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12.6   Overlapping Sub Squares (13 x 13)

12.6.1 Introduction

The 13th order Composed Magic Square shown below was previously published by William Symes Andrews (ref. Magic Squares and Cubes (1909), Fig. 394).

 157 13 23 147 109 31 111 138 36 66 102 100 72 145 25 17 153 61 139 59 32 134 104 68 98 70 16 154 144 26 57 56 30 112 136 99 105 60 110 22 148 156 14 113 114 140 58 34 65 71 133 37 97 73 94 76 151 18 21 89 146 135 35 29 141 79 91 78 92 27 82 150 155 11 63 107 33 137 74 96 75 95 143 159 15 20 88 115 55 101 69 90 80 93 77 19 24 81 149 152 54 116 103 67 164 6 3 167 85 142 158 12 28 64 106 108 62 7 163 168 86 1 132 44 39 125 50 48 118 124 162 8 84 2 169 38 126 131 45 120 122 52 46 5 83 161 10 166 129 43 40 128 123 117 49 51 87 165 9 160 4 41 127 130 42 47 53 121 119

This 13th order Magic Square J contains following Sub Squares:

• A Magic Center Square C of order 5;
• A Magic Corner Square G of order 5, one element overlapping with C;
• An embedded Semi Magic Square M of order 3, eccentric in G (right top);
• Four Magic Border Squares of order 4: A and B (left), D and E (bottom);
• Two each other overlapping Magic Squares of order 7:
- I with C in the left bottom corner and
- L with C in the right top corner;
• Two each other overlapping Magic Squares of order 9:
- F composed out of B (left top), G (left bottom), D (right bottom) and C (right top)
- H with eccentric embedded I (left bottom)and C (left bottom).
• An eccentric Magic Square K of order 11.

It can be proven that:

• None of the Magic Squares described above can be Pan Magic, except the Center Square C;
• The Semi Magic Square M can't be Magic;
• The value of the common element of the overlapping squares C and G is 85.

12.6.2 Analysis (Sub Squares)

As a consequence of the properties described in Section 12.6.1 above, the 13th order Magic Square J is composed out of:

• a Magic Center Square C with a Magic Sum s1 = 425 and
• 72 pairs, each summing to 170, distributed over two layers surrounding square C.

Magic Center Square C

Assuming the outer and inner border variables constant, the number of valid solutions for C can be determined based on the required variable values:

{11, 12, 15, 18, 19, 20, 21, 24, 27, 28, 81, 82, 85, 88, 89, 142, 143, 146, 149, 150, 151, 152, 158, 159}

which can be written as {ci} with i = 1 ... 25.

The required Magic Squares of the 5th order can be generated with:

1. either a routine comparable with MgcSqr5a2 (ref. Section 3.2) with c(25) = 85
2. or a routine comparable with CnstrSqrs5a (ref. Section 3.6) also with c(25) = 85.

The obtained squares have to be mirrored around the vertical axis to meet the required c(21) = 85.

The number of suitable squares is limited by the values of s21 = c(3) + c(9) + c(15), s22 = c(11) + c(17) + c(23) and c(5), as these values determine whether squares L, I, F and H are magic as required.

1. Routine MgcSqr5a4 combines the functionality of routine MgcSqr5a2, the required transformation to {ci} and reflection around the vertical axis.

For c(22) = 11 the routine MgcSqr5a4 produced 2185 Magic Squares with distinct integers {ci} and Magic Sum 425.

This number can be quadrupled by means of transposition around the diagonal c(21) - c(5) and complementation of both the generated and the transposed squares.

Note: The complement of a Magic Square {bi} = {sp - ai} for i = 1 ... 25 with sp = amin + amax.

From the collection of 4 * 2185 = 8740 Magic Squares, 36 Magic Center Squares could be filtered based on the requirements s21 = 264, s22 = 325 and c(5) = 146 which are shown in Attachment 12.6.1.

2. Routine CnstrSqrs5c combines the functionality of routine CnstrSqrs5a, the required transformation to {ci} and reflection around the vertical axis.

Routine CnstrSqrs5c produced 6912 Magic Squares with distinct integers {ci} and Magic Sum 425, of which 24 Magic Center Squares could be filtered based on the requirements mentioned above which are shown in Attachment 12.6.2.

