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The 5x5 Pan-Magic 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 pan-magic 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 Graeco-Latin square. It is the only possible Pan-Magic 5x5 Graeco-Latin square and it underlies all of the apparently quite different 5x5 Pan-Magic Squares. (Compare with the example given using a 2x1 and 3x1 knight's move.) In summary:

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

The Magic Carpet.

The pattern on the right shows the 5x5 Pan-Magic Carpet which underlies all 5x5 Pan-Magic 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 Graeco-Latin Square.

ABCDE
DEABC
BCDEA
EABCD
CDEAB
+
ADBEC
BECAD
CADBE
DBECA
ECADB
=
AaBdCbDeEc
DbEeAcBaCd
BcCaDdEbAe
EdAbBeCcDa
CeDcEaAdBb

08111922
16242513
71018214
23191215
14172036
Base 10
0013213442
3144021023
1220334104
4301142230
2432400311
Base 5

Graeco-Latin to Numerical

The relationship between the 5x5 Graeco-Latin square above and a numerical 5x5 Pan-Magic 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 Graeco-Latin 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 Graeco-Latin squares are possible for the larger prime number squares.

How Many 5x5 Pan-Magic Squares are there? - 28800.

Two other pages address this question. It is part of the more general question: "How Many Pan-Magic 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 Pan-Magic 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 order-5 Pan-Magic Squares.

All from one Latin Square.

All of these 28800 squares can be represented by the single Pan-Magic Graeco-Latin square above. Representation by a Graeco-Latin Square reveals the underlying structure and makes it easier to understand the number of possibilities that exist: One Graeco-Latin square is far easier to comprehend than 28800 different squares!


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