11.0 Magic Squares (11 x 11)
11.1.1 Concentric Magic Squares (11 x 11)
An 11^{th} order Concentric Magic Square consists of a Concentric Magic Square of the 9^{th} order, as discussed in Section 9.6.1, with a border around it.
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Based on the linear equations defining the border of a Concentric Magic Square of order 11:
a(111) = 671  a(112)  a(113)  a(114)  a(115)  a(116)  a(117)  a(118)  a(119)  a(120)  a(121)
a( 22) = 671  a( 11)  a( 33)  a( 44)  a( 55)  a( 66)  a( 77)  a( 88)  a( 99)  a(110)  a(121)
a( 1) = 122  a(121)
a(10) = 122  a(120)
a( 9) = 122  a(119)
a( 8) = 122  a(118)
a( 7) = 122  a(117)

a(6) = 122  a(116)
a(5) = 122  a(115)
a(4) = 122  a(114)
a(3) = 122  a(113)
a(2) = 122  a(112)

a(11) = 122  a(111)
a(12) = 122  a( 22)
a(23) = 122  a( 33)
a(34) = 122  a( 44)
a(45) = 122  a( 55)

a( 56) = 122  a( 66)
a( 67) = 122  a( 77)
a( 78) = 122  a( 88)
a( 89) = 122  a( 99)
a(100) = 122  a(110)

a routine can be written to generate the borders for subject Concentric Magic Squares
(ref. Priem11a).
Attachment 14.9.1 shows a few suitable borders for Concentric Magic Squares of order 11.
Each border shown corresponds with (9!)^{2} = 1,32 * 10^{11} borders with the same corner pairs, which can be obtained by permutation of the horizontal/vertical (non corner) pairs.
Note: The Concentric Magic Center Squares should be based on the consecutive integers 20, 21, ... 101.
11.1.2 Bordered Magic Squares (11 x 11), Miscellaneous Inlays
Also other Magic Squares of the 9^{th} order as described and constructed in
Section 9.1 thru Section 9.7, can be used as Center Squares for 11^{th} order Bordered Magic Squares.
The Embedded Magic Squares will have a Magic Sum s9 = 549 and can be based on the consecutive integers 20, 21, ... 101.
Attachment 14.9.2
contains  based on some of the described Magic Squares of order 9  examples of Bordered Magic Squares
for the same border.
11.1.3 Bordered Magic Squares (11 x 11), Split Border
Alternatively an 11^{th} order Bordered Magic Square with Magic Sum s11 = 671 can be constructed based on:

a Symmetric Magic Center Square of order 7 with Magic Sum s7 = 427;

36 pairs, each summing to 122, surrounding the (Concentric) Magic Center Square;

a split of the supplementary rows and columns into three parts:
two summing to s3 = 183 and one to s5 = 305.
as illustrated below:
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The supplementary rows and columns can be described by following linear equations:
Typical Corner Section (3 x 3):
a'(1) 
a'(2) 
a'(3) 
a'(4) 
a'(5) 
a'(6) 
a'(7) 
a'(8) 
 
 
a'(3) = 183  a'(2)  a'(1)
a'(4) = 122  a'(2)
a'(5) = 122  a'(1)
a'(6) = 122  a'(3)
a'(7) = 183  a'(4)  a'(1)
a'(8) = 122  a'(7)

Typical Border Rectangle (2 x 5):
a'(1) 
a'(2) 
a'(3) 
a'(4) 
a'(5) 
a'(6) 
a'(7) 
a'(8) 
a'(9) 
a'(10) 
 
a'( 5) = 305  a'(4)  a'(3)  a'(2)  a'(1)
a'( 6) = 122  a'(1)
a'( 7) = 122  a'(2)
a'( 8) = 122  a'(3)
a'( 9) = 122  a'(4)
a'(10) = 122  a'(5)

Based on the equations above, procedures can be developed:

to generate, based on the distinct integers {1 ... 121}, four Corner Squares (3 x 3);

to complete the exterior border with four Magic Rectangles (2 x 5);

to construct, based on the remaining 49 distinct integers, the border of the Concentric Center Square (7 x 7);

to construct, based on the remaining 25 distinct integers, the embedded Center Symmetric Magic Square of order 5.
The first occuring Bordered Magic Square is shown below:
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The border shown above corresponds with (4! * 8^{4}) * (4! * 240^{4})
= 7,83 * 10^{15} borders as:

The Corner Squares can be arranged in 4! ways and belong each to a collection of 8 Corner Squares;

