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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.

The applicable equations for Compact Associated Pan Magic Squares, with every third-row and third-column summing to one third of the magic constant, are:

```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
```
 a(40) = 82 - a(42) a(39) = 82 - a(43) a(38) = 82 - a(44) a(37) = 82 - a(45) a(36) = 82 - a(46) a(35) = 82 - a(47) a(34) = 82 - a(48) a(33) = 82 - a(49) a(32) = 82 - a(50) a(31) = 82 - a(51) a(30) = 82 - a(52) a(29) = 82 - a(53) a(28) = 82 - a(54) a(27) = 82 - a(55) a(26) = 82 - a(56) a(25) = 82 - a(57) a(24) = 82 - a(58) a(23) = 82 - a(59) a(22) = 82 - a(60) a(21) = 82 - a(61) a(20) = 82 - a(62) a(19) = 82 - a(63) a(18) = 82 - a(64) a(17) = 82 - a(65) a(16) = 82 - a(66) a(15) = 82 - a(67) a(14) = 82 - a(68) a(13) = 82 - a(69) a(12) = 82 - a(70) a(11) = 82 - a(71) a(10) = 82 - a(72) a( 9) = 82 - a(73) a( 8) = 82 - a(74) a( 7) = 82 - a(75) a( 6) = 82 - a(76) a( 5) = 82 - a(77) a( 4) = 82 - a(78) a( 3) = 82 - a(79) a( 2) = 82 - a(80) a( 1) = 82 - a(81)

which can be applied in an Excel spreadsheet (Ref. CnstrSngl9b2) and an appropriate guessing routine (MgcSqr9g).

Examples of Compact Associated Pan Magic Squares, with every third-row and third-column summing to one third of the magic constant, which can be obtained based on the equations deducted above, are shown in Attachment 9.4.4.

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 9.5.3.

9.4.4 Further Analysis, Partly Compact Magic Squares

Partly Compact means that, while starting with a 9th order square divided into nine 3th 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.

• For Compact Magic Squares the elements of 9 x 9 = 81 sub squares will sum to the Magic Sum;

• For Partly Compact Magic Squares the elements of only 3 * 9 + 3 * 6 = 45 sub squares 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 Section 9.5.

9.4.5 Further Analysis, Associated Magic Squares

Every third-row and third-column summing to s1/3

The defining equations for Associated Magic Squares for which every third-row and third-column sum to one third of the magic constant s1 are:

```a(79) =    s1/3 - a(80) - a(81)
a(76) =    s1/3 - a(77) - a(78)
a(73) =    s1/3 - a(74) - a(75)
a(70) =    s1/3 - a(71) - a(72)
a(67) =    s1/3 - a(68) - a(69)
a(64) =    s1/3 - a(65) - a(66)
a(63) =    s1/3 - a(72) - a(81)
a(62) =    s1/3 - a(71) - a(80)
a(61) = -  s1/3 + a(71) + a(72) + a(80) + a(81)
a(60) =    s1/3 - a(69) - a(78)
a(59) =    s1/3 - a(68) - a(77)
a(58) = -  s1/3 + a(68) + a(69) + a(77) + a(78)
a(57) =    s1/3 - a(66) - a(75)
a(56) =    s1/3 - a(65) - a(74)
a(55) = -  s1/3 + a(65) + a(66) + a(74) + a(75)
a(52) =    s1/3 - a(53) - a(54)
a(49) =    s1/3 - a(50) - a(51)
a(46) =    s1/3 - a(47) - a(48)
a(45) =  4*s1/9 - a(47) - a(48) - a(54)
a(44) =    s1/9 + a(47) - a(53)
a(43) = -2*s1/9 + a(48) + a(53) + a(54)
a(42) =  4*s1/9 - a(50) - 2 * a(51)
a(41) =    s1/9
```
 a(40) = 2*s1/9- a(42) a(39) = 2*s1/9- a(43) a(38) = 2*s1/9- a(44) a(37) = 2*s1/9- a(45) a(36) = 2*s1/9- a(46) a(35) = 2*s1/9- a(47) a(34) = 2*s1/9- a(48) a(33) = 2*s1/9- a(49) a(32) = 2*s1/9- a(50) a(31) = 2*s1/9- a(51) a(30) = 2*s1/9- a(52) a(29) = 2*s1/9- a(53) a(28) = 2*s1/9- a(54) a(27) = 2*s1/9- a(55) a(26) = 2*s1/9- a(56) a(25) = 2*s1/9- a(57) a(24) = 2*s1/9- a(58) a(23) = 2*s1/9- a(59) a(22) = 2*s1/9- a(60) a(21) = 2*s1/9- a(61) a(20) = 2*s1/9- a(62) a(19) = 2*s1/9- a(63) a(18) = 2*s1/9- a(64) a(17) = 2*s1/9- a(65) a(16) = 2*s1/9- a(66) a(15) = 2*s1/9- a(67) a(14) = 2*s1/9- a(68) a(13) = 2*s1/9- a(69) a(12) = 2*s1/9- a(70) a(11) = 2*s1/9- a(71) a(10) = 2*s1/9- a(72) a( 9) = 2*s1/9- a(73) a( 8) = 2*s1/9- a(74) a( 7) = 2*s1/9- a(75) a( 6) = 2*s1/9- a(76) a( 5) = 2*s1/9- a(77) a( 4) = 2*s1/9- a(78) a( 3) = 2*s1/9- a(79) a( 2) = 2*s1/9- a(80) a( 1) = 2*s1/9- a(81)

