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. |
Notations (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 10^{17} pan-magic order-7 squares. (See conclusions) |
summary | cells | equations | transpositions | improvements |
results | programs | files | conclusions | samples |