14.0 Special Magic Squares, Prime Numbers
14.9 Magic Squares (11 x 11)
14.9.1 Concentric Magic Squares (11 x 11)
An 11^{th} order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 9^{th} order with a border around it.
For Prime Number Concentric Magic Squares of order 11 with Magic Sum s11, it is convenient to split the supplementary rows and columns into
parts summing to s4 = 4 * s11 / 11 and s3 = 3 * s11 / 11:
a1 
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This results in following border equations:
a( 4) = s4  a( 3)  a( 2)  a(1)
a( 34) = s4  a(23)  a(12)  a(1)
a(121) = s4/2  a( 1)
a(112) = s4/2  a( 2)
a(113) = s4/2  a( 3)
a(114) = s4/2  a( 4)
a( 22) = s4/2  a(12)
a( 33) = s4/2  a(23)
a( 44) = s4/2  a(34)
a( 5) = s3  a(6)  a(7)
a(115) = s4/2  a(5)
a(116) = s4/2  a(6)
a(117) = s4/2  a(7)

a( 11) = s4  a(10)  a( 9)  a( 8)
a(110) = s4  a(99)  a(88)  a(11)
a(111) = s4/2  a( 11)
a(118) = s4/2  a( 8)
a(119) = s4/2  a( 9)
a(120) = s4/2  a( 10)
a( 78) = s4/2  a( 88)
a( 89) = s4/2  a( 99)
a(100) = s4/2  a(110)
a(55) = s3  a(66)  a(77)
a(45) = s4/2  a(55)
a(56) = s4/2  a(66)
a(67) = s4/2  a(77)

which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 11 (ref. Priem11a).
Miscellaneous Prime Number Concentric Magic Squares of order 11, based on 9^{th} order Concentric Magic Squares as discussed in Section 14.7.4, are shown in Attachment 14.9.1.
14.9.2 Bordered Magic Squares (11 x 11), Miscellaneous Inlays
Based on the collections of 9^{th} order Composed and miscellaneous Bordered Magic Squares, as discussed in
Section 14.7.1,
Section 14.7.5 and
14.7.10
also following 11^{th} order Bordered Magic Squares can be generated with routine Priem11a:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.9.3 Bordered Magic Squares (11 x 11), Split Border
Alternatively an 11^{th} order Bordered Magic Square with Magic Sum s11 can be constructed based on:

a Symmetric Magic Center Square of order 7 with Magic Sum s7 = 7 * s11 / 11;

36 pairs, each summing to 2 * s11 / 11, surrounding the (Concentric) Magic Center Square;

a split of the supplementary rows and columns into three parts:
two summing to s3 = 3 * s11 / 11 and one to s5 = 5 * s11 / 11.
as illustrated below:
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Based on the principles described in previous sections, a fast procedure (Priem11b) can be developed:

to read the previously generated Concentric Magic Squares (7 x 7);

to generate the four Magic Rectangles (2 x 5);

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

to transform these Magic Squares into suitable Corner Squares, thus completing the border.
Attachment 14.9.3 shows
based on 7^{th} order Concentric Magic Squares as discussed in Section 14.5.1,
one Prime Number Bordered Magic Square for some of the occurring Magic Sums.
Each square shown corresponds with numerous squares for the same Magic Sum.
14.9.4 Eccentric Magic Squares (11 x 11)
For Prime Number Eccentric Magic Squares of order 11 it is convenient to split the supplementary rows and columns into:
two parts summing to s4 = 4 * s11 / 11 and one part summing to s3 = 3 * s11 / 11.
a1 
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This enables, based on the same principles, the development of a set of fast procedures (ref. Priem11c):

to read the previously generated Eccentric Magic Squares of order 9;

to complete the Main Diagonal and determine the related Border Pairs;

to generate, based on the remainder of the available pairs, a suitable Corner Square of order 3;

to complete the Eccentric Magic Square of order 11 with the two remaining 2 x 4 Magic Rectangles.
Attachment 14.9.4 shows,
based on 9^{th} order Eccentric Magic Squares as discussed in Section 14.7.6,
one Prime Number Eccentric Magic Square for some of the occurring Magic Sums.
Each square shown corresponds with numerous squares for the same Magic Sum.
14.9.5 Bordered Magic Squares (11 x 11), Composed Border (1)
Order 11 Composed Border Magic Squares with Magic Sum s11 can be constructed based on:

a Border composed out of:
 4 Semi Magic Squares of order 3 with Magic Sum s3 = 3 * s11 / 11;
 4 Magic Rectangles order 3 x 5 with s3 = 3 * s11 / 11 and s5 = 5 * s11 / 11.