The Rectangles can be arranged in 4! ways and belong each to a collection of 2 * 5! = 240 Rectangles.
Based on the distinct integers applied in the border shown above, 470552 suitable sets of Corner Squares can be found.
11.1.4 Eccentric Magic Squares (11 x 11)
An 11^{th} order Eccentric Magic Square consists of one Magic Corner Square of the 9^{th} order, as discussed in Section 9.6.4, supplemented with two rows and two columns.
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Based on the linear equations defining the supplementary rows, columns and main diagonal of an Eccentroc Magic Square of order 11:
a(11) = 671  a(21)  a(31)  a(41)  a(51)  a(61)  a(71)  a(81)  a(91)  a(101)  a(111)
a(12) = 671  a(13)  a(14)  a(15)  a(16)  a(17)  a(18)  a(19)  a(20)  a( 21)  a( 22)
a(24) = 671  a( 2)  a(13)  a(35)  a(46)  a(57)  a(68)  a(79)  a(90)  a(101)  a(112)
a(112) = 122  a(111)
a(100) = 122  a(101)
a( 22) = 122  a( 11)
a( 10) = 122  a( 21)
a( 9) = 122  a( 20)

a(8) = 122  a(19)
a(7) = 122  a(18)
a(6) = 122  a(17)
a(5) = 122  a(16)
a(4) = 122  a(15)

a( 3) = 122  a(14)
a( 2) = 122  a(12)
a( 1) = 122  a(13)
a(89) = 122  a(90)
a(78) = 122  a(79)

a(67) = 122  a(68)
a(56) = 122  a(57)
a(45) = 122  a(46)
a(34) = 122  a(35)
a(23) = 122  a(24)

a routine can be written to generate subject Eccentric Magic Squares
(ref. Priem11c).
Attachment 14.9.4 shows a few 11^{th} order Eccentric Magic Squares,
based on 9^{th} order Eccentric Magic Squares.
Each Eccentric Magic Square shown corresponds with (7!)^{2} = 2,54 * 10^{7}
Eccentric Magic Squares with the same corner square and corner pairs, which can be obtained by permutation of the horizontal/vertical (non corner) pairs.
Note: The Eccentric Magic Corner Squares are based on the consecutive integers 20, 21, ... 101.
11.1.5 Bordered Magic Squares (11 x 11), Composed Border (1)
The 11^{th} order Composed Magic Square shown below, with Magic Sum s11 = 671, consists of:

a Border composed out of:
 4 Semi Magic Anti Symmetric Corner Squares of order 3 with Magic Sum s3 = 183;
 4 Anti Symmetric Magic Rectangles order 3 x 5 with s3 = 183 and s5 = 305.

a Center Symmetric Magic Center Square of order 5 with Magic Sum s5 = 305;
As a consequence of the defining properties mentioned above the 11^{th} order Composed Magic Square is Center Symmetric.
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Based on the principles described in previous sections, a fast procedure (Priem11d) can be developed:

to read the previously generated Center Symmetric Magic Squares (5 x 5);

to generate, based on the remainder of the pairs, the four Anti Symmetric Semi Magic Squares of order 3;

to complete the Composed Border of order 11 with the four 3 x 5 Anti Symmetric Magic Rectangles.
Attachment 14.9.7 shows miscellaneous order 11 Associated Composed Magic Squares.
11.1.6 Bordered Magic Squares (11 x 11), Composed Border (2)
The 11^{th} order Composed Magic Square shown below, with Magic Sum s11 = 671, consists of:

a Border composed out of:
 4 Associated Magic Squares of order 4 with Magic Sum s4 = 244
 4 Associated Magic Rectangles order 3 x 4 with s3 = 183 and s4 = 244

a Magic Center Square of order 3 with Magic Sum s3 = 183
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The Composed Square shown above corresponds with n11 = 8 * (4! * 384^{4}) * (4! * 16^{4})
= 6,57 * 10^{18} squares as:

The Center Square belongs to a collection of 8 Center Squares;

The Corner Squares can be arranged in 4! ways and belong each to a collection of 384 Corner Squares;

