Office Applications and Entertainment, Magic Squares

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

12.2.5 Introduction

The 12th order Most Perfect Pan Magic Square shown below 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. Any row or column can be divided in three parts summing to one third of the Magic Constant.

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

12.2.6 Analysis

The defining properties mentioned above result in a set of linear equations comparable with those shown in previous sections:

The numbers of the long diagonals and all the broken diagonals parallel to it sum to the s1 (Pan Magic):

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

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

Each 2 × 2 sub square sums to 2 * (n2 + 1) = s1/3 (Compact):

a(i) + a(i+1) + a(i+12) + a(i+13) = s1/3 with 1 =< i < 132 and i ≠ 12*n for n = 1, 2 ... 11

a(i) + a(i+1) + a(i+12) + a(i-11) = s1/3 with i = 12*n for n = 1, 2 ... 11

a(i) + a(i+1) + a(i+132) + a(i+133) = s1/3 with i = 1, 2 ... 11

a(1) + a(12)   + a(133)   + a(144)   = s1/3

All pairs of integers distant n/2 along a (main) diagonal sum to (n2 + 1) = s1/6 (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

Any row or column can be divided in three parts summing to s1/3:

a( 1) + a( 2) + a( 3) + a( 4) = s1/3
a(13) + a(14) + a(15) + a(16) = s1/3
a(25) + a(26) + a(27) + a(28) = s1/3
a(37) + a(38) + a(39) + a(40) = s1/3

a(49) + a(50) + a(51) + a(52) = s1/3
a(61) + a(62) + a(63) + a(64) = s1/3
a(73) + a(74) + a(75) + a(76) = s1/3
a(85) + a(86) + a(87) + a(88) = s1/3

a( 97)+ a( 98)+ a( 99)+ a(100)= s1/3
a(109)+ a(110)+ a(111)+ a(112)= s1/3
a(121)+ a(122)+ a(123)+ a(124)= s1/3
a(133)+ a(134)+ a(135)+ a(136)= s1/3

a( 5) + a( 6) + a( 7) + a( 8) = s1/3
a(17) + a(18) + a(19) + a(20) = s1/3
a(29) + a(30) + a(31) + a(32) = s1/3
a(41) + a(42) + a(43) + a(44) = s1/3

a(53) + a(54) + a(55) + a(56) = s1/3
a(65) + a(66) + a(67) + a(68) = s1/3
a(77) + a(78) + a(79) + a(80) = s1/3
a(89) + a(90) + a(91) + a(92) = s1/3

a(101)+ a(102)+ a(103)+ a(104)= s1/3
a(113)+ a(114)+ a(115)+ a(116)= s1/3
a(125)+ a(126)+ a(127)+ a(128)= s1/3
a(137)+ a(138)+ a(139)+ a(140)= s1/3

a( 9) + a(10) + a(11) + a(12) = s1/3
a(21) + a(22) + a(23) + a(24) = s1/3
a(33) + a(34) + a(35) + a(36) = s1/3
a(45) + a(46) + a(47) + a(48) = s1/3

a(57) + a(58) + a(59) + a(60) = s1/3
a(69) + a(70) + a(71) + a(72) = s1/3
a(81) + a(82) + a(83) + a(84) = s1/3
a(93) + a(94) + a(95) + a(96) = s1/3

a(105)+ a(106)+ a(107)+ a(108)= s1/3
a(117)+ a(118)+ a(119)+ a(120)= s1/3
a(129)+ a(130)+ a(131)+ a(132)= s1/3
a(141)+ a(142)+ a(143)+ a(144)= s1/3

a( 1) + a(13) + a(25) + a(37) = s1/3
a( 2) + a(14) + a(26) + a(38) = s1/3
a( 3) + a(15) + a(27) + a(39) = s1/3
a( 4) + a(16) + a(28) + a(40) = s1/3

a(49) + a(61) + a(73) + a(85) = s1/3
a(50) + a(62) + a(74) + a(86) = s1/3
a(51) + a(63) + a(75) + a(87) = s1/3
a(52) + a(64) + a(76) + a(88) = s1/3

a( 97)+ a(109)+ a(121)+ a(133)= s1/3
a( 98)+ a(110)+ a(122)+ a(134)= s1/3
a( 99)+ a(111)+ a(123)+ a(135)= s1/3
a(100)+ a(112)+ a(124)+ a(136)= s1/3

a( 5) + a(17) + a(29) + a(41) = s1/3
a( 6) + a(18) + a(30) + a(42) = s1/3
a( 7) + a(19) + a(31) + a(43) = s1/3
a( 8) + a(20) + a(32) + a(44) = s1/3

