Office Applications and Entertaiment, Magic Squares  
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9.4.2 Further Analysis, Compact Associated Pan Magic Squares
The symmetry applied in the deduction on previous page was limited to the Main Diagonals.
a(79) = 123  a(80)  a(81) a(76) = 123  a(77)  a(78) a(73) = 123  a(74)  a(75) a(70) = 123  a(71)  a(72) a(69) = a(72) + a(74) + a(75)  a(77)  2 * a(78) + a(81) a(68) = a(71)  a(74) + a(80) a(67) = 123  a(71)  a(72)  a(75) + a(77) + 2 * a(78)  a(80)  a(81) a(66) = a(72) + a(74)  a(80) a(65) = a(71)  2 * a(74) + 2 * a(80) a(64) = 123  a(71)  a(72) + a(74)  a(80) a(63) = 123  a(72)  a(81) a(62) = 123  a(71)  a(80) a(61) =123 + a(71) + a(72) + a(80) + a(81) a(60) = 123  a(72)  a(74)  a(75) + a(77) + a(78)  a(81) a(59) = 123  a(71) + a(74)  a(77)  a(80) a(58) =123 + a(71) + a(72) + a(75)  a(78) + a(80) + a(81) a(57) = 123  a(72)  a(74)  a(75) + a(80) a(56) = 123  a(71) + a(74)  2 * a(80) a(55) =123 + a(71) + a(72) + a(75) + a(80) a(54) = 82 + a(71) + a(72)  a(78) + a(80) + a(81) a(53) = 164  a(71)  a(77)  a(80) a(52) = 41  a(72) + a(77) + a(78)  1 * a(81) a(51) = 82 + a(71) + a(72) + a(80) a(50) = 164  a(71)  2 * a(80) a(49) = 41  a(72) + a(80) a(48) = 82 + a(71) + a(72) + a(75)  a(78) + a(80) a(47) = 164  a(71) + a(74)  a(77)  2 * a(80) a(46) = 41  a(72)  a(74)  a(75) + a(77) + a(78) + a(80) a(45) = 164  a(71)  2 * a(72)  a(74)  a(75) + a(77) + 2 * a(78)  a(81) a(44) = 41 + a(74)  a(80) a(43) = 82 + a(71) + 2 * a(72) + a(75)  a(77)  2 * a(78) + a(80) + a(81) a(42) = 164  a(71)  2 * a(72) a(41) = 41
which can be applied in an Excel spreadsheet (Ref. CnstrSngl9b2) and an appropriate guessing routine
(MgcSqr9g).
9.4.3 Further Analysis, Compact Pan Magic Squares
The applicable equations for Compact Pan Magic Squares are: a(73) = s1  a(74)  a(75)  a(76)  a(77)  a(78)  a(79)  a(80)  a(81) a(64) = s1  a(65)  a(66)  a(67)  a(68)  a(69)  a(70)  a(71)  a(72) a(61) = s1  a(62)  a(63)  a(70)  a(71)  a(72)  a(79)  a(80)  a(81) a(60) = s1  a(61)  a(62)  a(69)  a(70)  a(71)  a(78)  a(79)  a(80) a(59) = s1  a(60)  a(61)  a(68)  a(69)  a(70)  a(77)  a(78)  a(79) a(58) = s1  a(59)  a(60)  a(67)  a(68)  a(69)  a(76)  a(77)  a(78) a(57) = s1  a(58)  a(59)  a(66)  a(67)  a(68)  a(75)  a(76)  a(77) a(56) = s1  a(57)  a(58)  a(65)  a(66)  a(67)  a(74)  a(75)  a(76) a(55) = s1  a(56)  a(57)  a(64)  a(65)  a(66)  a(73)  a(74)  a(75) a(52) = s1  a(53)  a(54)  a(61)  a(62)  a(63)  a(70)  a(71)  a(72) a(51) = s1  a(52)  a(53)  a(60)  a(61)  a(62)  a(69)  a(70)  a(71) a(50) = s1  a(51)  a(52)  a(59)  a(60)  a(61)  a(68)  a(69)  a(70) a(49) = s1  a(50)  a(51)  a(58)  a(59)  a(60)  a(67)  a(68)  a(69) a(48) = s1  a(49)  a(50)  a(57)  a(58)  a(59)  a(66)  a(67)  a(68) a(47) = s1  a(48)  a(49)  a(56)  a(57)  a(58)  a(65)  a(66)  a(67) a(46) = s1  a(47)  a(48)  a(55)  a(56)  a(57)  a(64)  a(65)  a(66) a(43) = s1  a(44)  a(45)  a(52)  a(53)  a(54)  a(61)  a(62)  a(63) a(42) = s1  a(43)  a(44)  a(51)  a(52)  a(53)  a(60)  a(61)  a(62) a(41) = s1  a(42)  a(43)  a(50)  a(51)  a(52)  a(59)  a(60)  a(61) a(40) = s1  a(41)  a(42)  a(49)  a(50)  a(51)  a(58)  a(59)  a(60) a(39) = s1  a(40)  a(41)  a(48)  a(49)  a(50)  a(57)  a(58)  a(59) a(38) = s1  a(39)  a(40)  a(47)  a(48)  a(49)  a(56)  a(57)  a(58) a(37) = s1  a(38)  a(39)  a(46)  a(47)  a(48)  a(55)  a(56)  a(57) a(34) = s1  a(35)  a(36)  a(43)  a(44)  a(45)  a(52)  a(53)  a(54) a(33) = s1  a(34)  a(35)  a(42)  a(43)  a(44)  a(51)  a(52)  a(53) a(32) = s1  a(33)  a(34)  a(41)  a(42)  a(43)  a(50)  a(51)  a(52) a(31) = s1  a(32)  