Office Applications and Entertainment, Magic Squares

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

12.2.7 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 two parts summing to one half of the Magic Constant.

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

12.2.8 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 two parts summing to s1/2:

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

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

a( 1) + a(13) + a( 25) + a( 37) + a( 49) + a( 61) = s1/2
a(73) + a(85) + a( 97) + a(109) + a(121) + a(133) = s1/2
a( 2) + a(14) + a( 26) + a( 38) + a( 50) + a( 62) = s1/2
a(74) + a(86) + a( 98) + a(110) + a(122) + a(134) = s1/2
a( 3) + a(15) + a( 27) + a( 39) + a( 51) + a( 63) = s1/2
a(75) + a(87) + a( 99) + a(111) + a(123) + a(135) = s1/2
a( 4) + a(16) + a( 28) + a( 40) + a( 52) + a( 64) = s1/2
a(76) + a(88) + a(100) + a(112) + a(124) + a(136) = s1/2
a( 5) + a(17) + a( 29) + a( 41) + a( 53) + a( 65) = s1/2
a(77) + a(89) + a(101) + a(113) + a(125) + a(137) = s1/2
a( 6) + a(18) + a( 30) + a( 42) + a( 54) + a( 66) = s1/2
a(78) + a(90) + a(102) + a(114) + a(126) + a(138) = s1/2

a( 7) + a(19) + a( 31) + a( 43) + a( 55) + a( 67) = s1/2
a(79) + a(91) + a(103) + a(115) + a(127) + a(139) = s1/2
a( 8) + a(20) + a( 32) + a( 44) + a( 56) + a( 68) = s1/2
a(80) + a(92) + a(104) + a(116) + a(128) + a(140) = s1/2
a( 9) + a(21) + a( 33) + a( 45) + a( 57) + a( 69) = s1/2
a(81) + a(93) + a(105) + a(117) + a(129) + a(141) = s1/2
a(10) + a(22) + a( 34) + a( 46) + a( 58) + a( 70) = s1/2
a(82) + a(94) + a(106) + a(118) + a(130) + a(142) = s1/2
a(11) + a(23) + a( 35) + a( 47) + a( 59) + a( 71) = s1/2
a(83) + a(95) + a(107) + a(119) + a(131) + a(143) = s1/2
a(12) + a(24) + a( 36) + a( 48) + a( 60) + a( 72) = s1/2
a(84) + a(96) + a(108) + a(120) + a(132) + a(144) = s1/2

