The following values are not available for the parameter x:
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| 1 | because the number 1 can only occur once.
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| 25 | because n(4,4) = 25 is a precondition for the square to be self-complementary.
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| 49 | because n(6,7) = 49 or n(7,7) = 49 as 49 is the complement value of 1.
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| 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
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| a = T(1,2) + T(1,3) + ... + T(1,47)
| b = T(2,1) + T(3,1) + ... + T(47,1)
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| a = 1,920,790
| b = 1,444,324
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| 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
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| 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
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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 .
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| d = e / 8 = 6 · c = 20,190,684
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Notice that all 3,365,114 essentially different squares were saved on disk.
All other squares could be derived very easily.
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