|
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:
-
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 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.
-
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.
-
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
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:
→ →
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
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:
→ →
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
|
Attachment 12.6.4
|
|
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:
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.
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 13th order Composed Magic Squares and related sub squares, as deducted and discussed in previous sections, are summarized in following table:
|
