12.6 Overlapping Sub Squares (13 x 13)
12.6.1 Introduction
The 13^{th} 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 13^{th} 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 13^{th} 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 {c_{i}} with i = 1 ... 25.
The required Magic Squares of the 5^{th} order can be generated with:

either a routine comparable with MgcSqr5a2 (ref. Section 3.2) with c(25) = 85

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 s_{21} = c(3) + c(9) + c(15), s_{22} = c(11) + c(17) + c(23) and c(5), as these values determine whether squares
L,
I,
F and
H
are magic as required.

Routine MgcSqr5a4 combines the functionality of routine MgcSqr5a2, the required transformation to {c_{i}} and reflection around the vertical axis.
For c(22) = 11 the routine MgcSqr5a4 produced 2185 Magic Squares
with distinct integers {c_{i}} 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 {b_{i}} = {s_{p}  a_{i}} for i = 1 ... 25 with s_{p} = a_{min} + a_{max}.
From the collection of 4 * 2185 = 8740 Magic Squares, 36 Magic Center Squares could be filtered based on the requirements s_{21} = 264, s_{22} = 325 and c(5) = 146 which are shown in Attachment 12.6.1.

Routine CnstrSqrs5c combines the functionality of routine CnstrSqrs5a, the required transformation to {c_{i}} and reflection around the vertical axis.
Routine CnstrSqrs5c produced 6912 Magic Squares with distinct integers {c_{i}} 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 {g_{i}} 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 5^{th} 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
following equations should be added:
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:
→ →
A_{G} * 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 3^{th} 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:
{a_{i}} =
{ 13, 14, 16, 17, 22, 23, 25, 26, 144, 145, 147, 148, 153, 154, 156, 157 }
{b_{i}} =
{ 73, 74, 75, 76, 77, 78, 79, 80, 90, 91, 92, 93, 94, 95, 96, 97 }
{d_{i}} =
{ 38, 39, 40, 41, 42, 43, 44, 45, 125, 126, 127, 128, 129, 130, 131, 132 }
{e_{i}} =
{ 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 4^{th} 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
following equations should be added:
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:
→ →
A_{A} * 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

Attachment 12.6.4

A

{a_{i}}

a(3) + a(8) = 176 (K)

16

Section A

B

{b_{i}}

b(3) + b(8) = 186 (L)

16

Section B

D

{d_{i}}

d(3) + d(8) = 84 (L)

16

Section D

E

{e_{i}}

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 9^{th} 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 7^{th} 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 7^{th} 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:
{i_{m}} =
{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 9^{th} 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 {h_{m}}:
{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 11^{th} 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 16^{4} * 40 * 60 * 328 * n_{h} 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 13^{th} 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:
→ →
A_{J} * 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:
With n_{a} = n_{b} = n_{d} = n_{e} = n the total number of squares n_{j} can be written as:
n_{j} = n^{4} * n_{c} * n_{g} * n_{i} * n_{h} = 16^{4} * 60 * 1216 * 328 * n_{h}
It should be noted that n_{c} 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 {a_{m}} ... {h_{m}} comparable amounts of 13^{th} order Magic Squares can be found.
Attachment 12.6.7 shows for each of the 60 Magic Center Squares C one example of the 13^{th} 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 13^{th} 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 {c_{i}}, 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 s_{21} = c(3) + c(9) + c(15), s_{22} = 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 {s_{21} , s_{22}}, which are summarised in following table:
s_{22}

s_{21}

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 s_{22} require the application of other Border Squares
B and
D as applied in Section 12.6.2 above.
Without the limiting conditions s_{b} = b(3) + b(8) = 186 and s_{d} = 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 7^{th} order Magic Square
L is magic is now:
s_{22} = 595  (s_{b} + s_{d})
Attachment 12.6.10 shows the values s_{22} as a function of s_{b} and s_{d}.
The possible values of s_{22}, as summarised in the table above, are highlighted (hatched).
The number of combinations {s_{b} , s_{d}} which result in correct values of s_{22} are shown below:
Any of the required values for s_{b} and s_{d} corresponds with 16 related Border Squares
B and
D.
Any of the 16 values of s_{22} corresponds with 72 Pan Magic Squares C of the 5^{th} order.
To ensure that
I,
F and
H
are magic as required a suitable border for
I
should be selected for each occurring combination {s_{21} , c(5)}, which can be achieved with routine MgcSqr7h.
An example of a 13^{th} 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 n_{j} is comparable with the order of magnitude deducted in Section 12.6.4 above.
12.6.6 Spreadsheet Solutions
In order to ensure that the linear equations deducted in previous sections are correct, they have been applied in following Excel Spread Sheets:

CnstrSngl4d, Magic Border Square, 4 x 4, MC = 340

CnstrSngl5f, Magic Corner Square, 5 x 5, MC = 425

CnstrSngl7e, Eccentric Magic Square, 7 x 7, MC = 595

CnstrSngl9g, Eccentric Magic Square, 9 x 9, MC = 765

CnstrSngl13, Magic Square, 13 x 13, MC = 1105
Only the red figures have to be “guessed” to construct the applicable Magic Squares
(wrong solutions are obvious).
12.6.7 Summary (2)
The obtained results regarding the 13^{th} order Composed Magic Squares and related sub squares, as deducted and discussed in previous sections, are summarized in following table:
