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12.2   Most Perfect Magic Squares (12 x 12) based on Franklin Like Properties

12.2.3 Introduction

Also the 12th order Most Perfect Pan Magic Square shown below is constructed by Donald Morris and satisfies following properties:

1. Compact : Each 2 × 2 sub square sums to 2 * (n2 + 1) = s1/3;
2. Complete: All pairs of integers distant n/2 along a (main) diagonal sum to (n2 + 1) = s1/6;
3. The main bent diagonals and all the bent diagonals parallel to it sum to the Magic Constant s1 = 870.

 1 120 85 72 97 24 133 36 49 84 37 132 142 27 58 75 46 123 10 111 94 63 106 15 8 113 92 65 104 17 140 29 56 77 44 125 138 31 54 79 42 127 6 115 90 67 102 19 9 112 93 64 105 16 141 28 57 76 45 124 134 35 50 83 38 131 2 119 86 71 98 23 12 109 96 61 108 13 144 25 60 73 48 121 135 34 51 82 39 130 3 118 87 70 99 22 5 116 89 68 101 20 137 32 53 80 41 128 139 30 55 78 43 126 7 114 91 66 103 18 4 117 88 69 100 21 136 33 52 81 40 129 143 26 59 74 47 122 11 110 95 62 107 14

12.2.4 Analysis

The defining properties mentioned above result in a set of linear equations comparable with those shown in Section 12.2.2 with following exceptions:

The equations for the sum of the rows and columns of the 4 x 4 sub squares should be replaced by the sums of the rows and columns of the 12 x 12 square.

Following equations should be added to satisfy property 2 mentioned above (Complete).

 a( 1) + a(79) = s1/6 a( 2) + a(80) = s1/6 a( 3) + a(81) = s1/6 a( 4) + a(82) = s1/6 a( 5) + a(83) = s1/6 a( 6) + a(84) = s1/6 a( 7) + a(73) = s1/6 a( 8) + a(74) = s1/6 a( 9) + a(75) = s1/6 a(10) + a(76) = s1/6 a(11) + a(77) = s1/6 a(12) + a(78) = s1/6 a(13) + a(91) = s1/6 a(14) + a(92) = s1/6 a(15) + a(93) = s1/6 a(16) + a(94) = s1/6 a(17) + a(95) = s1/6 a(18) + a(96) = s1/6 a(19) + a( 85) = s1/6 a(20) + a( 86) = s1/6 a(21) + a( 87) = s1/6 a(22) + a( 88) = s1/6 a(23) + a( 89) = s1/6 a(24) + a( 90) = s1/6 a(25) + a(103) = s1/6 a(26) + a(104) = s1/6 a(27) + a(105) = s1/6 a(28) + a(106) = s1/6 a(29) + a(107) = s1/6 a(30) + a(108) = s1/6 a(31) + a( 97) = s1/6 a(32) + a( 98) = s1/6 a(33) + a( 99) = s1/6 a(34) + a(100) = s1/6 a(35) + a(101) = s1/6 a(36) + a(102) = s1/6 a(37) + a(115) = s1/6 a(38) + a(116) = s1/6 a(39) + a(117) = s1/6 a(40) + a(118) = s1/6 a(41) + a(119) = s1/6 a(42) + a(120) = s1/6 a(43) + a(109) = s1/6 a(44) + a(110) = s1/6 a(45) + a(111) = s1/6 a(46) + a(112) = s1/6 a(47) + a(113) = s1/6 a(48) + a(114) = s1/6 a(49) + a(127) = s1/6 a(50) + a(128) = s1/6 a(51) + a(129) = s1/6 a(52) + a(130) = s1/6 a(53) + a(131) = s1/6 a(54) + a(132) = s1/6 a(55) + a(121) = s1/6 a(56) + a(122) = s1/6 a(57) + a(123) = s1/6 a(58) + a(124) = s1/6 a(59) + a(125) = s1/6 a(60) + a(126) = s1/6 a(61) + a(139) = s1/6 a(62) + a(140) = s1/6 a(63) + a(141) = s1/6 a(64) + a(142) = s1/6 a(65) + a(143) = s1/6 a(66) + a(144) = s1/6 a(67) + a(133) = s1/6 a(68) + a(134) = s1/6 a(69) + a(135) = s1/6 a(70) + a(136) = s1/6 a(71) + a(137) = s1/6 a(72) + a(138) = s1/6