a(53) + a(65) + a(77) + a(89) = s1/3
a(54) + a(66) + a(78) + a(90) = s1/3
a(55) + a(67) + a(79) + a(91) = s1/3
a(56) + a(68) + a(80) + a(92) = s1/3

a(101)+ a(113)+ a(125)+ a(137)= s1/3
a(102)+ a(114)+ a(126)+ a(138)= s1/3
a(103)+ a(115)+ a(127)+ a(139)= s1/3
a(104)+ a(116)+ a(128)+ a(140)= s1/3

a( 9) + a(21) + a(33) + a(45) = s1/3
a(10) + a(22) + a(34) + a(46) = s1/3
a(11) + a(23) + a(35) + a(47) = s1/3
a(12) + a(24) + a(36) + a(48) = s1/3

a(57) + a(69) + a(81) + a(93) = s1/3
a(58) + a(70) + a(82) + a(94) = s1/3
a(59) + a(71) + a(83) + a(95) = s1/3
a(60) + a(72) + a(84) + a(96) = s1/3

a(105)+ a(117)+ a(129)+ a(141)= s1/3
a(106)+ a(118)+ a(130)+ a(142)= s1/3
a(107)+ a(119)+ a(131)+ a(143)= s1/3
a(108)+ a(120)+ a(132)+ a(144)= s1/3

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(141) =  4 * s1 / 12 - a(142) - a(143) - a(144)
a(139) =              - a(140) + a(143) + a(144)
a(137) =  4 * s1 / 12 - a(138) - a(143) - a(144)
a(136) =                a(138) - a(142) + a(144)
a(135) =              - a(138) + a(142) + a(143)
a(134) =                a(138) - a(140) + a(144)
a(133) =  4 * s1 / 12 - a(138) + a(140) - a(143) - 2 * a(144)
a(131) =  4 * s1 / 12 - a(132) - a(143) - a(144)
a(130) =                a(132) - a(142) + a(144)
a(129) =              - a(132) + a(142) + a(143)
a(128) =                a(132) - a(140) + a(144)
a(127) =  4 * s1 / 12 - a(132) + a(140) - a(143) - 2 * a(144)
a(126) =                a(132) - a(138) + a(144)
a(125) =              - a(132) + a(138) + a(143)
a(124) =                a(132) - a(138) + a(142)
a(122) =                a(132) - a(138) + a(140)
a(123) =  4 * s1 / 12 - a(132) + a(138) - a(142) - a(143) - a(144):
a(121) =              - a(132) + a(138) - a(140) + a(143) + a(144)
a(119) =              - a(120) + a(143) + a(144)
a(118) =                a(120) + a(142) - a(144)
a(117) =  4 * s1 / 12 - a(120) - a(142) - a(143)
a(116) =                a(120) + a(140) - a(144)
a(115) =              - a(120) - a(140) + a(143) + 2 * a(144)
a(114) =                a(120) + a(138) - a(144)
a(113) =  4 * s1 / 12 - a(120) - a(138) - a(143)
a(112) =                a(120) + a(138) - a(142)
a(111) =              - a(120) - a(138) + a(142) + a(143) + a(144)
a(110) =                a(120) + a(138) - a(140)
a(109) =  4 * s1 / 12 - a(120) - a(138) + a(140) - a(143) - a(144):
a(108) =  4 * s1 / 12 - a(120) - a(132) - a(144)
a(107) =                a(120) + a(132) - a(143)
a(106) =  4 * s1 / 12 - a(120) - a(132) - a(142)
a(105) = -4 * s1 / 12 + a(120) + a(132) + a(142) + a(143) + a(144)
a(104) =  4 * s1 / 12 - a(120) - a(132) - a(140)
a(103) =                a(120) + a(132) + a(140) - a(143) - a(144)
a(102) =  4 * s1 / 12 - a(120) - a(132) - a(138)
a(101) = -4 * s1 / 12 + a(120) + a(132) + a(138) + a(143) + a(144)
a(100) =  4 * s1 / 12 - a(120) - a(132) - a(138) + a(142) - a(144)
a( 99) =                a(120) + a(132) + a(138) - a(142) - a(143)
a( 98) =  4 * s1 / 12 - a(120) - a(132) - a(138) + a(140) - a(144)
a( 97) = -4 * s1 / 12 + a(120) + a(132) + a(138) - a(140) + a(143) + 2 * a(144)
a( 95) =              - a(96) + a(143) + a(144)
a( 94) =                a(96) + a(142) - a(144)
a( 93) =  4 * s1 / 12 - a(96) - a(142) - a(143)
a( 92) =                a(96) + a(140) - a(144)
a( 91) =              - a(96) - a(140) + a(143) + 2 * a(144)
a( 90) =                a(96) + a(138) - a(144)
a( 89) =  4 * s1 / 12 - a(96) - a(138) - a(143)
a( 88) =                a(96) + a(138) - a(142)
a( 87) =              - a(96) - a(138) + a(142) + a(143) + a(144)
a( 86) =                a(96) + a(138) - a(140)
a( 85) =  4 * s1 / 12 - a(96) - a(138) + a(140) - a(143) - a(144)
a( 84) =              - a(96) + a(132) + a(144)
a( 83) =  4 * s1 / 12 + a(96) - a(132) - a(143) - 2 * a(144)
a( 82) =              - a(96) + a(132) - a(142) + 2 * a(144)
a( 81) =                a(96) - a(132) + a(142) + a(143) - a(144)
a( 80) =              - a(96) + a(132) - a(140) + 2 * a(144)
a( 79) =  4 * s1 / 12 + a(96) - a(132) + a(140) - a(143) - 3 * a(144)
a( 78) =              - a(96) + a(132) - a(138) + 2 * a(144)
a( 77) =                a(96) - a(132) + a(138) + a(143) - a(144)
a( 76) =              - a(96) + a(132) - a(138) + a(142) + a(144)
a( 75) =  4 * s1 / 12 + a(96) - a(132) + a(138) - a(142) - a(143) - 2 * a(144)
a( 74) =              - a(96) + a(132) - a(138) + a(140) + a(144)
a( 73) =                a(96) - a(132) + a(138) - a(140) + a(143)

