Office Applications and Entertainment, Magic Squares

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12.0Special Squares, Higher Order

12.1Pan Magic Squares (12 x 12) composed of Pan Magic Sub Squares

12.1.1 Barink Squares, Introduction

The 8th order Pan Magic Squares described in section 8.5.2, consisting of 4 perfectly 4 x 4 Pan Magic Squares and satisfying the Franklin property that every 2 2 sub square sums to half the Magic Constant, are an example of Barink Squares.

The 12th order Barink Square shown below is also a Pan Magic Square and consists of 9 perfectly 4 x 4 Pan Magic Squares and satisfies the Franklin property that every 2 2 sub square sums to one third of the Magic Constant.

In addition to this any 4 consecutive numbers starting on any odd place in a row or column sum also to one third of the Magic Constant.

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

12.1.2 Barink Squares, Analysis

The properties mentioned in section 12.1.1 above result in following set of linear equations:

The nine sub squares are Pan Magic:

a(37) =290-a(38)-a(39)-a(40)
a(27) =290-a(28)-a(39)-a(40)
a(26) =a(28)-a(38)+a(40)
a(25) =-a(28)+a(38)+a(39)
a(16) =145-a(38)
a(15) = -145+a(38)+a(39)+a(40)
a(14) =145-a(40)
a(13) =145-a(39)
a( 4) =145-a(28)+a(38)-a(40)
a( 3) =145+a(28)-a(38)-a(39)
a( 2) =145-a(28)
a( 1) = -145+a(28)+a(39)+a(40)

a(41) =290-a(42)-a(43)-a(44)
a(31) =290-a(32)-a(43)-a(44)
a(30) =a(32)-a(42)+a(44)
a(29) =-a(32)+a(42)+a(43)
a(20) =145-a(42)
a(19) = -145+a(42)+a(43)+a(44)
a(18) =145-a(44)
a(17) =145-a(43)
a( 8) =145-a(32)+a(42)-a(44)
a( 7) =145+a(32)-a(42)-a(43)
a( 6) =145-a(32)
a( 5) = -145+a(32)+a(43)+a(44)

a(45) =290-a(46)-a(47)-a(48)
a(35) =290-a(36)-a(47)-a(48)
a(34) =a(36)-a(46)+a(48)
a(33) =-a(36)+a(46)+a(47)
a(24) =145-a(46)
a(23) = -145+a(46)+a(47)+a(48)
a(22) =145-a(48)
a(21) =145-a(47)
a(12) =145-a(36)+a(46)-a(48)
a(11) =145+a(36)-a(46)-a(47)
a(10) =145-a(36)
a( 9) = -145+a(36)+a(47)+a(48)

a(85) =290-a(86)-a(87)-a(88)
a(75) =290-a(76)-a(87)-a(88)
a(74) =a(76)-a(86)+a(88)
a(73) =-a(76)+a(86)+a(87)
a(64) =145-a(86)
a(63) = -145+a(86)+a(87)+a(88)
a(62) =145-a(88)
a(61) =145-a(87)
a(52) =145-a(76)+a(86)-a(88)
a(51) =145+a(76)-a(86)-a(87)
a(50) =145-a(76)
a(49) = -145+a(76)+a(87)+a(88)

a(89) =290-a(90)-a(91)-a(92)
a(79) =290-a(80)-a(91)-a(92)
a(78) =a(80)-a(90)+a(92)
a(77) =-a(80)+a(90)+a(91)
a(68) =145-a(90)
a(67) = -145+a(90)+a(91)+a(92)
a(66) =145-a(92)
a(65) =145-a(91)
a(56) =145-a(80)+a(90)-a(92)
a(55) =145+a(80)-a(90)-a(91)
a(54) =145-a(80)
a(53) = -145+a(80)+a(91)+a(92)

