Ultramagic Squares of Order 7
 Improvements

Confirmation of the Results
 On June 21st, 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."

The New Transposition W
 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, R2 and K3. That means, you get the original square by applying W twice.
 --->W --->W
The numbers in the white cells are fixed. Each other number is interchanged with a number in a cell of the same colour.

The New Transposition-Group
 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.

Are there more general transpositions?
 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.)

What's Next?
 - 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.)

Index
 summary cells equations transpositions improvements results programs files conclusions samples