The Border Rectangles can be arranged in 4! ways and belong each to a collection of 16 Rectangles.
Based on the principles described in previous sections, a fast procedure (Priem11e) can be developed:

to read the previously generated 3 x 3 Magic Center Square (850 unique);

to generate, based on the remainder of the pairs, four 4 x 4 Associated Magic Squares;

to complete the Composed Border of order 11 with four 3 x 4 Associated Magic Rectangles.
Attachment 14.9.6 shows miscellaneous order 11 Composed Border Magic Squares. Each (unique) square shown corresponds with n11 squares.
Note:
If the applied properties are changed to:

the opposite Semi Magic Corner Squares (4 x 4) are Anti Symmetric and Complementary;

the opposite Magic Rectangles (3 x 4) are Anti Symmetric and Complementary;

the Magic Center Square (3 x 3) is Center Symmetric (per definition);
the 11^{th} order Composed Magic Square will be associated
Attachment 14.9.8 shows miscellanous order 11 Associated Composed Magic Squares (ref. Priem11f).
11.2.1 Composed Magic Squares (11 x 11), Overlapping Sub Squares (1)
This 11^{th} order Composed Magic Square, with overlapping sub squares, is a variation on a well known 13^{th} order Composed Magic Square (Andrews, 1909) which will be discussed in
Section 12.6.
The 11^{th} order Magic Square K, with Magic Sum s11 = 671, contains following sub squares:

One 9^{th} order Eccentric Magic Square H (right top K):
 with embedded 7^{th} order Eccentric Magic I (left bottom H)
 with embedded 5^{th} order Magic Square C (left bottom I)

One 3^{th} order Semi Magic Square M, element a(91) common with C (left bottom K);

Four 2 x 4 Magic Rectangles: A and B (left), D and E (bottom);

Another 7^{th} order Magic Square
L with C in the right top corner (overlapping I);
as illustrated below:
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Attachment 14.8.5.07 shows,
miscellaneous of these 11^{th} order Composed Magic Squares
(ref. MgcSqr11).
11.2.2 Composed Magic Squares (11 x 11), Overlapping Sub Squares (2)
Following 11^{th} order Composed Magic Square, with overlapping sub squares, is a variation on
another higher order Composed Magic Square as discussed by William Symes Andrews (ref. Magic Squares and Cubes, Fig. 352).
The 11^{th} order Magic Square E, with Magic Sum s11 = 671, contains following sub squares:

One 3^{th} order Magic Center Square C;

Two each other overlapping 5^{th} order Eccentric Magic Squares A1 and A2;

Two each other overlapping 7^{th} order Eccentric Magic Squares B1 and B2;

Two 4^{th} order Pan Magic Squares PM1 and PM2;
as illustrated below:
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Attachment 14.9.8.1 shows, miscellaneous order 11 Composed Magic Squares
(ref. PriemE11).
The corresponding Composed Magic Squares of order 15 contain, in addition to the sub squares mentioned above, following Corner Squares:

Two 6^{th} order Eccentric Magic Squares
F1 and F2 with embedded
PM1 and PM2;

Two 9^{th} order Eccentric Magic Squares
D1 and D2 with embedded
B1 and B2;
Attachment 14.9.8.2 shows, for the sake of completeness, miscellanous Composed Magic Squares of order 15
(ref. PriemG15).
11.3.1 Associated Magic Squares (11 x 11)
Associated Square Inlays Order 5 and 6
Associated Magic Squares of order 11 with Square Inlays of order 5 and 6 can be obtained by means of transformation of order 11 Composed Magic Squares,
as illustrated in Section 14.7.4 for order 9 Magic Squares.
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The Associated Square shown above is composed out of:

One 5^{th} order Associated (Pan) Magic Square Inlay with Magic Sum s5 = 305,

One 6^{th} order Associated Magic Square Inlay with Magic Sum s6 = 366 and

Two Associated Magic Rectangle Inlays order 5 x 6 with s5 = 305 and s6 = 366
Based on this definition a routine can be developed to generate the required Composed Magic Squares (ref. Prime11c1).
Attachment 14.8.15 shows miscellaneous Composed Magic Square.
Attachment 14.8.16 shows the corresponding Associated Magic Squares with order 5 and 6 Square Inlays.
It should be noted that the reversed transformation is not necessarily possible because of the bottomleft / topright Main Diagonal.
11.3.2 Associated Magic Squares (11 x 11)
Associated Center Square Order 5
Associated Magic Squares of order 11 with an Associated Center Square of order 5 can be obtained by means of transformation of order 11 Composed Magic Squares as illustrated in Section 14.7.5 for order 9 Magic Squares.
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Attachment 14.8.17 shows the Associated Magic Squares with order 5 Associated Center Squares,
corresponding with the Composed Magic Squares as shown in Attachment 14.8.15.
It should be noted that the reversed transformation is not necessarily possible because of the bottomleft / topright Main Diagonal.
11.3.3 Associated Magic Squares (11 x 11)
Square Inlays Order 4 and 5 (Overlapping)
The 11^{th} order Associated Inlaid Magic Square shown below:
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45 
7 
64 
89 
37 
72 
22 
110 
78 
48 
117 
31 
34 
70 
6 
62 
97 
18 
57 
2 
32 
43 
46 
87 
112 
109 
98 
68 
17 


