Ultramagic Squares of Order 7 |
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On June 21^{st}, 2004 an email came in from the French engineer Francis Gaspalou
(email: francis.gaspalou@wanadoo.fr), who is living in Gif-sur-Yvette near Paris.
Here is a part of his email: "... I can confirm to you that I have found the same results, by a very similar method. The number of essentially different squares (3,365,114) can be divided by 2 : there are 96 transpositions for a given square, and not only 48. With a Duron 1,4 Ghz computer and a program in C language, I found in 12 days the 1,682,557 essentially different squares." |
Francis Gaspalou called the new transposition W. He found it in August, 2000 when he examined regular ultramagic 7x7-squares. Each ultramagic square is transformed to another ultramagic square by applying transposition W. W is an involutory transposition, like S, R^{2} and K^{3}. That means, you get the original square by applying W twice. |
---> W |
---> W |
The transpositions R, S, K and W generate a group of 96 different transpositions.
Thus there are only 20,190,684 · 8 / 96 = 1,682,557 essentially different ultramagic order-7 squares. Now the computation should take only half the time. |
No! As 1,682,557 is a prime the set of essentially different ultramagic order-7 squares can't be invariant relative to any further transposition. (Otherwise this transposition would generate a complete set of essentially different ultramagic order-7 squares and any pair of such squares would define such a transposition - that's not true.) |
- The ideas of Francis Gaspalou will be explained in more detail. - The transpositions should be described in a better way. - The results can be arranged in a new order. (Probably all that will take some months.) |
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