a(72) = s1/6 - a(138)
a(71) = s1/6 - a(137)
a(70) = s1/6 - a(136)
a(69) = s1/6 - a(135)
a(68) = s1/6 - a(134)
a(67) = s1/6 - a(133)
a(66) = s1/6 - a(144)
a(65) = s1/6 - a(143)
a(64) = s1/6 - a(142)
a(63) = s1/6 - a(141)
a(62) = s1/6 - a(140)
a(61) = s1/6 - a(139)
a(60) = s1/6 - a(126)
a(59) = s1/6 - a(125)
a(58) = s1/6 - a(124)
a(57) = s1/6 - a(123)
a(56) = s1/6 - a(122)
a(55) = s1/6 - a(121)

a(54) = s1/6 - a(132)
a(53) = s1/6 - a(131)
a(52) = s1/6 - a(130)
a(51) = s1/6 - a(129)
a(50) = s1/6 - a(128)
a(49) = s1/6 - a(127)
a(48) = s1/6 - a(114)
a(47) = s1/6 - a(113)
a(46) = s1/6 - a(112)
a(45) = s1/6 - a(111)
a(44) = s1/6 - a(110)
a(43) = s1/6 - a(109)
a(42) = s1/6 - a(120)
a(41) = s1/6 - a(119)
a(40) = s1/6 - a(118)
a(39) = s1/6 - a(117)
a(38) = s1/6 - a(116)
a(37) = s1/6 - a(115)

a(36) = s1/6 - a(102)
a(35) = s1/6 - a(101)
a(34) = s1/6 - a(100)
a(33) = s1/6 - a( 99)
a(32) = s1/6 - a( 98)
a(31) = s1/6 - a( 97)
a(30) = s1/6 - a(108)
a(29) = s1/6 - a(107)
a(28) = s1/6 - a(106)
a(27) = s1/6 - a(105)
a(26) = s1/6 - a(104)
a(25) = s1/6 - a(103)
a(24) = s1/6 - a( 90)
a(23) = s1/6 - a( 89)
a(22) = s1/6 - a( 88)
a(21) = s1/6 - a( 87)
a(20) = s1/6 - a( 86)
a(19) = s1/6 - a( 85)

a(18) = s1/6 - a(96)
a(17) = s1/6 - a(95)
a(16) = s1/6 - a(94)
a(15) = s1/6 - a(93)
a(14) = s1/6 - a(92)
a(13) = s1/6 - a(91)
a(12) = s1/6 - a(78)
a(11) = s1/6 - a(77)
a(10) = s1/6 - a(76)
a( 9) = s1/6 - a(75)
a( 8) = s1/6 - a(74)
a( 7) = s1/6 - a(73)
a( 6) = s1/6 - a(84)
a( 5) = s1/6 - a(83)
a( 4) = s1/6 - a(82)
a( 3) = s1/6 - a(81)
a( 2) = s1/6 - a(80)
a( 1) = s1/6 - a(79)

The solutions can be obtained by guessing:

   a(96), a(120), a(132), a(138), a(140) and a(142) ... 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 ... 119, 121 ... 131, 133 ... 137, 139, 141

a(i) ≠ a(j)           for i ≠ j

An optimized guessing routine (MgcSqr12e) produced, with a(144) = 1 and a(143) = 138, 3072 Most Perfect Magic Squares within 1010 seconds, of which the first 384 are shown in Attachment 12.2e.

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