contains following inlays:

two each 5^{th} order Pan Magic Squares  Magic Sums s(1) = 341 and s(4) = 269  with the center element in common,

two each 4^{th} order Simple Magic Squares with Magic Sums s(2) = 214 and s(3) = 274.
The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:
s(1) = 10 * s11 / 11  s(4)
s(2) = 8 * s11 / 11  s(3)
With s11 = 671 the Magic Sum of the 11^{th} order Inlaid Magic Square.
The Associated Border can be described by following linear equations:
a(115) = s11/11 + a(117)  s(3) + s(4)
a(114) = s11/11 + a(118)  s(3) + s(4)
a(113) = s11/11 + a(119)  s(3) + s(4)
a(112) = s11/11 + a(120)  s(3) + s(4)
a(111) = 15*s11/11  a(116)  2 * a(117)  2 * a(118)  2 * a(119)  2 * a(120)  a(121) + 4 * s(3)  4 * s(4)
a(100) = s11  a(110)  s(3)  s(4)
a( 89) = s11  a( 99)  s(3)  s(4)
a( 78) = s11  a( 88)  s(3)  s(4)
a( 67) = s11  a( 77)  s(3)  s(4)
a( 66) = 60*s11/11  2*a(77)  2*a(88)  2*a(99)  2*a(110)  a(116)  2*a(117)  2*a(118) +
 2*a(119)  2*a(120)  2*a(121)  8*s(4)
a(56) = 2 * s11/11  a(66)
a(55) = 2 * s11/11  a(67)
a(45) = 2 * s11/11  a(77)
a(44) = 2 * s11/11  a(78)
a(34) = 2 * s11/11  a(88)
a(33) = 2 * s11/11  a(89)

a(23) = 2 * s11/11  a( 99)
a(22) = 2 * s11/11  a(100)
a(12) = 2 * s11/11  a(110)
a(11) = 2 * s11/11  a(111)
a(10) = 2 * s11/11  a(112)
a( 9) = 2 * s11/11  a(113)
a( 8) = 2 * s11/11  a(114)

a(7) = 2 * s11/11  a(115)
a(6) = 2 * s11/11  a(116)
a(5) = 2 * s11/11  a(117)
a(4) = 2 * s11/11  a(118)
a(3) = 2 * s11/11  a(119)
a(2) = 2 * s11/11  a(120)
a(1) = 2 * s11/11  a(121)

Which can be incorporated in an optimised guessing routine MgcSqr11k2.
The Magic Center Squares can be constructed by means of suitable selected Latin Squares
(ref. MgcSqr11k1),
based on resp. order 4 and 5 Magic Lines for the integers 0 ... 10 as shown in
Attachment 14.9.9a.
Attachment 14.9.9b shows a few 11^{th} order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).
Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.
11.3.4 Associated Magic Squares (11 x 11)
Diamond Inlays Order 5 and 6
The 11^{th} order Associated Inlaid Magic Square shown below:
116 
110 
10 
8 
26 
113 
120 
46 
60 
38 
24 
92 
64 
32 
108 
59 
25 
49 
18 
88 
66 
70 
20 
82 
44 
67 
117 
23 
27 
17 
100 
94 
80 
16 
72 
91 
45 
21 
79 
81 
89 
37 
86 
54 
48 
35 
107 
65 
83 
119 
19 
109 
53 
29 
4 
1 
11 
115 
47 
51 
61 
71 
75 
7 
111 
121 
118 
93 
69 
13 
103 
3 
39 
57 
15 
87 
74 
68 
36 
85 
33 
41 
43 
101 
77 
31 
50 
106 
42 
28 
22 
105 
95 
99 
5 
55 
78 
40 
102 
52 
56 
34 
104 
73 
97 
63 
14 
90 
58 
30 
98 
84 
62 
76 
2 
9 
96 
114 
112 
12 
6 
contains following Diamond Inlays:

one each 5^{th} order Associated Diamond Inlay with Magic Sum s5 = 305,

one each 6^{th} order Associated Diamond Inlay with Magic Sum s6 = 366.
As the order 5 and 6 Diamond Inlays contain only odd numbers, the Associated Inlaid Magic Square is a Lozenge Square.
The method to generate order 11 Associated Lozenge Squares with order 5 and 6 Diamond Inlays will be discussed in Section 18.5.3.
11.4 Summary
The obtained results regarding the miscellaneous types of order 11 Magic Squares as deducted and discussed in previous sections are summarized in following table:
