Office Applications and Entertaiment, Magic Squares

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11.0   Magic Squares (11 x 11)

11.1.1 Concentric Magic Squares (11 x 11)

An 11th order Concentric Magic Square consists of a Concentric Magic Square of the 9th order, as discussed in Section 9.6.1, with a border around it.

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 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 * 1011 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 9th order as described and constructed in Section 9.1 thru Section 9.7, can be used as Center Squares for 11th 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 11th 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:

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

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:

2 62 119 10 11 69 107 108 114 6 63
60 120 3 112 111 53 15 14 8 59 116
121 1 93 49 41 90 89 37 28 118 4
98 24 80 88 87 47 45 38 42 106 16
97 25 40 39 50 51 79 86 82 104 18
52 70 30 58 48 61 74 64 92 55 67
31 91 44 36 43 71 72 83 78 21 101
27 95 46 84 77 75 35 34 76 19 103
117 5 94 73 81 32 33 85 29 115 7
9 65 109 20 22 68 96 99 105 12 66
57 113 13 102 100 54 26 23 17 56 110

The border shown above corresponds with (4! * 84) * (4! * 2404) = 7,83 * 1015 borders as:

  1. The Corner Squares can be arranged in 4! ways and belong each to a collection of 8 Corner Squares;
  2. 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 11th order Eccentric Magic Square consists of one Magic Corner Square of the 9th order, as discussed in Section 9.6.4, supplemented with two rows and two columns.

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 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 11th order Eccentric Magic Squares, based on 9th order Eccentric Magic Squares.

Each Eccentric Magic Square shown corresponds with (7!)2 = 2,54 * 107 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 11th 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 11th order Composed Magic Square is Center Symmetric.

70 104 9 24 21 42 102 116 19 95 69
108 7 68 40 44 76 79 66 67 17 99
5 72 106 119 118 65 2 1 97 71 15
11 58 114 109 84 37 28 47 110 39 34
30 41 112 63 31 29 86 96 100 48 35
89 49 45 32 60 61 62 90 77 73 33
87 74 22 26 36 93 91 59 10 81 92
88 83 12 75 94 85 38 13 8 64 111
107 51 25 121 120 57 4 3 16 50 117
23 105 55 56 43 46 78 82 54 115 14
53 27 103 6 20 80 101 98 113 18 52

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 11th 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
112 111 20 1 60 30 93 109 106 25 4
9 12 103 120 90 36 57 14 15 96 119
2 19 110 113 65 86 32 3 26 107 108
121 102 11 10 29 92 62 118 97 16 13
83 49 42 70 72 53 58 91 51 35 67
48 54 68 74 47 61 75 37 45 77 85
52 80 73 39 64 69 50 55 87 71 31
99 94 43 8 59 40 84 105 100 33 6
24 27 78 115 81 46 56 18 21 88 117
7 44 95 98 66 76 41 5 34 101 104
114 79 28 23 38 82 63 116 89 22 17

The Composed Square shown above corresponds with n11 = 8 * (4! * 3844) * (4! * 164) = 6,57 * 1018 squares as:

  1. The Center Square belongs to a collection of 8 Center Squares;
  2. The Corner Squares can be arranged in 4! ways and belong each to a collection of 384 Corner Squares;
  3. 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 11th 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 11th order Composed Magic Square, with overlapping sub squares, is a variation on a well known 13th order Composed Magic Square (Andrews, 1909) which will be discussed in Section 12.6.

The 11th order Magic Square K, with Magic Sum s11 = 671, 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 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:

92 30 83 49 51 90 47 46 98 21 64
31 91 39 73 71 32 75 76 24 58 101
36 86 100 23 28 93 95 19 69 104 18
85 37 22 99 94 29 27 53 103 77 45
114 8 4 106 59 15 121 111 11 72 50
52 70 13 119 5 108 60 25 97 34 88
12 110 109 62 14 117 3 79 43 44 78
66 56 118 1 107 63 16 38 84 82 40
81 41 61 17 120 2 105 102 20 57 65
48 68 67 9 112 7 116 33 80 35 96
54 74 55 113 10 115 6 89 42 87 26

Attachment 14.8.5.07 shows, miscellaneous of these 11th order Composed Magic Squares (ref. MgcSqr11).

