Semi-Magic Square Knight Tours
On 11/1/2005, I re-illustrated and grouped all 63 closed semi-magic square knight tours from MagicTour.free.fr into the following sections:
Section 1 - (17 - 180° Rotation-Symmetric tours)
Section 2 - (1 group of 5 tours with identical knight moves)
Section 3 - (7 groups of 4 tours each with identical knight moves per group)
Section 4 - (6 groups of 2 tours each with identical knight moves per group)
Section 5 - (1 tour that does not fit in the previous sections)
Two tours from Section 1 (smkt40 and smkt121) could also be added to Section 4. There is 1 group of 10 tours with identical knight moves within top-half of tour.
Of the 63 closed semi-magic square knight tours, there are only 31 "geometry classes," or "geometrically distinct" tours.
For a listing of all 63 individual closed tours, check out the SMKT Table.
Of the 77 open semi-magic square knight tours, there are only 15 open tours with vertical number pairs adding up to 49 and 81 that alternate entirely throughout each tour. In the example below on the left, each vertical number pair shaded in gray adds up to 49, while each vertical number pair with a white background adds up to 81.
With rotation, the alternating pairs of numbers may be horizontal as shown in the above example on the right. Also, with rotation, reflection, and reversing, these 15 tours may yield 240 separate tours. The first half of the name of the graphic (i.e. SMKT3-00_1.gif) corresponds to one of the 140 tours shown on MKTS.GIF, while the second half of the name (i.e. SMKT3-00_1.gif) corresponds to one of the 280 tours shown on Ydenef.club.fr.
You might notice that I rotated or flipped over some of the original tours to show their similarity. All the columns and rows of each tour add up to 260. Each graphic makes two separate semi-magic knight tours with one path being the reverse of the original path. Keep in mind that with rotation, reflection, and reversing, it is actually possible to turn each unique tour into 16 separate tours.
The first move begins where the blue and red lines connect. Follow the blue lines away from the red line (in either direction) to complete the semi-magic tour. The name of the graphic, which corresponds to one of the 140 tours shown on MKTS.GIF, will appear by placing your cursor over a tour.
There are 11 reflection-symmetric halftours within 19 of the 140 open and closed tours listed in MKTS.GIF.
Thanks goes to Guenter Stertenbrink and George Jelliss for their suggestions and corrections in helping to make this page more accurate.
www.BordersChess.org/SMKT.htm modified 2006.12.14