which can be applied in an Excel spreadsheet (Ref. CnstrSngl9b3) and an appropriate guessing routine as discussed in Section 9.5.2.

Examples of Associated Magic Squares, with every third-row and third-column summing to one third of the magic constant, which can be obtained based on the equations deducted above, are shown in Attachment 9.4.5.

9.4.6 Further Analysis, Associated Magic Squares

Regular Sub Squares summing to s1

The defining equations for Associated Magic Squares for which the elements of the regular sub squares (9) sum to the magic constant s1 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(58) =     s1   - a(59) - a(60) - a(67) - a(68) - a(69) - a(76) - a(77) - a(78)
a(55) =     s1   - a(56) - a(57) - a(64) - a(65) - a(66) - a(73) - a(74) - a(75)
a(46) =     s1   - a(47) - a(48) - a(49) - a(50) - a(51) - a(52) - a(53) - a(54)
a(45) =     s1/9 + a(46) - a(54) + a(55) - a(63) + a(64) - a(72) + a(73) - a(81)
a(44) =     s1/9 + a(47) - a(53) + a(56) - a(62) + a(65) - a(71) + a(74) - a(80)
a(43) = - 8*s1/9 + a(48) - a(52) + a(57) + a(62) + a(63) + a(66) + a(71) + a(72) + a(75) + a(80) + a(81)
a(42) =  10*s1/9 + a(49) - a(51) - a(59) - 2 * a(60) - a(68) - 2 * a(69) - a(77) - 2 * a(78)
a(41) =     s1/9
```
 a(40) = 2*s1/9- a(42) a(39) = 2*s1/9- a(43) a(38) = 2*s1/9- a(44) a(37) = 2*s1/9- a(45) a(36) = 2*s1/9- a(46) a(35) = 2*s1/9- a(47) a(34) = 2*s1/9- a(48) a(33) = 2*s1/9- a(49) a(32) = 2*s1/9- a(50) a(31) = 2*s1/9- a(51) a(30) = 2*s1/9- a(52) a(29) = 2*s1/9- a(53) a(28) = 2*s1/9- a(54) a(27) = 2*s1/9- a(55) a(26) = 2*s1/9- a(56) a(25) = 2*s1/9- a(57) a(24) = 2*s1/9- a(58) a(23) = 2*s1/9- a(59) a(22) = 2*s1/9- a(60) a(21) = 2*s1/9- a(61) a(20) = 2*s1/9- a(62) a(19) = 2*s1/9- a(63) a(18) = 2*s1/9- a(64) a(17) = 2*s1/9- a(65) a(16) = 2*s1/9- a(66) a(15) = 2*s1/9- a(67) a(14) = 2*s1/9- a(68) a(13) = 2*s1/9- a(69) a(12) = 2*s1/9- a(70) a(11) = 2*s1/9- a(71) a(10) = 2*s1/9- a(72) a( 9) = 2*s1/9- a(73) a( 8) = 2*s1/9- a(74) a( 7) = 2*s1/9- a(75) a( 6) = 2*s1/9- a(76) a( 5) = 2*s1/9- a(77) a( 4) = 2*s1/9- a(78) a( 3) = 2*s1/9- a(79) a( 2) = 2*s1/9- a(80) a( 1) = 2*s1/9- a(81)

which can be applied in an Excel spreadsheet (Ref. CnstrSngl9b4) and an appropriate guessing routine as discussed in Section 9.5.2.

Examples of Associated Magc Squares, for which the elemnets of the regular sub squares sum to the magic constant, which can be obtained based on the equations deducted above, are shown in Attachment 9.4.6.