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14.0   Special Magic Squares, Prime Numbers

14.9   Magic Squares (11 x 11)

14.9.1 Concentric Magic Squares (11 x 11)

An 11th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 9th 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 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121

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 9th 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 9th order Composed and miscellaneous Bordered Magic Squares, as discussed in Section 14.7.1, Section 14.7.5 and 14.7.10 also following 11th order Bordered Magic Squares can be generated with routine Priem11a:

• Center Squares with Overlapping Sub Squares     (ref. Attachment 14.9.10)

• Concentric Center Squares with Diamond Inlay    (ref. Attachment 14.9.11)

• Bordered Center Square, Embedded Ultra Magic Square with:

- Conc. Square and Square Inlay (a)             (ref. Attachment 14.9.14)
- Conc. Square and Square Inlay (b)             (ref. Attachment 14.9.15)
- Square Inlays                                 (ref. Attachment 14.9.16)
- Square and Diamond Inlay (a)                  (ref. Attachment 14.9.17)
- Square and Diamond Inlay (b)                  (ref. Attachment 14.9.18)
- Conc. Square and Diamond Inlay                (ref. Attachment 14.9.19)

• Double Bordered Center Squares with:

- Embedded Ultra Magic Square                   (ref. Attachment 14.9.21)
- Embedded Square with Square Inlay             (ref. Attachment 14.9.23)
- Embedded Square with Square and Diamond Inlay (ref. Attachment 14.9.24)

• Composed Associated Magic Center Squares        (ref. Attachment 14.9.42)

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 11th 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:

 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121

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 7th 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 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121

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 9th 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 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121

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 11th 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 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121

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 11th 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 11th order Composed Magic Square, with overlapping sub squares, is a sub square of a well known 13th order Composed Magic Square (Andrews, 1909) which will be developed for Prime Numbers in Next Section.

The 11th order Magic Square K, with Magic Sum s11, contains following sub squares:

• One 9th order Eccentric Magic Square H (right top K):
- with embedded 7th order Eccentric Magic I (left bottom H)
- with embedded 5th order Pan Magic Square C (left bottom I)
• One 3th 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 7th order Magic Square L with C in the right top corner (overlapping I);

as illustrated below:

 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121

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 11th 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 11th order Magic Square E, with Magic Sum s11, contains following sub squares:

• One 3th order Magic Center Square C;
• Two each other overlapping 5th order Eccentric Magic Squares A1 and A2;
• Two each other overlapping 7th order Eccentric Magic Squares B1 and B2;
• Two 4th order Pan Magic Squares PM1 and PM2;

as illustrated below:

 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121

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 6th order Eccentric Magic Squares F1 and F2 with embedded PM1 and PM2;
• Two 9th 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 5th order Associated (Pan) Magic Square Inlay with Magic Sum s5 = 5 * s1 / 11,
• One 6th 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 bottom-left / top-right 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:

 Type Characteristics Subroutine Results Concentric - Bordered Miscellaneous Inlays - Split Border Lines Composed Border (1) Composed Border (2) Eccentric Split Border Lines Associated Composed Border (1) Composed Border (2) Associated Square Inlays Order 5 and 6 - Composed Overlapping Sub Squares (1) - Overlapping Sub Squares (2), Order 11 Overlapping Sub Squares (2), Order 15 Associated Corner Squares and - Rectangles - - - -
 Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 13, which will be described in following sections.