The 13x13 Regular Pan-Magic Squares

Magic Carpets/Latin Squares.

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.

Made with a Sequence of Knight's Moves.

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.

Notes:

1. Non Pan-magic.

   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.

2. Congruent Group 1.

   comprises 2x1, 6x1, 7x1, and 11x1 Knight's moves Latin Squares.

3. Congruent Group 2.

   comprises 3x1, 4x1, 9x1, and 10x1 Knight's moves Latin Squares.

4. Congruent Group 3.

   comprises 5x1 and 8x1 Knight's moves Latin Squares.

Congruent Groups.

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.

Fewest Graeco-Latin squares.

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.

All Graeco-latin squares.

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.