Grogono Photo Pages
Size:  Index   3x3   4x4   5x5   6x6   7x7   8x8   9x9   10x10   11x11   12x12   13x13 

The 7x7 Magic Square Index Regular   Irregular 

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 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
ABCDEFG
FGABCDE
DEFGABC
BCDEFGA
GABCDEF
EFGABCD
CDEFGAB
2
ABCDEFG
EFGABCD
BCDEFGA
FGABCDE
CDEFGAB
GABCDEF
DEFGABC
3
ABCDEFG
DEFGABC
GABCDEF
CDEFGAB
FGABCDE
BCDEFGA
EFGABCD
4
ABCDEFG
CDEFGAB
EFGABCD
GABCDEF
BCDEFGA
DEFGABC
FGABCDE

The Six derived Graeco-Latin Squares

From 1 & 2
AaBbCcDdEeFfGg
FeGfAgBaCbDcEd
DbEcFdGeAfBgCa
BfCgDaEbFcGdAe
GcAdBeCfDgEaFb
EgFaGbAcBdCeDf
CdDeEfFgGaAbBc
From 1 & 3
AaBbCcDdEeFfGg
FdGeAfBgCaDbEc
DgEaFbGcAdBeCf
BcCdDeEfFgGaAb
GfAgBaCbDcEdFe
EbFcGdAeBfCgDa
CeDfEgFaGbAcBd
From 1 & 4
AaBbCcDdEeFfGg
FcGdAeBfCgDaEb
DeEfFgGaAbBcCd
BgCaDbEcFdGeAf
GbAcBdCeDfEgFa
EdFeGfAgBaCbDc
CfDgEaFbGcAdBe
From 2 & 3
AaBbCcDdEeFfGg
EdFeGfAgBaCbDc
BgCaDbEcFdGeAf
FcGdAeBfCgDaEb
CfDgEaFbGcAdBe
GbAcBdCeDfEgFa
DeEfFgGaAbBcCd
From 2 & 4
AaBbCcDdEeFfGg
EcFdGeAfBgCaDb
BeCfDgEaFbGcAd
FgGaAbBcCdDeEf
CbDcEdFeGfAgBa
GdAeBfCgDaEbFc
DfEgFaGbAcBdCe
From 3 & 4
AaBbCcDdEeFfGg
DcEdFeGfAgBaCb
GeAfBgCaDbEcFd
CgDaEbFcGdAeBf
FbGcAdBeCfDgEa
BdCeDfEgFaGbAc
EfFgGaAbBcCdDe

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 Regular   Irregular 

 
 
 
 



Copyright © Mar 2010 Magic Squares
Website
Updated
Mar 6, 2010