Magic Squares (31a)

Office Applications and Entertainment, Magic Squares
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12.0 Special Squares, Higher Order

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

12.1.4 Hendricks Squares, Introduction

The 12^th order Pan Magic Square shown below is constructed by John Hendricks.

94 57 76 63
75 64 93 58
69 82 51 88
52 87 70 81

142 9 124 15
123 16 141 10
21 130 3 136
4 135 22 129

118 33 100 39
99 40 117 34
45 106 27 112
28 111 46 105

96 59 74 61
73 62 95 60
71 84 49 86
50 85 72 83

144 11 122 13
121 14 143 12
23 132 1 134
2 133 24 131

120 35 98 37
97 38 119 36
47 108 25 110
26 109 48 107

92 55 78 65
77 66 91 56
67 80 53 90
54 89 68 79

140 7 126 17
125 18 139 8
19 128 5 138
6 137 20 127

116 31 102 41
101 42 115 32
43 104 29 114
30 113 44 103

The square is composed out of nine 4^th order Pan Magic Sub Squares and contains - in addition to this - sixteen 4^th order Embeddeb Magic Squares.

12.1.5 Hendricks Squares, Analysis

The properties mentioned in section 12.1.4 above result in following set of linear equations:

Pan Magic Sub Squares (9 ea):

a(i1)	a(i2)	a(i3)	a(i4)
a(i5)	a(i6)	a(i7)	a(i8)
a(i9)	a(i10)	a(i11)	a(i12)
a(i13)	a(i14)	a(i15)	a(i16)

a(i13) = 290 - a(i14) - a(i15) - a(i16)
a(i11) = 290 - a(i12) - a(i15) - a(i16)
a(i10) = a(i12) - a(i14) + a(i16)
a(i9 ) = - a(i12) + a(i14) + a(i15)
a(i8 ) = 145 - a(i14)
a(i7 ) =-145 + a(i14) + a(i15) + a(i16)
a(i6 ) = 145 - a(i16)
a(i5 ) = 145 - a(i15)
a(i4 ) = 145 - a(i12) + a(i14) - a(i16)
a(i3 ) = 145 + a(i12) - a(i14) - a(i15)
a(i2 ) = 145 - a(i12)
a(i1 ) =-145 + a(i12) + a(i15) + a(i16)

with i1 ... i16 the indices in the main square for the applicable Pan Magic Sub Square.

Embedded Magic Square (16 ea):

a(i1)	a(i2)	a(i3)	a(i4)
a(i5)	a(i6)	a(i7)	a(i8)
a(i9)	a(i10)	a(i11)	a(i12)
a(i13)	a(i14)	a(i15)	a(i16)

a(i13) = 290 - a(i14) - a(i15) - a(i16)
a(i9 ) = 290 - a(i10) - a(i11) - a(i12)
a(i7 ) =       a(i8 ) - a(i10) + a(i12) - a(i13) + a(i16)
a(i6 ) = 290 - a(i8 ) - a(i11) - a(i12) + a(i13) - a(i16)
a(i5 ) =     - a(i8 ) + a(i10) + a(i11)
a(i4 ) = 290 - a(i7 ) - a(i10) - a(i13)
a(i3 ) =-290 - a(i8 ) + a(i9 ) + 2*a(i10) + 2*a(i13) + a(i14)
a(i2 ) =       a(i8 ) - a(i9 ) - 2*a(i10) + a(i15) + 2*a(i16)
a(i1 ) =       a(i8 ) + a(i12) - a(i13)

with i1 ... i16 the indices in the main square for the applicable Embedded Magic Square.

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

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

→ →
A_H * 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
a(82)  =       a(96) + a(132) - a(142)
a(81)  =       a(96) - a(132) + a(142) + a(143) - a(144)
a(80)  =     - a(96) + a(132) + a(140)
a(79)  = 290 - a(96) - a(132) - a(140) - a(143) + 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)
a(75)  = 290 - a(96) - a(132) - a(136) - a(143) + 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)
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)
a(60)  = 145 - a(96) - a(132) + a(142)
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)
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)
a(53)  =-145 + a(96) + a(132) + a(140) + a(143) - 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)
a(49)  =-145 + a(96) + a(132) + a(136) + a(143) - 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)
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)
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)
a(34)  =       a(48) + a(132) - a(142)
a(33)  =       a(48) - a(132) + a(142) + a(143) - a(144)
a(32)  =     - a(48) + a(132) + a(140)
a(31)  = 290 - a(48) - a(132) - a(140) - a(143) + 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)
a(27)  = 290 - a(48) - a(132) - a(136) - a(143) + 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)
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)
a(12)  = 145 - a(48) - a(132) + a(142)
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)
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)
a( 5)  =-145 + a(48) + a(132) + a(140) + a(143) - 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)
a( 1)  =-145 + a(48) + a(132) + a(136) + a(143) - 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 the bottom/left corner square constant, an optimized guessing routine (MgcSqr12d), produced 128 Composed Pan Magic Squares within 18,0 seconds, which are shown in Attachment 12.3.

12.1.6 Spreadsheet Solutions

The linear equations, deducted in Section 12.1.2 and Section 12.1.5 above, have been applied in following Excel Spread Sheets:

CnstrSngl12a, Barrink Squares Order 12
CnstrSngl12d, Hendricks Squares Order 12

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

Index

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