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12.0   Special Squares, Higher Order

12.1   Pan 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) =< 144       for 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 ).