11.2.2 Composed Magic Squares (11 x 11), Overlapping Sub Squares (2)

Following 11th 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 11th order Magic Square E, with Magic Sum s11 = 671, 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:

38 35 83 88 15 86 41 102 104 16 63
78 93 33 40 107 36 81 20 18 59 106
39 34 84 87 1 117 4 116 67 108 14
89 82 44 29 121 5 118 55 6 24 98
96 26 113 9 65 58 60 13 109 100 22
27 95 10 112 56 61 66 119 3 103 19
31 91 101 21 62 64 57 2 120 17 105
90 32 12 53 114 11 115 46 47 74 77
23 99 69 110 8 111 7 71 80 43 50
92 54 37 28 70 49 97 48 45 76 75
68 30 85 94 52 73 25 79 72 51 42

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 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, 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 a transformation of order 11 Composed Magic Squares, as illustrated in Section 14.7.4 for order 9 Magic Squares.

102 108 23 42 39 22 91 36 26 97 85
114 121 59 120 10 57 111 4 60 3 12
24 58 95 49 81 84 33 44 93 70 40
53 54 51 32 79 9 21 94 55 116 107
34 17 75 92 76 77 72 103 74 16 35
13 5 66 7 104 61 18 115 56 117 109
87 106 48 19 50 45 46 30 47 105 88
15 6 67 28 101 113 43 90 71 68 69
82 52 29 78 89 38 41 73 27 64 98
110 119 62 118 11 65 112 2 63 1 8
37 25 96 86 31 100 83 80 99 14 20

The Associated Square shown above is composed out of:

  • One 5th order Associated (Pan) Magic Square Inlay with Magic Sum s5 = 305,
  • One 6th 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 bottom-left / top-right Main Diagonal.

11.3.2 Associated Magic Squares (11 x 11)
       Square Inlays Order 4 and 5 (Overlapping)


The 11th order Associated Inlaid Magic Square shown below:

105 54 24 13 10 35 76 79 90 120 65
104 25 60 116 52 88 91 5 74 44 12
100 50 85 33 58 115 77 41 93 3 16
15 66 113 49 83 30 38 69 11 96 101
14 82 28 63 121 47 8 99 36 71 102
19 118 55 80 27 61 95 42 67 4 103
20 51 86 23 114 75 1 59 94 40 108
21 26 111 53 84 92 39 73 9 56 107
106 119 29 81 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
341 214
274 269

contains following inlays:

  • two each 5th order Pan Magic Squares - Magic Sums s(1) = 341 and s(4) = 269 - with the center element in common,
  • two each 4th 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 11th 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 11th 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.

11.4   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:

Type

Characteristics

Subroutine

Results

Concentric

-

Priem11a

Attachment 14.9.1

Bordered

Miscellaneous Inlays

-

Attachment 14.9.2

Composed Border

Priem11e

Attachment 14.9.6

Eccentric

-

Priem11c

Attachment 14.9.4

Associated

Composed Border (1)

Priem11d

Attachment 14.9.7

Composed Border (2)

Priem11f

Attachment 14.9.8

Square Inlays Order 5 and 6

-

Attachment 14.8.16

Square Inlays Order 4 and 5 (Overlapping)

MgcSqr11k

Attachment 14.9.9b

Composed

Overlapping Sub Squares (1)

MgcSqr11

Attachment 14.8.5.7

Overlapping Sub Squares (2), Order 11

PriemE11

Attachment 14.9.8.1

Overlapping Sub Squares (2), Order 15

PriemG15

Attachment 14.9.8.2

Associated Corner Squares and - Rectangles

Prime11c1

Attachment 14.8.15

-

-

-

-

Comparable routines as listed above, can be used to generate Magic Squares of order 12 and higher, which will be described in following sections.


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