Magic Corner Square G

Assuming the variables outside the Corner Square constant, the number of valid solutions for G can be determined based on the required variable values:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 83, 84, 85, 86, 87, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169}

which can be written as {gi} with i = 1 ... 25.

The Magic Corner Square G can be represented as:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25)

In addition to the defining equations of a 5th order Magic Square:

a( 1) + a( 2) + a( 3) + a( 4) + a( 5) = s1
a( 6) + a( 7) + a( 8) + a( 9) + a(10) = s1
a(11) + a(12) + a(13) + a(14) + a(15) = s1
a(16) + a(17) + a(18) + a(19) + a(20) = s1
a(21) + a(22) + a(23) + a(24) + a(25) = s1

a( 1) + a( 6) + a(11) + a(16) + a(21) = s1
a( 2) + a( 7) + a(12) + a(17) + a(22) = s1
a( 3) + a( 8) + a(13) + a(18) + a(23) = s1
a( 4) + a( 9) + a(14) + a(19) + a(24) = s1
a( 5) + a(10) + a(15) + a(20) + a(25) = s1

a( 1) + a( 7) + a(13) + a(19) + a(25) = s1
a( 5) + a( 9) + a(13) + a(17) + a(21) = s1

 a( 1) + a( 2) = 170 a( 3) + a( 4) = 170 a( 6) + a( 7) = 170 a(11) + a(12) = 170 a(18) + a(23) = 170 a(19) + a(24) = 170 a(20) + a(25) = 170 a(10) + a(15) = 170 a( 8) + a(14) = 170 a( 9) + a(13) = 170 a(16) + a(22) = 170 a(17) + a(21) = 170

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

AG * a = s

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

a(21) = 425 - a(22) - a(23) - a(24) - a(25)
a(20) = 170 - a(25)
a(19) = 170 - a(24)
a(18) = 170 - a(23)
a(17) = 170 - a(21)
a(16) = 170 - a(22)
a(13) = 255 - a(14) - a(15)
a(11) = 170 - a(12)
a(10) = 170 - a(15)
a( 9) = 170 - a(13)
a( 8) = 170 - a(14)
a( 7) =(340 - a(12) - a(13) + a(21) - a(22) + a(24) - a(25)) / 2
a( 6) = 170 - a( 7)
a( 5) = 85
a( 4) = 340 - 2 * a(14) - a(15)
a( 3) = 170 - a( 4)
a( 2) = 340 - a( 6) - a(14) - a(15) - a(24) + a(25)
a( 1) = 170 - a( 2)

It should be noted that the common element of the overlapping Corner Square G and Center Square C is fully determined by the defining properties of the Corner Square.

With an optimized guessing routine (MgcSqr5g), based on the equations above, 1216 Magic Corner Squares could be generated within 150 seconds, which are shown in Attachment 12.6.3.

Semi Magic Square M

The 3th order Semi Magic Square M, embedded in the Magic Corner Square G as discussed above, is a consequence of the defining properties of the Magic Corner Square G.

The 1216 Corner Squares shown in Attachment 12.6.3 contain 40 different embedded Semi Magic Squares.

Magic Border Squares A, B, D, E

Assuming the variables outside the applicable Border Square constant, the number of valid solutions for A, B, D and E can be determined based on the applicable variable values:

{ai} = { 13, 14, 16, 17, 22, 23, 25, 26, 144, 145, 147, 148, 153, 154, 156, 157 }
{bi} = { 73, 74, 75, 76, 77, 78, 79, 80, 90, 91, 92, 93, 94, 95, 96, 97 }
{di} = { 38, 39, 40, 41, 42, 43, 44, 45, 125, 126, 127, 128, 129, 130, 131, 132 }
{ei} = { 46, 47, 48, 49, 50, 51, 52, 53, 117, 118, 119, 120, 121, 122, 123, 124 }

The Magic Border Square A can be represented as:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16)

In addition to the defining equations of a 4th order Magic Square:

a( 1) + a( 2) + a( 3) + a( 4) = s1
a( 5) + a( 6) + a( 7) + a( 8) = s1
a( 9) + a(10) + a(11) + a(12) = s1
a(13) + a(14) + a(15) + a(16) = s1

a( 1) + a( 5) + a( 9) + a(13) = s1
a( 2) + a( 6) + a(10) + a(14) = s1
a( 3) + a( 7) + a(11) + a(15) = s1
a( 4) + a( 8) + a(12) + a(16) = s1

a( 1) + a( 6) + a(11) + a(16) = s1
a( 4) + a( 7) + a(10) + a(13) = s1

 a( 1) + a( 2) = 170 a( 5) + a( 6) = 170 a( 9) + a(10) = 170 a(13) + a(14) = 170 a( 3) + a( 4) = 170 a( 7) + a( 8) = 170 a(11) + a(12) = 170 a(15) + a(16) = 170

