Knight's Tour with 180 Degrees Symmetry
While making designs on the chessboard with four mini-knight's tour closed circuits, I realized I could construct a complete closed knight's tour covering all 64 squares with one circuit that has 180 degrees symmetry. Check out the following steps I used in making this knight's tour.
Step 1:
I first made one complete circuit consisting of 16 moves then copied the circuit and rotated it 90 degrees left to make the second circuit.
Step 2:
I made additional copies of the first mini-knight's tour for the third circuit rotated 90 degrees right and the fourth tour rotated 180 degrees from the original tour.
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), deleting one line, and adding one line. In the following image, I changed the colors of seven knight moves. I added red, orange, yellow, green, lavender, aqua blue, black and added a brown line to complete the tour.
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, added the brown line then recolored all the lines in the tour back to blue. The following single closed circuit with 180 degrees rotational symmetry (either horizontally or vertically) is the result of moving only three lines, deleting one line and adding one line.
Step 6:
After successfully making a single closed knight's tour circuit out of four mini-knight's tour circuits, I colored each original mini-knight's tour circuit edges blue, red, green, and orange.
Step 7:
I then colored in the tour patterns with the same colors to make additional images. Try to figure out who or what the colored-in mini-knight's tours resemble from the images below.