The 5x5 PanMagic Squares
Discovery: Only 144 Squares; Only One Underlying Pattern
Just as with the Order four squares, early workers focussed on ennumerating how many squares there are. The focus here is on how few there are and how each one of these 144 is actually a derivation of one single underlying pattern. As with the order four squares, I would appreciate notification of any earlier recognition of these findings.
Summary
There are 28,800 order five panmagic squares, but only144 that are uniquely different. More surprising, these 144 can all be derived from the same underying "Magic Carpet" or Latin Square. When two reflections of this one Magic Carpet are combined, they make a single GraecoLatin square. It is the only possible PanMagic 5x5 GraecoLatin square and it underlies all of the apparently quite different 5x5 PanMagic Squares. (Compare with the example given using a 2x1 and 3x1 knight's move.) In summary:
 There are 144 pan magic squares of order five.
 They are based on one underlying panmagic carpet, or Latin square.
 Two reflections of this carpet combined make a GraecoLatin Square.
 The one GraecoLatin Square underlies all of the 5x5 panmagic squares.
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The Magic Carpet.
The pattern on the right shows the 5x5 PanMagic Carpet which underlies all 5x5 PanMagic Squares. The letters used, and their sequence, is immaterial because they will be replaced by numbers. The sample below, with its reflection, shows how two versions of the one derived Latin Square are combined to make a 5x5 GraecoLatin Square.
A  B  C  D  E  D  E  A  B  C  B  C  D  E  A  E  A  B  C  D  C  D  E  A  B 


+ 
A  D  B  E  C  B  E  C  A  D  C  A  D  B  E  D  B  E  C  A  E  C  A  D  B 


= 
Aa  Bd  Cb  De  Ec  Db  Ee  Ac  Ba  Cd  Bc  Ca  Dd  Eb  Ae  Ed  Ab  Be  Cc  Da  Ce  Dc  Ea  Ad  Bb 


0  8  11  19  22  16  24  2  5  13  7  10  18  21  4  23  1  9  12  15  14  17  20  3  6  Base 10 
00  13  21  34  42  31  44  02  10  23  12  20  33  41  04  43  01  14  22  30  24  32  40  03  11  Base 5 
GraecoLatin to Numerical
The relationship between the 5x5 GraecoLatin square above and a numerical 5x5 PanMagic square is best understood by looking at the two squares to the right. For simplicity the conversion in both squares is A=0, B=1, C=2, D=3, E=4.
Knight's Moves: 2x1 and 3x1.
The construction above shows a single Latin square being used a second time, as a reflection, to make the GraecoLatin Square. Inspection of the two Latin squares clearly shows the classic Knight's Move construction. In the first square there is the conventional two moves sideways and one down. In the second square there is an extended Knight's move, three moves sideways and one down. The importance of understanding such Knight's moves here is that this approach is used when considering how many distinct GraecoLatin squares are possible for the larger prime number squares.
How Many 5x5 PanMagic Squares are there?  28800.
Two other pages address this question. It is part of the more general question: "How Many PanMagic Squares are there?" for all the Prime Number Order Squares?", and for the order 5 square it is answered in detail with a complete list of all 144 Order 5 PanMagic Squares. Each of the 144 unique squares has 25 translocations with four rotations and two reflections, for a total of 200 x 25 x 4 x 2 = 28800 order5 PanMagic Squares.
All from one Latin Square.
All of these 28800 squares can be represented by the single PanMagic GraecoLatin square above. Representation by a GraecoLatin Square reveals the underlying structure and makes it easier to understand the number of possibilities that exist: One GraecoLatin square is far easier to comprehend than 28800 different squares!
Copyright © Mar 2010 
Magic Squares Website 
Updated
Mar 6, 2010