The resulting number of equations can be written in the matrix representation as:

AM * a = s

which can be reduced, by means of row and column manipulations, and results in following set of linear equations:

```a(138) =(435 - a(139) + a(140) - a(141) + a(142) - a(143) - 2*a(144))/3
a(137) = 290 - a(138) - a(143) - a(144)
a(136) = 290 - a(137) - a(142) - a(143)
a(135) = 290 - a(136) - a(141) - a(142)
a(134) = 290 - a(135) - a(140) - a(141)
a(133) = 870 - a(134) - a(135) - a(136) - a(137) - a(138) - a(139) - a(140) - a(141) - a(142) - a(143) -  a(144)
a(131) = 290 - a(132) - a(143) - a(144)
a(130) = 290 - a(131) - a(142) - a(143)
a(129) = 290 - a(130) - a(141) - a(142)
a(128) = 290 - a(129) - a(140) - a(141)
a(127) = 290 - a(128) - a(139) - a(140)
a(126) = 290 - a(127) - a(138) - a(139)
a(125) = 290 - a(126) - a(137) - a(138)
a(124) = 290 - a(125) - a(136) - a(137)
a(123) = 290 - a(124) - a(135) - a(136)
a(122) = 290 - a(123) - a(134) - a(135)
a(121) = 290 - a(122) - a(133) - a(134)
a(119) = 290 - a(120) - a(131) - a(132)
a(118) = 290 - a(119) - a(130) - a(131)
a(117) = 290 - a(118) - a(129) - a(130)
a(116) = 290 - a(117) - a(128) - a(129)
a(115) = 290 - a(116) - a(127) - a(128)
a(114) = 290 - a(115) - a(126) - a(127)
a(113) = 290 - a(114) - a(125) - a(126)
a(112) = 290 - a(113) - a(124) - a(125)
a(111) = 290 - a(112) - a(123) - a(124)
a(110) = 290 - a(111) - a(122) - a(123)
a(109) = 290 - a(110) - a(121) - a(122)
a(107) = 290 - a(108) - a(119) - a(120)
a(106) = 290 - a(107) - a(118) - a(119)
a(105) = 290 - a(106) - a(117) - a(118)
a(104) = 290 - a(105) - a(116) - a(117)
a(103) = 290 - a(104) - a(115) - a(116)
a(102) = 290 - a(103) - a(114) - a(115)
a(101) = 290 - a(102) - a(113) - a(114)
a(100) = 290 - a(101) - a(112) - a(113)
a( 99) = 290 - a(100) - a(111) - a(112)
a( 98) = 290 - a( 99) - a(110) - a(111)
a( 97) = 290 - a( 98) - a(109) - a(110)
a( 95) = 290 - a( 96) - a(107) - a(108)
a( 94) = 290 - a( 95) - a(106) - a(107)
a( 93) = 290 - a( 94) - a(105) - a(106)
a( 92) = 290 - a( 93) - a(104) - a(105)
a( 91) = 290 - a( 92) - a(103) - a(104)
a( 90) = 290 - a( 91) - a(102) - a(103)
a( 89) = 290 - a( 90) - a(101) - a(102)
a( 88) = 290 - a( 89) - a(100) - a(101)
a( 87) = 290 - a( 88) - a( 99) - a(100)
a( 86) = 290 - a( 87) - a( 98) - a( 99)
a( 85) = 290 - a( 86) - a( 97) - a( 98)
a( 84) = 435 + a( 96) - a(108) + a(120) - a(132) - a(139) + a(140) - a(141) + a(142) - a(143) - 4*a(144)
a( 83) = 290 - a( 84) - a( 95) - a( 96)
