Office Applications and Entertainment, Magic Squares  
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6.0 Magic Squares (6 x 6)
Magic squares of order 6 can be represented as follows:
For Pan Magic Squares the numbers a(i), i = 1 ... 36, in all rows, columns, main and broken diagonals should sum to the same constant, which results in following linear equations:
a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) = s1
which can be reduced, by means of row and column manipulations, to:
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(25) = s1  a(26)  a(27)  a(28)  a(29)  a(30) a(19) = s1  a(20)  a(21)  a(22)  a(23)  a(24) a(17) = 2/3 * s1  a(18) + a(20) + a(21)  a(23)  a(24) + a(26) + a(27)  a(29)  a(30)  a(35)  a(36) a(16) = 2 * s1 + a(18)  a(20)  2*a(21)  2*a(22)  a(23)  a(26)  2*a(27)  2*a(28)  a(29)  a(34) + a(36) a(15) = 2/3 * s1  a(18)  a(33)  a(36) a(14) = a(18)  a(20)  a(21) + a(23) + a(24)  a(26)  a(27) + a(29) + a(30)  a(32) + a(36) a(13) = 4/3 * s1  a(18) + a(20) + 2*a(21) + 2*a(22) + a(23) + a(26) + 2*a(27) + 2*a(28) + a(29)  a(31)  a(36) a(12) = 11/6 * s1  a(20)  a(21)  a(22)  a(26)  2*a(27)  a(28)  a(32)  a(33)  a(34) a(11) = 7/6 * s1 + a(22) + a(23) + a(24)  a(26) + a(28) + a(29) + a(30) + a(34) + a(35) + a(36) a(10) = 7/6 * s1 + a(21) + a(22) + a(23)  a(25) + a(27) + a(28) + a(29) + a(33) + a(34) + a(35) a(9) = 7/6 * s1 + a(20) + a(21) + a(22) + a(26) + a(27) + a(28)  a(30) + a(32) + a(33) + a(34) a(8) = 11/6 * s1  a(22)  a(23)  a(24)  a(28)  2*a(29)  a(30)  a(34)  a(35)  a(36) a(7) = 11/6 * s1  a(21)  a(22)  a(23)  a(27)  2*a(28)  a(29)  a(33)  a(34)  a(35) a(6) = 5/6 * s1  a(18) + a(20) + a(21) + a(22)  a(24) + a(26) + 2*a(27) + a(28)  a(30)+a(32)+a(33)+a(34)a(36) a(5) = 1/2 * s1 + a(18) + a(19) + a(24)  a(27)  a(28)  a(29)  a(34)  a(35) a(4) = 1/6 * s1  a(18) + a(20) + a(21)  a(28)  a(29)  a(30) + a(31) + a(32) a(3) = 1/2 * s1 + a(18)  a(20)  2*a(21)  a(22)  a(26)  2*a(27)  a(28) + a(30) + a(31) + a(35) + 2*a(36) a(2) = 1/6 * s1  a(18) + a(21) + a(22)  a(25)  a(26)  a(30) + a(34) + a(35) a(1) = 1/2 * s1 + a(18) + a(23) + a(24)  a(25)  a(26)  a(27)  a(31)  a(32)
Although the linear equations shown above can be solved for any magic constant s1, it is obvious that no solution can be found for the consecutive integers {1 ... 36}, as the related magic constant s1 = 111 is odd.
Solutions for non consecutive integers, resulting in other magic constants, will be discussed in Section 6.09.
6.2 Analytic Solution, Simple Magic Squares
For solutions with consecutive integers we moderate the definition to all rows, columns and main diagonals sum to the same constant, which results in following linear equations:
a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) = s1 which can be reduced, by means of row and column manipulations, to: 
a(31) = s1  a(32)  a(33)  a(34)  a(35)  a(36) a(25) = s1  a(26)  a(27)  a(28)  a(29)  a(30) a(19) = s1  a(20)  a(21)  a(22)  a(23)  a(24) a(13) = s1  a(14)  a(15)  a(16)  a(17)  a(18) a(11) = + a(12)  a(16) + a(18)  a(21) + a(24)  a(26) + a(30)  a(31) + a(36) a( 8) = { a( 9)  a(10)  2*a(11)  a(14)  2*a(15)  2*a(16)  a(17) + 2*a(19) + a(20) + a(23) + 2*a(24) + 2*a(25) + a(27) + a(28) + 2*a(30)}/2 a( 7) = a( 8)  a(13) + a(15)  a(19) + a(22)  a(25) + a(29)  a(31) + a(36) a( 6) = s1  a(11)  a(16)  a(21)  a(26)  a(31) a( 5) = s1  a(11)  a(17)  a(23)  a(29)  a(35) a( 4) = s1  a(10)  a(16)  a(22)  a(28)  a(34) a( 3) = s1  a( 9)  a(15)  a(21)  a(27)  a(33) a( 2) = s1  a( 8)  a(14)  a(20)  a(26)  a(32) a( 1) = s1  a( 8)  a(15)  a(22)  a(29)  a(36)
The linear equations shown above are ready to be solved, for the magic constant 111.
0 < a(i) =< 36 for i = 1, 2, ... 8, 11, 13, 19, 25
which can be incorporated in a guessing routine, which can be used to generate  if not all  at least collections of 6^{th} order squares with distinct integers within a reasonable time (MgcSqr6b).
With the highlighted variables constant, above mentioned guessing routine (MgcSqr6b), produced 856 Simple Magic Squares of the 6^{th} order within 56 minutes, which are shown in Attachment 6.5.1.
An alternative routine  in which after the bottom row, first the main diagonals and afterward the remaining rows and columns are calculated  appeared to be much faster (MgcSqr6a).
A Magic Square is Bimagic if it remains magic after each of the numbers have been squared.
It has been proven that Bimagic Squares of order 6 can't exist.
However Simple Magic Squares of order 6 can have Bimagic Diagonals.
Although Bimagic Squares of order 6 can't exist, it is possible to construct Partly Bimagic Squares of order 6 with following procedure:
The procedure described above is illustrated below for the first occurring Partly Bimagc Square based on Generator 1:
and is a variation on the methods used by e.g. Achille Rilly (1901) for the construction of Bimagic Squares of order 8 based on limited amounts of Bimagic Series (ref. Section 15.3.1).
For simple magic squares, related squares can be found by means of rotation and/or reflection (ref. Attachment 6.5.2).
Based on this set of transformations and the eight squares which can be found by means of rotation and/or reflection (ref. Attachment 6.5.2) any 6^{th} order simple magic square corresponds with a Class of 8 * 24 = 192 magic squares.

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