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

AA * a = s

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

a(15) = 170 - a(16)
a(13) = 170 - a(14)
a(11) = 170 - a(12)
a( 9) = 170 - a(10)
a( 8) =(340 + a(10) - a(12) - a(14) - a(16))/2
a( 7) = 170 - a( 8)
a( 6) = 170 - a( 7) - a(10) + a(12)
a( 5) = 170 - a( 6)
a( 4) = 170 - a( 5) - a(12) + a(14)
a( 3) = 170 - a( 4)
a( 2) = 340 - a( 4) - a(10) - a(12)
a( 1) = 170 - a( 2)

In addition to this the limiting condition a(3) + a(8) = 176 should be taken into consideration as this value determines whether square K is magic as required.

With an optimized guessing routine (MgcSqr4f), based on the equations above, suitable Magic Border Squares A could be generated.

Comparable results, taken the applicable conditions into account, could be obtained for Border Squares B, D and E.

The applicable conditions and results are summarized below:

 Border Square Variable Values Limiting Condition (Square) Results A {ai} a(3) + a(8) = 176 (K) 16 Section A B {bi} b(3) + b(8) = 186 (L) 16 Section B D {di} d(3) + d(8) =  84 (L) 16 Section D E {ei} e(3) + e(8) = 164 (K) 16 Section E

It should be noted that for D and E the equations deducted above require a transposition around the a(13) - a(4) axis.

Magic Square F

The 9th order Magic Square F is composed out of B (left top), G (left bottom), D (right bottom) and C (right top).

Under the restrictions made in previous sections with regard to the variable values per square and the application of only 60 Center Squares it is already possible to construct 16 * 1216 * 16 * 60 = 18677760 solutions for F.

Magic Square L

The 7th order Magic Square L is determined by B (left top), M (left bottom), D (right bottom) and C (right top).

Under the same restrictions mentioned above it is possible to construct 16 * 40 * 16 * 60 = 614400 solutions for L, which are all included in the number mentioned above for F.

Magic Square I

The 7th order Magic Square I is an Eccentric Square as discussed in Section 7.5.2. which can be represented as:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40) a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48) a(49)

Assuming the variables within Magic Square C and outside Magic Square I constant, the number of valid solutions can be determined based on the applicable border variable values:

{im} = {30,34,35,54,55,56,57,58,63,64,65,71,99,105,106,107,112,113,114,115,116,135,136,140}

The supplementary rows and columns (hatched) can be described by following linear equations:

a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) + a( 7) = 595
a( 8) + a( 9) + a(10) + a(11) + a(12) + a(13) + a(14) = 595
a( 6) + a(13) + a(20) + a(27) + a(34) + a(41) + a(48) = 585
a( 7) + a(14) + a(21) + a(28) + a(35) + a(42) + a(49) = 595
a( 1) + a( 9) + a(17) + a(25) + a(33) + a(41) + a(49) = 595
a( 1) + a( 8) = 170
a( 2) + a( 9) = 170
a( 3) + a(10) = 170
a( 4) + a(11) = 170
a( 5) + a(12) = 170
a( 6) + a(14) = 170
a( 7) + a(13) = 170
a(20) + a(21) = 170
a(27) + a(28) = 170
a(34) + a(35) = 170
a(41) + a(42) = 170
a(48) + a(49) = 170

Which can be reduced to:

a(48) = 170 -  a(49)
a(41) = 170 -  a(42)
a(34) = 170 -  a(35)
a(27) = 170 -  a(28)
a(20) = 170 -  a(21)
a(13) =-425 +  a(14) + a(21) + a(28) + a(35) + a(42) + a(49)
a( 9) = 510 - (a(10) + a(11) + a(12) + a(13) + a(14) + a(17) + a(25) + a(33) + a(41) + a(49))/2
a( 8) = 595 -  a(9) - a(10) - a(11) - a(12) - a(13) - a(14)
a( 7) = 170 -  a(13)
a( 6) = 170 -  a(14)
a( 5) = 170 -  a(12)
a( 4) = 170 -  a(11)
a( 3) = 170 -  a(10)
a( 2) = 170 -  a( 9)
a( 1) = 170 -  a( 8)

In addition to this the limiting condition s3 = a(3) + a(11) + a(19) + a(27) + a(35) = 352 should be taken into consideration, as this value determines whether square H is magic as required.