a Symmetric Magic Center Square of order 5 with Magic Sum s5 = 5 * s11 / 11;
as illustrated below:
a1 
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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 Semi Magic Squares of order 3;

to complete the Composed Border of order 11 with the four 3 x 5 Magic Rectangles.
Following Attachments show, for some of the occuring Magic Sums, one Bordered Magic Square per Magic Sum:
Each square shown corresponds with numerous squares for the same Magic Sum.
Note:
As a consequence of the applied properties (ref. Priem11d):

The opposite Semi Magic Corner Squares are Anti Symmetric and Complementary;

The Magic Center Square might be Associated;
In this case the 11^{th} order Composed Magic Square will be associated, if the opposite Magic Rectangles (3 x 5) are Anti Symmetric and Complementary as well.
Attachment 14.9.7 shows one Associated Composed Magic Square for some of the occurring Magic Sums (ref. Priem11d3).
14.9.6 Bordered Magic Squares (11 x 11), Composed Border (2)
Alternatively order 11 Composed Border Magic Squares with Magic Sum s11 can be constructed based on:

a Border composed out of:
 4 (Semi) Magic Squares of order 4 with Magic Sum s4 = 4 * s11 / 11;
 4 Associated Magic Rectangles order 3 x 4 with s3 = 3 * s11 / 11 and s4 = 4 * s11 / 11.

a Magic Center Square of order 3 with Magic Sum s3 = 3 * s11 / 11;
as illustrated below:
a1 
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Based on the principles described in previous sections, a fast procedure (Priem11e) can be developed:

to generate the 3 x 3 Magic Center Square;

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 one Prime Number Composed Border Magic Square for some of the occurring Magic Sums.
Each square shown corresponds with numerous squares for the same Magic Sum.
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 one Associated Composed Magic Square for some of the occurring Magic Sums (ref. Priem11f).
14.9.7 Composed Magic Squares (11 x 11), Overlapping Sub Squares (1)
This 11^{th} order Composed Magic Square, with overlapping sub squares, is a sub square of a well known 13^{th} order Composed Magic Square (Andrews, 1909) which will be developed for Prime Numbers in Next Section.
The 11^{th} order Magic Square K, with Magic Sum s11, 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 Pan 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, for the sake of completeness, one Prime Number Composed Magic Square of order 11 for some of the occurring Magic Sums.
Each square shown corresponds with numerous squares for the same Magic Sum.
14.9.8 Composed Magic Squares (11 x 11), Overlapping Sub Squares (2)
Following 11^{th} order Composed Magic Square, with overlapping sub squares, is also a sub square of
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, 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, one Prime Number Composed Magic Square of order 11 for some of the occurring Magic Sums
(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, the Prime Number Composed Magic Squares of order 15 for some of the occurring Magic Sums (ref. PriemG15).
Each square shown corresponds with numerous squares for the same Magic Sum.
14.9.9 Associated Magic Squares (11 x 11) with 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 a transformation of order 11 Composed Magic Squares,
as illustrated in Section 14.7.11 for order 9 Magic Squares.
MC = 36619
4517 
6521 
167 
2609 
401 
1187 
5039 
2879 
4679 
3449 
5171 
6653 
5477 
2777 
2399 
557 
2711 
6599 
4421 
3299 
1637 
89 
3191 
2897 
5441 
1871 
4937 
4349 
2939 
1259 
1559 
6269 
1907 
2957 
2111 
719 
3821 
6311 
3167 
2741 
3617 
947 
3929 
6299 
797 
569 
4871 
5507 
5639 
4451 
2999 
5849 
4241 
269 
1427 
107 
1307 
4079 
5147 
5801 
3329 
857 
1511 
2579 
5351 
6551 
5231 
6389 
2417 
809 
3659 
2207 
1019 
1151 
1787 
6089 
5861 
359 
2729 
5711 
3041 
3917 
3491 
347 
2837 
5939 
4547 
3701 
4751 
389 
5099 
5399 
3719 
2309 
1721 
4787 
1217 
3761 
3467 
6569 
5021 
3359 
2237 
59 
3947 
6101 
4259 
3881 
1181 
5 
1487 
3209 
1979 
3779 
1619 
5471 
6257 
4049 
6491 
137 
2141 
The Associated Square shown above is composed out of:

One 5^{th} order Associated (Pan) Magic Square Inlay with Magic Sum s5 = 5 * s1 / 11,

One 6^{th} order Associated Magic Square Inlay with Magic Sum s6 = 6 * s1 / 11 and

Two Associated Magic Rectangle Inlays order 5 x 6 with s5 = 5 * s1 / 11 and s6 = 6 * s1 / 11
Based on this definition a routine can be developed to generate the required Composed Magic Squares (ref. Prime11c).
Attachment 14.8.15 shows for miscellaneous Magic Sums the first occurring Prime Number 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.
14.9.10 Summary
The obtained results regarding the miscellaneous types of order 11 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table:
