The 7x7 Regular PanMagic Squares
Magic Carpets/Latin Squares.
For Order7 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 panmagic order7 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 panmagic.
 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 Order7 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.)
1A  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 


2A  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 


3A  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 


4A  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 GraecoLatin Squares
From 1 & 2Aa  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 & 3Aa  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 & 4Aa  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 & 3Aa  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 & 4Aa  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 & 4Aa  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 GraecoLatin 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 panmagic 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 GraecoLatin squares, potentially, produces 720 x 720 = 518,400 panmagic squares. However, reflection and translocation shows that there are actually 259,200 pairs. Moreover, although the six GraecoLatin 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 GraecoLatin 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, panmagic 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, panmagic squares.
Copyright © Mar 2010 
Magic Squares Website 
Updated
Mar 6, 2010