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(139) =  s1 / 2 - a(140) - a(141) - a(142) - a(143) - a(144)
a(137) =  s1 / 3 - a(138) - a(143) - a(144)
a(136) =           a(138) - a(142) + a(144)
a(135) =  s1 / 3 - a(138) - a(141) - a(144)
a(134) =           a(138) - a(140) + a(144)
a(133) = -s1 / 6 - a(138) + a(140) + a(141) + a(142) + a(143)
a(131) =  s1 / 3 - a(132) - a(143) - a(144)
a(130) =           a(132) - a(142) + a(144)
a(129) =  s1 / 3 - a(132) - a(141) - a(144)
a(128) =           a(132) - a(140) + a(144)
a(127) = -s1 / 6 - a(132) + a(140) + a(141) + a(142) + a(143)
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(123) = -a(132) + a(138) + a(141)
a(122) =  a(132) - a(138) + a(140)
a(121) =  s1 / 2 - a(132) + a(138) - a(140) - a(141) - a(142) - a(143) - a(144)
a(119) =         - a(120) + a(143) + a(144)
a(118) =           a(120) + a(142) - a(144)
a(117) =         - a(120) + a(141) + a(144)
a(116) =           a(120) + a(140) - a(144)
a(115) =  s1 / 2 - a(120) - a(140) - a(141) - a(142) - a(143)
a(114) =           a(120) + a(138) - a(144)
a(113) =  s1 / 3 - a(120) - a(138) - a(143)
a(112) =           a(120) + a(138) - a(142)
a(111) =  s1 / 3 - a(120) - a(138) - a(141)
a(110) =  a(120) + a(138) - a(140)
a(109) =          -s1 / 6 - a(120) - a(138) + a(140) + a(141) + a(142) + a(143) + a(144)
a(107) =  s1 / 3 - a(108) - a(143) - a(144)
a(106) =           a(108) - a(142) + a(144)
a(105) =  s1 / 3 - a(108) - a(141) - a(144)
a(104) =           a(108) - a(140) + a(144)
a(103) = -s1 / 6 - a(108) + a(140) + a(141) + a(142) + a(143)
a(102) =           a(108) - a(138) + a(144)
a(101) =         - a(108) + a(138) + a(143)
a(100) =           a(108) - a(138) + a(142)
a( 99) =         - a(108) + a(138) + a(141)
a( 98) =           a(108) - a(138) + a(140)
a( 97) =  s1 / 2 - a(108) + a(138) - a(140) - a(141) - a(142) - a(143) - a(144)
a( 95) =         - a(96) + a(143) + a(144)
a( 94) =           a(96) + a(142) - a(144)
a( 93) =         - a(96) + a(141) + a(144)
a( 92) =           a(96) + a(140) - a(144)
a( 91) =  s1 / 2 - a(96) - a(140) - a(141) - a(142) - a(143)
a( 90) =           a(96) + a(138) - a(144)
a( 89) =  s1 / 3 - a(96) - a(138) - a(143)
a( 88) =           a(96) + a(138) - a(142)
a( 87) =  s1 / 3 - a(96) - a(138) - a(141)
a( 86) =           a(96) + a(138) - a(140)
a( 85) = -s1 / 6 - a(96) - a(138) + a(140) + a(141) + a(142) + a(143) + a(144)
a( 84) =  s1 / 2 - a(96) - a(108) - a(120) - a(132) - a(144)
a( 83) = -s1 / 6 + a(96) + a(108) + a(120) + a(132) - a(143)
a( 82) =  s1 / 2 - a(96) - a(108) - a(120) - a(132) - a(142)
a( 81) = -s1 / 6 + a(96) + a(108) + a(120) + a(132) - a(141)
a( 80) =  s1 / 2 - a(96) - a(108) - a(120) - a(132) - a(140)
a( 79) = -2 * s1 / 3 + a(96) + a(108) + a(120) + a(132) + a(140) + a(141) + a(142) + a(143) + a(144)
a( 78) =  s1 / 2 - a(96) - a(108) - a(120) - a(132) - a(138)
a( 77) = -s1 / 2 + a(96) + a(108) + a(120) + a(132) + a(138) + a(143) + a(144)
a( 76) =  s1 / 2 - a(96) - a(108) - a(120) - a(132) - a(138) + a(142) - a(144)
a( 75) = -s1 / 2 + a(96) + a(108) + a(120) + a(132) + a(138) + a(141) + a(144)
a( 74) =  s1 / 2 - a(96) - a(108) - a(120) - a(132) - a(138) + a(140) - a(144)
a( 73) =           a(96) + a(108) + a(120) + a(132) + a(138) - a(140) - a(141) - a(142) - 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(108), a(120), a(132), a(138), a(140) ... 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 ... 137, 139

a(i) ≠ a(j)           for i ≠ j

An optimized guessing routine (MgcSqr12f) produced, with a(144) = 1, a(143) = 36 and a(142) = 73 2880 Most Perfect Magic Squares within 103 seconds, of which the first 384 are shown in Attachment 12.2f.

12.2.9 Spreadsheet Solutions

The linear equations, deducted in Section 12.2.2, Section 12.2.4, Section 12.2.6 and Section 12.2.8 above, have been applied in following Excel Spread Sheets:

  • CnstrSngl12b, Morris Squares, Franklin Properties

  • CnstrSngl12c, Most Perfect Magic Squares, Franklin Properties

  • CnstrSngl12e, Most Perfect Magic Squares, 1/3 Rows and 1/3 Columns sum to s1/3

  • CnstrSngl12f, Most Perfect Magic Squares, 1/2 Rows and 1/2 Columns sum to s1/2

The red figures have to be “guessed” to construct a (Most Perefect) Pan Magic Square of the 12th order (wrong solutions are obvious).

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