With an optimized guessing routine (MgcSqr7h), based on the equations above, 328 suitable Eccentric Magic Squares I could be generated (ref. Attachment 12.6.5).

Magic Square H

The 9th order Magic Square H is an Eccentric Square as discussed in Section 9.5.2. which can be represented as:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40) a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48) a(49) a(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

Assuming the variables within Magic Square I and outside Magic Square H constant, the number of valid solutions can be determined based on the applicable border variable values {hm}:

{29,31,32,33,36,37,59,60,61,62,66,67,68,69,70,72,98,100,101,102,103,104,108,109,110,111,133,134,137,138,139,141}

The supplementary rows and columns can be described by following linear equations:

a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) + a( 7) + a( 8) + a( 9) = 765
a(10) + a(11) + a(12) + a(13) + a(14) + a(15) + a(16) + a(17) + a(18) = 765
a( 9) + a(18) + a(27) + a(36) + a(45) + a(54) + a(63) + a(72) + a(81) = 765
a( 8) + a(17) + a(26) + a(35) + a(44) + a(53) + a(62) + a(71) + a(80) = 765
a( 1) + a(11) + a(21) + a(31) + a(41) + a(51) + a(61) + a(71) + a(81) = 765
a( 1) + a(10) = 170
a( 2) + a(11) = 170
a( 3) + a(12) = 170
a( 4) + a(13) = 170
a( 5) + a(14) = 170
a( 6) + a(15) = 170
a( 7) + a(16) = 170
a( 8) + a(18) = 170
a( 9) + a(17) = 170
a(26) + a(27) = 170
a(35) + a(36) = 170
a(44) + a(45) = 170
a(53) + a(54) = 170
a(62) + a(63) = 170
a(71) + a(72) = 170
a(80) + a(81) = 170

Which can be reduced to:

a(80) = 170 -  a(81)
a(71) = 170 -  a(72)
a(62) = 170 -  a(63)
a(53) = 170 -  a(54)
a(44) = 170 -  a(45)
a(35) = 170 -  a(36)
a(26) = 170 -  a(27)
a(17) = 595 +  a(18) - a(26) - a(35) - a(44) - a(53) - a(62) - a(71) - a(80)
a(11) = 680 - (a(12) + a(13) + a(14) + a(15) + a(16) + a(17) + a(18) +
+ a(21) + a(31) + a(41) + a(51) + a(61) + a(71) + a(81))/2
a(10) = 765 -  a(11) - a(12) - a(13) - a(14) - a(15) - a(16) - a(17) - a(18)
a( 9) = 170 -  a(17)
a( 8) = 170 -  a(18)
a( 7) = 170 -  a(16)
a( 6) = 170 -  a(15)
a( 5) = 170 -  a(14)
a( 4) = 170 -  a(13)
a( 3) = 170 -  a(12)
a( 2) = 170 -  a(11)
a( 1) = 170 -  a(10)

The number of possible solutions for H is determined by the sum of the values of the key variables s3 = a(21) + a(31) + a(41) + a(51) + a(61) = 352 and is quite high.

An optimized guessing routine (MgcSqr9f) produced, based on the equations above, 240 suitable Eccentric Magic Squares H while varying the variables a(12) through a(16), which are shown in Attachment 12.6.6.

While varying 4 more variables - a(18), a(27), a(36) and a(45) - already 168480 Eccentric Magic Squares could be generated.

Magic Square K

The 11th order Magic Square K is determined by A (left top), B (left center), M (left bottom), D (bottom center), E (bottom right), C (eccentric), border I (top right) and border H (top right).

Under the restrictions made in previous sections with regard to the variable values per square and the application of only 60 Center Squares it is possible to construct 164 * 40 * 60 * 328 * nh solutions for K, which are all included in the number mentioned in Section 12.6.4 below for J.

