Knight's Tour Cube

(8x8x8 Closed Tour, by Dan Thomasson, 11-19-2005)

On 11-19-2005, I decided to see if I could make an 8x8x8 Closed Knight's Tour Cube. Without the aid of a computer, it only took me a few minutes to derive at a solution. However, entering 512 numbers on paper to test my theory, and then entering them into the computer to make graphical pictures took much longer. I first made a simple 4x4x4 Knight's Tour Cube, and used its move sequence as a key for solving the 8x8x8 tour. I had to occasionally flip the four 4x4 levels either vertically, or horizontally, to derive at the correct solution for the 8x8x8 tour. This method can be used for easily constructing any size Knight's Tour Cube if it is a multiple of the original 4x4x4 cube size. Look at the following key sequences:

In the process of making the key sequences, I also ended up with the following nice little closed 4x4x4 Knight's Tour Cube that I did not use in the construction of the 8x8x8 tour.

Here are the eight levels of the 8x8x8 Knight's Tour Cube. The accuracy of the numbers were confirmed by Guenter Stertenbrink on 11-20-2005. You may notice that I built the cube in modular sections, using a one-eighth section at a time. Overall, I used eight 4x4x4 cubes in which several have identical knight moves. You may download a plain-text copy of the following tour:   Closed Knight's Tour Cube.

    

    

    

    

Guenter Stertenbrink suggested another possible way of solving 8x8x8 Knight Tours.

"You could also make a 8*8*8 closed tour by just taking one closed 8*8 tour replicated 8 times. Walk through plane 1, jump to plane 3, walk through the whole plane 3, jump to plane 5, walk through the whole plane 5, jump to plane 7, walk through the whole plane 7, jump to plane 8,... planes 1,3,5,7,8,6,4,2 and back to the start-position."

Guenter's suggestion is very plausible. However, I was only able to make such a tour with the same 8x8 tour replicated seven times with one additional 8x8 tour slightly modified (see Level 2). I made a solution key for solving such 8x8x8 tours. The following graphic shows the 4x4 middle squares of a chessboard. Of course, these same 4x4 squares could also be any matching quadrant of the eight levels (planes) of the 8x8x8 cube (i.e. all top left quadrants). The lower number in each level represents the first move on the 8x8 board, while the higher number represents the very last move on that same board (i.e. on Level 1: 1 = move 1 and 2 = move 64; Level 3: 3 = move 65 and 4 = move 128, ...).

If you would like a copy of this Closed Knight's Tour Cube page in Microsoft Word format, please download the following file:

Closed Knight's Tour Cube

You may make copies of the doc file for publication, distribution, or for posting on your websites as long as you credit my name.

www.BordersChess.org/ClosedKTCube.htm   modified 2006.12.14