For order-13 squares, like all prime number squares, the magic carpet and the Latin square are one and the same. The Latin Square cannot be further subdivided into component carpets.
There are ten pan-magic order-13 Latin squares. They are made using the ten Knight's moves: 2x1 to 11x1. These moves have the effect of distributing the similar letters, e.g., all the "A"s, so that each square is different and, in each square, there is one of each letter in every row, column, and diagonal.
The remaining two Latin Squares (made with a 1x1 step and a 12x1 step) both produce diagonal rows of similar letters and cannot, therefore, be pan-magic.
comprises 2x1, 6x1, 7x1, and 11x1 Knight's moves Latin Squares.
comprises 3x1, 4x1, 9x1, and 10x1 Knight's moves Latin Squares.
comprises 5x1 and 8x1 Knight's moves Latin Squares.
In Congruent Groups, any one Latin square produces all the Knight's moves found in the group (also see Example on the 11x11 page. The example here shows that the 2x1 Knight's move produces the 6x1, 7x1, and 11x1 as well.
When Graeco-Latin squares are being created, to obtain at least one example of each unique type, it is sufficient to pair one example of each group with the remaining nine squares, i.e., for order 13: 3 x 9 = 27.
To obtain all possible order-13 Graeco-Latin squares, all ten Latin squares are paired with all the remaining nine, i.e., 10 x 9 = 90.
|Copyright © Mar 2010
Mar 6, 2010