Knight's Tours Categories
Closed Tours
On 11/1/2005, I re-illustrated and categorized all 63 Closed Semi-Magic Knight's Tours from the 140 Semi-Magic Knight's Tours graphic (MKTS.GIF) into the following sections:
Of the 63 Closed Semi-Magic Knight's Tours, there are only 31 "Geometry Classes," or "Geometrically Distinct" Tours. There is also 1 group of 10 Knight's Tours with identical Top-half tours put in Section 6.
Open Tours
Of the 77 Open Semi-Magic Knight's Tours, there are only 15 Open Knight's Tours with vertical number pairs adding up to 49 and 81 that alternate entirely throughout each tour. In the first example below, each vertical number pair shaded in gray adds up to 49, while each vertical number pair with a white background adds up to 81.
On 11/1/2005, I re-illustrated and categorized all 63 Closed Semi-Magic Knight's Tours from the 140 Semi-Magic Knight's Tours graphic (MKTS.GIF) 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 group of 1 tour unique that does not fit in the previous sections)
Of the 63 Closed Semi-Magic Knight's Tours, there are only 31 "Geometry Classes," or "Geometrically Distinct" Tours. There is also 1 group of 10 Knight's Tours with identical Top-half tours put in Section 6.
Open Tours
Of the 77 Open Semi-Magic Knight's Tours, there are only 15 Open Knight's Tours with vertical number pairs adding up to 49 and 81 that alternate entirely throughout each tour. In the first example below, each vertical number pair shaded in gray adds up to 49, while each vertical number pair with a white background adds up to 81.
Alternating vertical pairs of 81 and 49 totals.
Alternating horizontal pairs of 49 and 81 totals.
With rotation, the alternating pairs of numbers may be horizontal as shown in the above. Also, with rotation, reflection, and reversing, these 15 tours may yield 240 separate tours.
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's 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.
There are 11 Reflection-Symmetric Knight's Half-tours 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.
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's 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.
There are 11 Reflection-Symmetric Knight's Half-tours 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.