Ultramagic Squares of Order 7 |
Results |
The calculation was divided into 90 parts.
In each part n(1,1) and n(2,1) were fixed. | |
T(1,x) is the total number of squares with n(1,1) = 1 and n(2,1) = x and n(1,2) > x | |
T(x,1) is the total number of squares with n(1,1) = x and n(2,1) = 1 and n(4,6) > x |
The following values are not available for the parameter x: | |
1 | because the number 1 can only occur once. |
25 | because n(4,4) = 25 is a precondition for the square to be self-complementary. |
49 | because n(6,7) = 49 or n(7,7) = 49 as 49 is the complement value of 1. |
48 | because in the case n(1,1) = 1 is x = n(2,1) < n(1,2) < 49 because in the case n(2,1) = 1 is x = n(1,1) < n(4,6) < 49 |
x | T(1,x) | T(x,1) | x | T(1,x) | T(x,1) | x | T(1,x) | T(x,1) | |||
2 | 10971 | 11032 | 17 | 54163 | 33714 | 33 | 56225 | 40940 | |||
3 | 14805 | 12388 | 18 | 54928 | 39271 | 34 | 54913 | 45353 | |||
4 | 17422 | 15318 | 19 | 58294 | 35674 | 35 | 53394 | 40684 | |||
5 | 21499 | 16172 | 20 | 58114 | 40675 | 36 | 49525 | 43648 | |||
6 | 24188 | 19825 | 21 | 60299 | 37195 | 37 | 45498 | 35885 | |||
7 | 28539 | 19968 | 22 | 59509 | 40657 | 38 | 43142 | 40704 | |||
8 | 30060 | 23778 | 23 | 60892 | 36256 | 39 | 40530 | 34942 | |||
9 | 34409 | 22698 | 24 | 58887 | 41212 | 40 | 35467 | 36479 | |||
10 | 36486 | 27437 | 26 | 68566 | 47458 | 41 | 31393 | 30900 | |||
11 | 40332 | 25558 | 27 | 69538 | 42359 | 42 | 27436 | 31202 | |||
12 | 41629 | 30730 | 28 | 66469 | 47145 | 43 | 23581 | 25281 | |||
13 | 45905 | 29409 | 29 | 64984 | 42210 | 44 | 19033 | 23581 | |||
14 | 46978 | 34453 | 30 | 61937 | 47938 | 45 | 14082 | 16880 | |||
15 | 49326 | 30872 | 31 | 63046 | 42199 | 46 | 9854 | 13013 | |||
16 | 51005 | 37091 | 32 | 58975 | 46927 | 47 | 4562 | 7213 |
a = T(1,2) + T(1,3) + ... + T(1,47) | b = T(2,1) + T(3,1) + ... + T(47,1) |
a = 1,920,790 | b = 1,444,324 |
There are c = a + b so-called essentially different squares, i.e. it is not possible to derive one square from another by one of the 48 transpositions. | |
c = a + b = 3,365,114 | |
2a is the total number of squares with n(1,1) = 1, including reflections at the downward main diagonal, i.e. without the condition n(2,1) < n(1,2). | 2b is the total number of squares with n(2,1) = 1, including K^{3}-transposed squares, i.e. without the condition n(1,1) < n(4,6). |
2a = 3,841,580 | 2b = 2,888,648 |
There are 24 transpositions that move the number from cell (1,1)
to any of the 24 axis-cells. Thus you get 24 · 2·a = 48·a different squares with number 1 in any axis-cell. | There are 24 transpositions that move the number from cell (2,1)
to any of the 24 filling-cells. Thus you get 24 · 2·b = 48·b different squares with number 1 in any filling-cell. |
Finally there are e = 48·c squares including rotated and reflected squares. Each of these squares could be generated from one of the c essentially different squares by one of the 48 transpositions. | |
e = 48 · c = 161,525,472 | |
As it is not usual to count reflected and rotated squares, the total number of different self-complementary pan-magic 7x7-squares is d = e / 8 . | |
d = e / 8 = 6 · c = 20,190,684 | |
Notice that all 3,365,114 essentially different squares were saved on disk. All other squares could be derived very easily. |
Of course I can not garantee that the results are correct. But I'm rather sure they are. - All calculated magic squares had been tested with regard to the demanded properties. - There are no doublets within the calculated squares. - It could be shown that there are 3456 regular pandiagonal symmetric magic 7x7-squares. My computer-program found exactly that amount of these special squares. - In June, 2004 Francis Gaspalou from Gif-sur-Yvette, France told me about his own computations. He got the same total number of ultramagic order-7 squares. See improvements. |
Prof. Grogono describes the properties of regular pan-magic squares. |
Can anybody confirm one of the partial results in the above table? |
summary | cells | equations | transpositions | improvements |
results | programs | files | conclusions | samples |