Knight's Tour Semi-magic Square Construction

While making designs on the chessboard with four mini-knight's tour closed circuits, I realized I could construct a complete semi-magic knight's tour covering all 64 squares with one circuit.  Check out the following steps I used in making the knight's tour.

Step 1:

I first made one complete circuit consisting of 16 moves then copied the circuit and flipped it horizontally to make the second circuit.

Step 2:

I designed a third 16 move circuit, copied it, then flipped it horizontally as in Step 1.

Step 3:

I combined all four mini-knight's tours on one board that ended up making a very nice geometric and symmetrical pattern.

Step 4:

After combining all four mini-knight's tours, I wanted to keep most of the symmetrical pattern but make a single 64 move knight's tour.  I was able to do this by moving only three lines (knight moves) and deleting one line.  In the following image, I changed the colors of seven knight moves.  I added red, orange, yellow, green, lavender, aqua blue, and black.

Step 5:

In this step, I moved the red line over the top of the orange line, yellow over to green, lavender to aqua blue, deleted the black line then recolored all the lines in the tour back to blue.  The following single circuit is the result of moving only three lines and deleting one line.

Step 6:

After successfully making a single knight's tour circuit out of four mini-knight's tour circuits, I colored each original mini-knight's tour circuit patterns blue, red, green, and orange.

Step 7:

I wanted to find out if this knight's tour would make a Latin square so I numbered each move 1 to 8 and repeated the same number order of 1 to 8 throughout the entire circuit.  Though the circuit does not make a Latin square, the numbers in the knight's tour create a 180 degree rotation symmetric numerical pattern.  Dividing the tour in half either horizontally or vertically and rotating either half 180 degrees results in all the numbers in the squares matching each half.

Step 8:

Finally, I renumbered the entire circuit from 1 to 64 starting at a5 with 1 and ending at e2 with 64.  I realized that each row added to 260, each column added to 260, and the diagonals added to 200 and 320 respectively.  Therefore, the following knight's tour is known as a semi-magic knight's tour with all rows and columns adding to 260.

Step 9:

I verified the results of this semi-magic tour with the 140 semi-magic knight's tours found by me and a host of others during 2003.  Guenter Stertenbrink made graphic images showing all 140 tours on a single page at magictour.free.fr/MKTS.GIF.  To match my tour with one of the 140 tours listed on the page, I had to rotate my tour 90 degrees right then flip it horizontally.  I found my tour in the MKTS.GIF page located at g1, if using algebraic chess notation, or (13, 6) if using matrix notation where (0, 0) is the top left of the page and (13, 9) is the bottom right of the page.  I placed the moves on a board to match the rest of the knight's tour graphics on this page.

Step 10:

I wanted to check to see if the individual mini-knight's tour patterns could make a tessellation piece when combined together but was unsuccessful.  I will show the process here anyway.  Using the same techniques I provide will be possible to find other knight's tours that can make tessellation pieces resulting in nice tessellations that can be used on book covers, website backgrounds, tiles, or other types of art.