a(33)  a(40)  a(41)  a(42)  a(49)  a(50)  a(51) a(30) = s1  a(31)  a(32)  a(39)  a(40)  a(41)  a(48)  a(49)  a(50) a(29) = s1  a(30)  a(31)  a(38)  a(39)  a(40)  a(47)  a(48)  a(49) a(28) = s1  a(29)  a(30)  a(37)  a(38)  a(39)  a(46)  a(47)  a(48) a(27) = s1 / 3  a(54)  a(75)  a(78) + a(81) a(26) = s1 / 3  a(53)  a(74)  a(77) + a(80) a(25) = s1  a(26)  a(27)  a(34)  a(35)  a(36)  a(43)  a(44)  a(45) a(24) = s1  a(25)  a(26)  a(33)  a(34)  a(35)  a(42)  a(43)  a(44) a(23) = s1  a(24)  a(25)  a(32)  a(33)  a(34)  a(41)  a(42)  a(43) a(22) = s1  a(23)  a(24)  a(31)  a(32)  a(33)  a(40)  a(41)  a(42) a(21) = s1  a(22)  a(23)  a(30)  a(31)  a(32)  a(39)  a(40)  a(41) a(20) = s1  a(21)  a(22)  a(29)  a(30)  a(31)  a(38)  a(39)  a(40) a(19) = s1  a(20)  a(21)  a(28)  a(29)  a(30)  a(37)  a(38)  a(39) a(18) = s1 / 3  a(45)  a(66)  a(69) + a(72) a(17) = s1 / 3  a(44)  a(65)  a(68) + a(71) a(16) = s1  a(17)  a(18)  a(25)  a(26)  a(27)  a(34)  a(35)  a(36) a(15) = s1  a(16)  a(17)  a(24)  a(25)  a(26)  a(33)  a(34)  a(35) a(14) = s1  a(15)  a(16)  a(23)  a(24)  a(25)  a(32)  a(33)  a(34) a(13) = s1  a(14)  a(15)  a(22)  a(23)  a(24)  a(31)  a(32)  a(33) a(12) = s1  a(13)  a(14)  a(21)  a(22)  a(23)  a(30)  a(31)  a(32) a(11) = s1  a(12)  a(13)  a(20)  a(21)  a(22)  a(29)  a(30)  a(31) a(10) = s1  a(11)  a(12)  a(19)  a(20)  a(21)  a(28)  a(29)  a(30) a( 9) = s1  a(17)  a(25)  a(33)  a(41)  a(49)  a(57)  a(65)  a(73) a( 8) = s1  a(16)  a(24)  a(32)  a(40)  a(48)  a(56)  a(64)  a(81) a( 7) = s1  a( 8)  a( 9)  a(16)  a(17)  a(18)  a(25)  a(26)  a(27) a( 6) = s1  a( 7)  a( 8)  a(15)  a(16)  a(17)  a(24)  a(25)  a(26) a( 5) = s1  a( 6)  a( 7)  a(14)  a(15)  a(16)  a(23)  a(24)  a(25) a( 4) = s1  a( 5)  a( 6)  a(13)  a(14)  a(15)  a(22)  a(23)  a(24) a( 3) = s1  a( 4)  a( 5)  a(12)  a(13)  a(14)  a(21)  a(22)  a(23) a( 2) = s1  a( 3)  a( 4)  a(11)  a(12)  a(13)  a(20)  a(21)  a(22) a( 1) = s1  a( 2)  a( 3)  a(10)  a(11)  a(12)  a(19)  a(20)  a(21)
Subject equations will be used for the generation of Sudoku Comparable Squares as discussed in Section 9.5 below,
9.4.4 Further Analysis, Partly Compact Magic Squares
Partly Compact means that, while starting with a 9^{th} order square divided into nine 3^{th} order sub squares, only the elements of the sub squares obtained by moving a window either horizontally or vertically will sum to the Magic Sum.
Although Partly Compact Magic Squares will not be considered in this section, the definition is useful to understand some of the pre selection criteria used in following sections.
9.5 More Solutions, Sudoku Comparable
Any number m = 0 ... 80 can be written as m = b_{1} + 9 * b_{2} with
b_{i} = 0, 1, ... 8 for i = 1, 2.
9.5.2 Sudoku Comparable Squares
Rows, columns and 3^{th} order sub squares
of the matrices B1 and B2
containing the nine integers 0, 1, ... 8 will sum to the Magic Sum 36.
Subject matrices will be further referred to as Sudoku Comparable Squares.
The number of independent variables (24) for the equations deducted in Section 9.4.3 above is too high to obtain results within a reasonable time, even for the integers 0, 1 ... 8.
Numerous (Pan) Magic Squares can be generated by selecting combinations of Sudoku Comparable Squares (B1, B2) while ensuring that the resulting square M contains all integers 1 thru 81 (CnstrSqrs9a):
Attachment 9.6.5 shows a few examples of four Ternary squares, two related Sudoku Comparable Squares, the resulting Magic Square and the corresponding Magic Square of squared elements.

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