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 The 7x7 Magic Square Index

# The 7x7 Regular Pan-Magic Squares

### Magic Carpets/Latin Squares.

For Order-7 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 four pan-magic order-7 Latin squares. They are made using a sequence of Knight's moves: 2x1, 3x1, 4x1, and 5x1. These moves have the effect of distributing the letters, e.g., the "A"s, so that there is one of each letter in every row, column, and diagonal. In addition, each square is different

Notes:

• The remaining two Latin Squares (made with a 1x1 step and a 6x1 step) both produce diagonal rows, e.g., of the letter "A", and cannot, therefore, be pan-magic.
• The 4x1 and 5x1 Latin Squares could be obtained by reflections of the 2x1 and 3x1.
• In fact all four Latin Squares actually contain a 2x1 step and can, therefore, be made congruent by reflection, rotation, and substitution.

### The Four Order-7 Latin Squares.

The following sequence makes it easy to visualize and understand the four possibilities. Observe the increasing "step" between the letter "A" in each square. (Also note that in every square there is, in fact, a 2x1 Knight's move for the letter "A"; it is in a different direction in each square.)

1
 A B C D E F G F G A B C D E D E F G A B C B C D E F G A G A B C D E F E F G A B C D C D E F G A B
2
 A B C D E F G E F G A B C D B C D E F G A F G A B C D E C D E F G A B G A B C D E F D E F G A B C
3
 A B C D E F G D E F G A B C G A B C D E F C D E F G A B F G A B C D E B C D E F G A E F G A B C D
4
 A B C D E F G C D E F G A B E F G A B C D G A B C D E F B C D E F G A D E F G A B C F G A B C D E

### The Six derived Graeco-Latin Squares

From 1 & 2
 Aa Bb Cc Dd Ee Ff Gg Fe Gf Ag Ba Cb Dc Ed Db Ec Fd Ge Af Bg Ca Bf Cg Da Eb Fc Gd Ae Gc Ad Be Cf Dg Ea Fb Eg Fa Gb Ac Bd Ce Df Cd De Ef Fg Ga Ab Bc
From 1 & 3
 Aa Bb Cc Dd Ee Ff Gg Fd Ge Af Bg Ca Db Ec Dg Ea Fb Gc Ad Be Cf Bc Cd De Ef Fg Ga Ab Gf Ag Ba Cb Dc Ed Fe Eb Fc Gd Ae Bf Cg Da Ce Df Eg Fa Gb Ac Bd
From 1 & 4
 Aa Bb Cc Dd Ee Ff Gg Fc Gd Ae Bf Cg Da Eb De Ef Fg Ga Ab Bc Cd Bg Ca Db Ec Fd Ge Af Gb Ac Bd Ce Df Eg Fa Ed Fe Gf Ag Ba Cb Dc Cf Dg Ea Fb Gc Ad Be
From 2 & 3
 Aa Bb Cc Dd Ee Ff Gg Ed Fe Gf Ag Ba Cb Dc Bg Ca Db Ec Fd Ge Af Fc Gd Ae Bf Cg Da Eb Cf Dg Ea Fb Gc Ad Be Gb Ac Bd Ce Df Eg Fa De Ef Fg Ga Ab Bc Cd
From 2 & 4
 Aa Bb Cc Dd Ee Ff Gg Ec Fd Ge Af Bg Ca Db Be Cf Dg Ea Fb Gc Ad Fg Ga Ab Bc Cd De Ef Cb Dc Ed Fe Gf Ag Ba Gd Ae Bf Cg Da Eb Fc Df Eg Fa Gb Ac Bd Ce
From 3 & 4
 Aa Bb Cc Dd Ee Ff Gg Dc Ed Fe Gf Ag Ba Cb Ge Af Bg Ca Db Ec Fd Cg Da Eb Fc Gd Ae Bf Fb Gc Ad Be Cf Dg Ea Bd Ce Df Eg Fa Gb Ac Ef Fg Ga Ab Bc Cd De

### Making the Numerical Squares.

For each of the six Graeco-Latin squares, the numerals 0 - 6 are substituted for both the capital, and the lower case, letters. To simplify this process, while still make all the possible numerical squares, zero is assigned both to "A" and to "a". (This is appropriate because we are making pan-magic squares which can always be rearranged later by translocation to put the zero in any position.) The remaing six numeral can then be assigned to any letter. This provides 6 x 5 x 4 x 3 x 2 x 1 = 720 combinations for the capital letters and the same number for the lower case letters.

### Duplication.

Each of the Graeco-Latin squares, potentially, produces 720 x 720 = 518,400 pan-magic squares. However, reflection and translocation shows that there are actually 259,200 pairs. Moreover, although the six Graeco-Latin squares appear to be very different, they actually produce additional duplicate copies of each square.

When all of the possible combinations are used for all six Graeco-Latin squares, a total of four duplicates of every square are produced.

### Total Number.

Therefore there are 6 x 518,400 / 4 = 777,600 unique, regular, order 7, pan-magic squares. Each of these unique squares represents 49 which are obtainable by translation; and each of these has eight versions obtainable by reflection and rotation. The grand total therefore is 777,600 x 49 x 8 = 304,819,200 regular, order 7, pan-magic squares.

 The 7x7 Magic Square Index

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