12.6.3 Analysis (Complete Square)

For the sake of completeness, the full set of equations - describing the whole 13th order Magic Square J as defined in Section 12.6.1 above - has been deducted.

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40) a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48) a(49) a(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81) a(82) a(83) a(84) a(85) a(86) a(87) a(88) a(89) a(90) a(91) a(92) a(93) a(94) a(95) a(96) a(97) a(98) a(99) a(100) a(101) a(102) a(103) a(104) a(105) a(106) a(107) a(108) a(109) a(110) a(111) a(112) a(113) a(114) a(115) a(116) a(117) a(117) a(119) a(120) a(121) a(122) a(123) a(124) a(125) a(126) a(127) a(128) a(129) a(130) a(131) a(132) a(133) a(134) a(135) a(136) a(137) a(138) a(139) a(140) a(141) a(142) a(143) a(144) a(145) a(146) a(147) a(148) a(149) a(150) a(151) a(152) a(153) a(154) a(155) a(156) a(157) a(158) a(159) a(160) a(161) a(162) a(163) a(164) a(165) a(166) a(167) a(168) a(169)

Based on the defining equations of: the Magic Square J (13 x 13); the Magic Square K (11 x 11); the Magic Squares F and H (9 x 9); the Magic Squares I and L (7 x 7); the Magic Squares C and G (5 x 5); the Magic Squares A, B, D and E (4 x 4) and the Semi Magic Square M (3 x 3)

a matrix equation can be composed:

AJ * a = s

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

```a(166) =  340 - a(167) - a(168) - a(169)
a(162) =  340 - a(163) - a(164) - a(165)
a(157) =  425 - a(158) - a(159) - a(160) - a(161)
a(156) =  170 - a(169)
a(155) =  170 - a(168);  a(154) =   170  - a(167); a(153) = 170 - a(166); a(152) = 170 - a(165)
a(151) =  170 - a(164);  a(150) =   170  - a(163); a(149) = 170 - a(162); a(148) = 170 - a(161)
a(147) =  170 - a(160);  a(146) =   170  - a(159); a(145) = 170 - a(157); a(144) = 170 - a(158)
a(142) =        a(143) - a(166) + a(167)
a(141) =  340 - a(143) - a(167) - a(169)
a(140) =  340 - a(143) - a(167) - a(168)
a(138) =        a(139) - a(162) + a(163)
a(137) =      - a(138) + a(163) + a(164)
a(136) =  340 - a(137) - a(138) - a(139)
a(133) =  255 - a(134) - a(135)
a(131) =  170 - a(132);  a(130) =   170  - a(143); a(129) = 170 - a(142); a(128) = 170 - a(141)
a(127) =  170 - a(140);  a(126) =   170  - a(139); a(125) = 170 - a(138); a(124) = 170 - a(137)
a(123) =  170 - a(136);  a(122) =   170  - a(135); a(121) = 170 - a(133)
a(120) =  170 - a(134)
a(119) = (510 - a(132) + a(134) + a(135) - 2 * a(158) - a(159) - 2 * a(161)) / 2
a(118) =  170 - a(119)
a(116) =  170 - a(117)
a(114) =  170 - a(115)
a(110) =  340 - a(111) - a(112) - a(113)
a(108) =  340 - 2 * a(134) - a(135)
a(107) =  170 - a(108)
a(106) =  255 - a(119) - a(132) + a(136) - a(137) + a(157) - a(158) + a(164) - a(165)
a(105) =  170 - a(106)
a(103) =  170 - a(104)
a(101) =  170 - a(102)
a( 96) =  425 - a(97) - a(98) - a(99) - a(100)
a( 94) =  170 - a(95)
a( 92) =  170 - a(93)
a( 90) =  170 - a(91)
a( 88) =  170 - a(89)
a( 83) =  425 - a(84) - a(85) - a(86) - a(87)
a( 81) =  170 - a(82)
a( 80) =  255 - a(83) + a(93) - a(97) - a(111) + a(138) - a(139)
a( 79) =  170 - a(80)
a( 77) =  170 - a(78)
a( 75) =  170 - a(76)
a( 73) =  -85 + a(74) - a(85) + a(87) - a(97) + a(100) + a(113)
a( 71) =(1020 - a(72) - 2 * a(74) + a(83) - a(87) - a(98) - 2 * a(99) - 2 * a(100) - 2 * a(113))/2
a( 70) =  425 - a(71)-a(72)-a(73)-a(74)
a( 69) = (595 - a(82) - a(83) - a(95) - a(97) - a(111) + a(138) - a(139))/2
a( 68) =  170 - a(69)
a( 67) =  340 - a(68) - a(80) - a(81)
a( 66) =  170 - a(67)
a( 64) =  170 - a(65)
a( 62) =  170 - a(63)
a( 61) =  425 - a(73) - a(85) - a(97) - a(109)
a( 60) =  425 - a(73) - a(86) - a(99) - a(112)
a( 59) =  425 - a(72) - a(85) - a(98) - a(111)
a( 58) =  425 - a(71) - a(84) - a(97) - a(110)
a( 57) =  425 - a(58) - a(59) - a(60) - a(61)
a( 56) =  340 - a(68) - a(80) - a(92)
a( 55) =  170 - a(56)
a( 54) =  340 - a(67) - a(80) - a(93)
a( 53) =  170 - a(54)
a( 51) =  170 - a(52)
a( 49) = -425 + a(50) + a(63) + a(76) + a(89) + a(102) + a(115)
a( 45) = (935 - a(46) - a(47) - a(48) - 2 * a(50) - a(63) + a(72) - a(74) - a(76) + 2 * a(85) - 2 * a(87) +
- a(89) + a(97) + a(98) - a(100) + a(111) - a(113) - 2 * a(115))/ 2
a( 44) =  595 - a(45) - a(46) - a(47) - a(48)  - a(49) - a(50)
a( 42) =  170 - a(43)
a( 40) =  170 - a(41);  a( 38) =   170 - a(39); a(37) =  170 - a(49); a(36) =  170 - a(50)
a( 35) =  170 - a(48);  a( 34) =   170 - a(47); a(33) =  170 - a(46); a(32) =  170 - a(45)
a( 31) =  170 - a(44);  a( 29) =   170 - a(30);
a( 28) =  170 + a(41) - a(129) - a(143)
a( 27) =  170 - a(28)
a( 25) = -595 + a(26) + a(39) + a(52) + a(65) + a(78) + a(91) + a(104) + a(117)
a( 19) = (425 - a(20) - a(21) - a(22) - a(23) - a(24) - a(25) - a(26) + a(46) - a(47) + a(74) + a(76) + a(87) +
- a(89) + a(100) + a(104) + a(113) - a(117))/2
a( 18) =  765 - a(19) - a(20) - a(21) - a(22) - a(23) - a(24) - a(25) - a(26)
a( 17) = (510 - a(27) - a(30) - a(41) - a(43))/2
a( 16) =  170 - a(17)
a( 15) =  340 - a(16) - a(28) - a(29)
a( 14) =  170 - a(15)
a( 13) =  170 - a(25);  a(12) =  170 - a(26); a(11) =  170 - a(24); a(10) =  170 - a(23)
a(  9) =  170 - a(22);  a( 8) =  170 - a(21); a( 7) =  170 - a(20); a( 6) =  170 - a(19)
a(  5) =  170 - a(18)
a(  4) =        a(15) - a(30) + a(41)
a(  3) =  170 - a( 4)
a(  2) =        a( 4) - a(41) + a(43)
a(  1) =  170 - a( 2)
```

The equations deducted above have been applied in an Excel Spreadsheet (CnstrSngl13) which illustrates the mutual dependency of the individual sub squares discussed in previous sections.

12.6.4 Summary (1)

Magic Squares of order 13 as defined in Section 12.6.1 above can be constructed based on independent from each other generated sub squares and border sections for corresponding predefined variable values as summarised below:

 Square Border Section Variable Values Number Results A - {am} m = 1 ... 16 na = 16 Attachment 12.6.4 Sect A B - {bm} m = 1 ... 16 nb = 16 Attachment 12.6.4 Sect B C - {cm} m = 1 ... 25 nc = 60 D - {dm} m = 1 ... 16 nd = 16 Attachment 12.6.4 Sect D E - {em} m = 1 ... 16 ne = 16 Attachment 12.6.4 Sect E G - {gm} m = 1 ... 25 ng = 1216 - I {im} m = 1 ... 24 ni = 328 - H {hm} m = 1 ... 32 nh > 168480

With na = nb = nd = ne = n the total number of squares nj can be written as:

nj = n4 * nc * ng * ni * nh = 164 * 60 * 1216 * 328 * nh

It should be noted that nc is limited to those Center Squares (60 ea) which could be found within a reasonable time, which is only a fraction of all possible Center Squares.