a(93) =290-a(94)-a(95)-a(96)
a(83) =290-a(84)-a(95)-a(96)
a(82) =a(84)-a(94)+a(96)
a(81) =-a(84)+a(94)+a(95)
a(72) =145-a(94)
a(71) = -145+a(94)+a(95)+a(96)
a(70) =145-a(96)
a(69) =145-a(95)
a(60) =145-a(84)+a(94)-a(96)
a(59) =145+a(84)-a(94)-a(95)
a(58) =145-a(84)
a(57) = -145+a(84)+a(95)+a(96)

a(133) =290-a(134)-a(135)-a(136)
a(123) =290-a(124)-a(135)-a(136)
a(122) =a(124)-a(134)+a(136)
a(121) =-a(124)+a(134)+a(135)
a(112) =145-a(134)
a(111) = -145+a(134)+a(135)+a(136)
a(110) =145-a(136)
a(109) =145-a(135)
a(100) =145-a(124)+a(134)-a(136)
a( 99) =145+a(124)-a(134)-a(135)
a( 98) =145-a(124)
a( 97) = -145+a(124)+a(135)+a(136)

a(137) =290-a(138)-a(139)-a(140)
a(127) =290-a(128)-a(139)-a(140)
a(126) =a(128)-a(138)+a(140)
a(125) =-a(128)+a(138)+a(139)
a(116) =145-a(138)
a(115) = -145+a(138)+a(139)+a(140)
a(114) =145-a(140)
a(113) =145-a(139)
a(104) =145-a(128)+a(138)-a(140)
a(103) =145+a(128)-a(138)-a(139)
a(102) =145-a(128)
a(101) = -145+a(128)+a(139)+a(140)

a(141) =290-a(142)-a(143)-a(144)
a(131) =290-a(132)-a(143)-a(144)
a(130) =a(132)-a(142)+a(144)
a(129) =-a(132)+a(142)+a(143)
a(120) =145-a(142)
a(119) = -145+a(142)+a(143)+a(144)
a(118) =145-a(144)
a(117) =145-a(143)
a(108) =145-a(132)+a(142)-a(144)
a(107) =145+a(132)-a(142)-a(143)
a(106) =145-a(132)
a(105) = -145+a(132)+a(143)+a(144)

This includes that the numbers of the main diagonals sum to the Magic Constant. Following equations ensure that also the numbers of all the broken diagonals parallel to it sum to the Magic Constant as well:

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

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

Every 2 2 sub square sums to one third of the Magic Constant:

a(i) + a(i+1) + a(i+12) + a(i+13) = 290 with 1 =< i < 132 and i ≠ 12*n for n = 1, 2 ... 11

a(i) + a(i+1) + a(i+12) + a(i-11) = 290 with i = 12*n for n = 1, 2 ... 11

a(i) + a(i+1) + a(i+132) + a(i+133) = 290 with i = 1, 2 ... 11

a(1) + a(12)+ a(133)+ a(144)= 290

Any 4 consecutive numbers starting on any odd place in a row or column sum also to one third of the Magic Constant.

For i = 1, 5 and 9 (i = row or column number) this is already covered by the equations for the 4 x 4 Pan Magic squares.
The remaining equations are:

a( 3) + a (4) + a( 5) + a( 6) = 290
a(15) + a(16) + a(17) + a(18) = 290
a(27) + a(28) + a(29) + a(30) = 290
a(39) + a(40) + a(41) + a(42) = 290

a(51) + a(52) + a(53) + a(54) = 290
a(63) + a(64) + a(65) + a(66) = 290
a(75) + a(76) + a(77) + a(78) = 290
a(87) + a(88) + a(89) + a(90) = 290

a(99) + a(100)+ a(101)+ a(102)= 290
a(111)+ a(112)+ a(113)+ a(114)= 290
a(123)+ a(124)+ a(125)+ a(126)= 290
a(135)+ a(136)+ a(137)+ a(138)= 290

a( 7) + a( 8) + a( 9) + a(10) = 290
a(19) + a(20) + a(21) + a(22) = 290
a(31) + a(32) + a(33) + a(34) = 290
a(43) + a(44) + a(45) + a(46) = 290