a( 82) = 290 - a( 83) - a( 94) - a( 95)
a( 81) = 290 - a( 82) - a( 93) - a( 94)
a( 80) = 290 - a( 81) - a( 92) - a( 93)
a( 79) = 290 - a( 80) - a( 91) - a( 92)
a( 78) = 290 - a( 79) - a( 90) - a( 91)
a( 77) = 290 - a( 78) - a( 89) - a( 90)
a( 76) = 290 - a( 77) - a( 88) - a( 89)
a( 75) = 290 - a( 76) - a( 87) - a( 88)
a( 74) = 290 - a( 75) - a( 86) - a( 87)
a( 73) = 290 - a( 74) - a( 85) - a( 86)
```
 a(72) = 145 - a(138) a(71) = 145 - a(137) a(70) = 145 - a(136) a(69) = 145 - a(135) a(68) = 145 - a(134) a(67) = 145 - a(133) a(66) = 145 - a(144) a(65) = 145 - a(143) a(64) = 145 - a(142) a(63) = 145 - a(141) a(62) = 145 - a(140) a(61) = 145 - a(139) a(60) = 145 - a(126) a(59) = 145 - a(125) a(58) = 145 - a(124) a(57) = 145 - a(123) a(56) = 145 - a(122) a(55) = 145 - a(121) a(54) = 145 - a(132) a(53) = 145 - a(131) a(52) = 145 - a(130) a(51) = 145 - a(129) a(50) = 145 - a(128) a(49) = 145 - a(127) a(48) = 145 - a(114) a(47) = 145 - a(113) a(46) = 145 - a(112) a(45) = 145 - a(111) a(44) = 145 - a(110) a(43) = 145 - a(109) a(42) = 145 - a(120) a(41) = 145 - a(119) a(40) = 145 - a(118) a(39) = 145 - a(117) a(38) = 145 - a(116) a(37) = 145 - a(115) a(36) = 145 - a(102) a(35) = 145 - a(101) a(34) = 145 - a(100) a(33) = 145 - a( 99) a(32) = 145 - a( 98) a(31) = 145 - a( 97) a(30) = 145 - a(108) a(29) = 145 - a(107) a(28) = 145 - a(106) a(27) = 145 - a(105) a(26) = 145 - a(104) a(25) = 145 - a(103) a(24) = 145 - a( 90) a(23) = 145 - a( 89) a(22) = 145 - a( 88) a(21) = 145 - a( 87) a(20) = 145 - a( 86) a(19) = 145 - a( 85) a(18) = 145 - a(96) a(17) = 145 - a(95) a(16) = 145 - a(94) a(15) = 145 - a(93) a(14) = 145 - a(92) a(13) = 145 - a(91) a(12) = 145 - a(78) a(11) = 145 - a(77) a(10) = 145 - a(76) a( 9) = 145 - a(75) a( 8) = 145 - a(74) a( 7) = 145 - a(73) a( 6) = 145 - a(84) a( 5) = 145 - a(83) a( 4) = 145 - a(82) a( 3) = 145 - a(81) a( 2) = 145 - a(80) a( 1) = 145 - a(79)

The solutions can be obtained by guessing:

a(96), a(108), a(120), a(132) and a(139) ... a(144)

and filling out these guesses in the abovementioned equations.

For distinct integers also following inequalities should be applied:

0 < a(i) =< 144       for i = 1, 2 ... 95, 97 ... 107, 109 ... 119, 121 ... 131, 133 ... 138

a(i) ≠ a(j)           for i ≠ j

With a(139) ... a(144) constant, an optimized guessing routine (MgcSqr12c), produced 480 (= 4 * 120) Most Perfect Magic Squares within 126 (= 4 * 31,5) seconds (ref. Attachment 12.2b).