For other defined variable ranges {am} ... {hm} comparable amounts of 13th order Magic Squares can be found.

Attachment 12.6.7 shows for each of the 60 Magic Center Squares C one example of the 13th order Magic Square J, completed with randomly selected sub squares (A, B, D, E, G) and border sections (I, H).

12.6.5 Pan Magic Center Squares

If to the defining equations of the 13th order Magic Square J, as discussed in Section 12.6.3 above, the equations of Pan Diagonals for Center Square C are added, the resulting equations suggest that valid solutions are possible.

It requires only minor additional effort to construct possible solutions based on the methods discussed and applied in previous sections.

With routine MgcSqr5a5 - comparable with MgcSqr5a3 (ref. Section 7.7.2) - applied on the applicable variable values {ci}, 1152 Pan Magic Center Squares with corner element 85 (bottom left) can be generated (ref. Attachment 12.6.8).

As mentioned in Section 12.6.2 above, the number of suitable Center Squares C is limited by the values of s21 = c(3) + c(9) + c(15), s22 = c(11) + c(17) + c(23) and c(5), as these values determine whether squares L, I, F and H are magic as required.

The 1152 Pan Magic Squares shown in Attachment 12.6.8 contain 144 combinations {s21 , s22}, which are summarised in following table:

 s22 s21 181 117 123 124 239 245 246 248 254 255 182 117 124 125 239 246 247 248 255 256 188 123 124 131 245 246 253 254 255 262 189 124 125 131 246 247 253 255 256 262 190 117 123 124 248 254 255 257 263 264 191 117 124 125 248 255 256 257 264 265 197 123 124 131 254 255 262 263 264 271 198 124 125 131 255 256 262 264 265 271 312 239 245 246 248 254 255 379 385 386 313 239 246 247 248 255 256 379 386 387 319 245 246 253 254 255 262 385 386 393 320 246 247 253 255 256 262 386 387 393 321 248 254 255 257 263 264 379 385 386 322 248 255 256 257 264 265 379 386 387 328 254 255 262 263 264 271 385 386 393 329 255 256 262 264 265 271 386 387 393

The 16 possible values of s22 require the application of other Border Squares B and D as applied in Section 12.6.2 above.

Without the limiting conditions sb = b(3) + b(8) = 186 and sd = d(3) + d(8) = 84 the routine MgcSqr4f will produce 384 Border Squares B and 384 Border Squares D, which are shown in Attachment 12.6.9.

The limiting condition to ensure that the 7th order Magic Square L is magic is now:

s22 = 595 - (sb + sd)

Attachment 12.6.10 shows the values s22 as a function of sb and sd. The possible values of s22, as summarised in the table above, are highlighted (hatched).

The number of combinations {sb , sd} which result in correct values of s22 are shown below: Any of the required values for sb and sd corresponds with 16 related Border Squares B and D.

Any of the 16 values of s22 corresponds with 72 Pan Magic Squares C of the 5th order.

To ensure that I, F and H are magic as required a suitable border for I should be selected for each occurring combination {s21 , c(5)}, which can be achieved with routine MgcSqr7h.

An example of a 13th order Magic Square J with overlapping Sub Squares and a Pan Magic Center Square C is shown below:

 26 144 22 148 139 61 134 32 59 102 66 100 72 14 156 16 154 31 109 36 138 111 68 104 98 70 147 23 145 25 71 56 55 113 106 58 136 60 110 153 17 157 13 99 114 115 57 64 34 112 133 37 91 79 92 78 158 12 24 89 142 105 65 29 141 76 94 73 97 28 81 149 152 15 35 135 33 137 77 93 80 90 143 155 19 20 88 107 63 101 69 96 74 95 75 11 27 82 146 159 116 54 103 67 1 169 8 162 85 150 151 18 21 140 30 108 62 3 167 161 84 10 38 127 45 130 122 120 46 52 166 4 86 9 160 132 43 125 40 48 50 124 118 168 83 163 6 5 126 41 131 42 123 117 49 51 87 2 7 164 165 44 129 39 128 47 53 121 119

The order of magnitude of nj is comparable with the order of magnitude deducted in Section 12.6.4 above.