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
|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 K3-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.