Ultramagic Squares of Order 7
Total Number
6 · (1,920,790 + 1,444,324) =
= 20,190,684

On these pages you will find many details about transposing and calculating ultramagic order-7 squares.
These squares are pandiagonal and center symmetrical (associative).
Please look for other sites first if you are not familar with fundamental properties of magic squares.
There are extensive magic-squares sites by Holger Danielsson or by Harvey Heinz.

(x,y) is the cell in column x and row y of the square, with x, y from 1 to 7.
n(x,y) is the natural number (from 1 to 49) in cell (x,y).
There are a = 1,920,790 squares with n(1,1) = 1 and n(2,1) < n(1,2).
There are b = 1,444,324 squares with n(2,1) = 1 and n(1,1) < n(4,6).
This makes 2a squares with n(1,1) = 1 and 2b squares with n(2,1) = 1.
There are special transpositions that move the number 1 from (1,1) to 23 other cells
without disturbing the special properties of the square.
Additionally the number 1 could be transposed from (2,1) to 23 new cells.
This makes 24 · (2a + 2b) squares.
As it is not usual to count rotated and reflected squares, you have to devide by 8 and get:
24 · (2a + 2b) / 8 = 6 · (1,920,790 + 1,444,324) = 20,190,684
In 2004 Francis Gaspalou (France) told me about another transposition for ultramagic squares.
It enables him to decrease the number of essentially different squares by the factor 2.
See Improvements.
The numbers a und b were calculated by a small computer-program.
The cells of the square were filled with numbers from -24 to 24.
Only 24 cells had to be inspected.
There are 12 equations connecting the numbers of these cells.
I saved all (a + b) = 3,365,114 essentially different squares on disk.
This data can be used for further investigations.
First examinations of the data show that they contain 576 regular pan-magic squares.
Thus there are 6 · 576 = 3456 regular ultramagic squares.
This is exactly what the theory predicts.
Notice that each of the 20,190,684 squares generates 49 panmagic squares.
Thus you get a lower bound for the total number of panmagic squares:
(20,190,684 - 3456) · 49 + 38,102,400 = 1,027,276,572
(38,102,400 is the wellknown total of regular pan-magic 7x7-squares.)
But this is very weak, as there are more than 1017 pan-magic order-7 squares. (See conclusions)

summary cells equations transpositions improvements
results programs files conclusions samples

Walter Trump, Nürnberg, Germany, (c) 2001-04-25 (last modified: 2004-06-27)
(Nürnberg (Nuernberg, Nuremberg) was the home of Albrecht Dürer.)