The last Knight move on level 1 is shown with a highlighted dark grey square that connects to the dark grey square on level 2. As the Knight moves from level to level, dark grey squares connect to dark grey squares while light grey squares connect to light grey squares.
The following graphic shows the expected Top-Down view of the 3d Cubic Open Knights Tour.
The 3x4 M-O-D-U-L-A-R Knight Tour patterns "A" and "B" can also be used to make nice picture frame borders, borders around web pages, or trim around walls. See the following Closed Knight's Tour border sample. To increase the size of the frame or border, insert additional module "B" patterns between the module "A" patterns.
Check out my 8x8x8 Closed Knight's Tour page for information on how to create different types of 8x8x8 closed tours. I connected eight 4x4x4 Knight Tour Cubes together to make one continuous tour of 512 moves with the last move connecting back to the first move.
