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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).
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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
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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
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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
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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
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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)
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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)
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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)
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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)
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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)
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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).
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