a(55) + a(56) + a(57) + a(58) = 290
a(67) + a(68) + a(69) + a(70) = 290
a(79) + a(80) + a(81) + a(82) = 290
a(91) + a(92) + a(93) + a(94) = 290

a(103)+ a(104)+ a(105)+ a(106)= 290
a(115)+ a(116)+ a(117)+ a(118)= 290
a(127)+ a(128)+ a(129)+ a(130)= 290
a(139)+ a(140)+ a(141)+ a(142)= 290

a(11) + a(12) + a( 1) + a( 2) = 290
a(23) + a(24) + a(13) + a(14) = 290
a(35) + a(36) + a(25) + a(26) = 290
a(47) + a(48) + a(37) + a(38) = 290

a(59) + a(60) + a(49) + a(50) = 290
a(71) + a(72) + a(61) + a(62) = 290
a(83) + a(84) + a(73) + a(74) = 290
a(95) + a(96) + a(85) + a(86) = 290

a(107)+ a(108)+ a(97) + a(98) = 290
a(119)+ a(120)+ a(109)+ a(110)= 290
a(131)+ a(132)+ a(121)+ a(122)= 290
a(143)+ a(144)+ a(133)+ a(134)= 290

a(25) + a(37) + a(49) + a(61) = 290
a(26) + a(38) + a(50) + a(62) = 290
a(27) + a(39) + a(51) + a(63) = 290
a(28) + a(40) + a(52) + a(64) = 290

a(29) + a(41) + a(53) + a(65) = 290
a(30) + a(42) + a(54) + a(66) = 290
a(31) + a(43) + a(55) + a(67) = 290
a(32) + a(44) + a(56) + a(68) = 290

a(33) + a(45) + a(57) + a(69) = 290
a(34) + a(46) + a(58) + a(70) = 290
a(35) + a(47) + a(59) + a(71) = 290
a(36) + a(48) + a(60) + a(72) = 290

a(73) + a(85) + a(97) + a(109)= 290
a(74) + a(86) + a(98) + a(110)= 290
a(75) + a(87) + a(99) + a(111)= 290
a(76) + a(88) + a(100)+ a(112)= 290

a(77) + a(89) + a(101)+ a(113)= 290
a(78) + a(90) + a(102)+ a(114)= 290
a(79) + a(91) + a(103)+ a(115)= 290
a(80) + a(92) + a(104)+ a(116)= 290

a(81) + a(93) + a(105)+ a(117)= 290
a(82) + a(94) + a(106)+ a(118)= 290
a(83) + a(95) + a(107)+ a(119)= 290
a(84) + a(96) + a(108)+ a(120)= 290

a(121) + a(133)+ a( 1)+ a(13) = 290
a(122) + a(134)+ a( 2)+ a(14) = 290
a(123) + a(135)+ a( 3)+ a(15) = 290
a(124) + a(136)+ a( 4)+ a(16) = 290

a(125) + a(137)+ a( 5)+ a(17) = 290
a(126) + a(138)+ a( 6)+ a(18) = 290
a(127) + a(139)+ a( 7)+ a(19) = 290
a(128) + a(140)+ a( 8)+ a(20) = 290

a(129) + a(141)+ a( 9)+ a(21) = 290
a(130) + a(142)+ a(10)+ a(22) = 290
a(131) + a(143)+ a(11)+ a(23) = 290
a(132) + a(144)+ a(12)+ a(24) = 290

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


AB * a = s

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

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

The solutions can be obtained by guessing a(48), a(96), a(132), a(136), a(140), 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) =< 144for i = 1, 2, ... 47, 49 ... 95, 97 ... 131, 133 ... 135, 137 ... 139 and 141
a(i) ≠ a(j)for i ≠ j

With a(144) = 72 and a(143) = 74 , an optimized guessing routine (MgcSqr12a), produced 768 Pan Magic Squares for a(142) = 11, 17, 19, 21 and 23 (ref. Attachment 12.1 ).


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