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# The 11x11 Regular Pan-Magic Squares

### Magic Carpets/Latin Squares.

For order-11 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 eight pan-magic order-11 Latin squares. They are made using the eight Knight's moves: 2x1 to 9x1. 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.

1. ### Non Pan-magic.

The remaining two Latin Squares (made with a 1x1 step and a 10x1 step) both produce diagonal rows of similar letters and cannot, therefore, be pan-magic.

2. ### Congruent Group 1.

comprises 2x1, 5x1, 6x1, and 9x1 Knight's moves Latin Squares.

3. ### Congruent Group 2.

comprises 3x1, 4x1, 7x6, and 8x1 Knight's moves Latin Squares.

Congruent Group 1.
 A B C D E F G H I J K J K A B C D E F G H I H I J K A B C D E F G F G H I J K A B C D E D E F G H I J K A B C B C D E F G H I J K A K A B C D E F G H I J I J K A B C D E F G H G H I J K A B C D E F E F G H I J K A B C D C D E F G H I J K A B

### Congruent Groups.

In the order-11 Latin square, the 2x1 Knight's move creates a pattern which also contains the 5x1, 6x1, and 9x1 moves (see square on the left). The 3x1 Knight's move square contains the 4x1, 7x1, and 8x1 moves (see square on the right).

### Fewest Graeco-Latin squares.

When Graeco-Latin squares are being created, to obtain at least one sample of each unique type, it is sufficient to pair one example of each set with the remaining seven squares, i.e., for order 11: 2 x 7 = 14.

### All Graeco-latin squares.

To obtain all possible order-11 Graeco-Latin squares, all eight Latin squares are paired with all the remaining seven, i.e., 8 x 7 = 56.

Congruent Group 2.
 A B C D E F G H I J K I J K A B C D E F G H F G H I J K A B C D E C D E F G H I J K A B K A B C D E F G H I J H I J K A B C D E F G E F G H I J K A B C D B C D E F G H I J K A J K A B C D E F G H I G H I J K A B C D E F D E F G H I J K A B C

### Numerical Squares.

Order-11, regular, pan-magic squares are made by combining two Latin squares and substituting numbers for the letters. The total possible number is discussed on the How Many Page

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