Office Applications and Entertainment, Latin Squares

Vorige Pagina Volgende Pagina Index About the Author

11.0   Latin Squares (11 x 11)

A Latin Square of order 11 is an 11 x 11 square filled with 11 different symbols, each occurring only once in each row and only once in each column.

11.1   Latin Diagonal Squares (11 x 11)

Latin Diagonal Squares are Latin Squares for which the 11 different symbols occur also only once in each of the main diagonals.

11.2   Magic Squares, Natural Numbers

11.2.1 Pan Magic Squares

Pan Magic Square M of order 11 with the integers 1 ... 121 can be written as M = A + 11 * B + 1 where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.

Consequently order 11 Pan Magic Squares can be based on pairs of Orthogonal Latin Diagonal Squares (A, B).

The required Orthogonal Latin Diagonal Squares (A, B) for Pan Magic Squares can be constructed as follows:

  1. Fill the first row of square A and square B with the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.
    While starting with 0 there are 10! = 3628800 possible combinations for each square.

  2. Complete square A and B by copying the first row into the following rows of the applicable square,
    according to one of the following 28 schemes:

    A/B L2 R2 L3 R3 L4 R4 L5 R5
    L2 - y y y y y y y
    L3 y y - y y y y y
    L4 y y y y - y y y
    L5 y y y y y y - y

    Ln = shift n columns to the left  (n = 2, 3, 4, 5)
    Rn = shift n columns to the center (n = 2, 3, 4, 5)

Attachment 11.2.1 shows the eight types Latin Diagonal Squares based on the construction method described above.

An example of such a pair (A, B) and the resulting Pan Magic Square M is shown below:

A (L2)
0 1 2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 0 1
4 5 6 7 8 9 10 0 1 2 3
6 7 8 9 10 0 1 2 3 4 5
8 9 10 0 1 2 3 4 5 6 7
10 0 1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 10 0
3 4 5 6 7 8 9 10 0 1 2
5 6 7 8 9 10 0 1 2 3 4
7 8 9 10 0 1 2 3 4 5 6
9 10 0 1 2 3 4 5 6 7 8
B (R2)
0 1 2 3 4 5 6 7 8 9 10
9 10 0 1 2 3 4 5 6 7 8
7 8 9 10 0 1 2 3 4 5 6
5 6 7 8 9 10 0 1 2 3 4
3 4 5 6 7 8 9 10 0 1 2
1 2 3 4 5 6 7 8 9 10 0
10 0 1 2 3 4 5 6 7 8 9
8 9 10 0 1 2 3 4 5 6 7
6 7 8 9 10 0 1 2 3 4 5
4 5 6 7 8 9 10 0 1 2 3
2 3 4 5 6 7 8 9 10 0 1
M = A + 11 * B + 1
1 13 25 37 49 61 73 85 97 109 121
102 114 5 17 29 41 53 65 77 78 90
82 94 106 118 9 21 33 34 46 58 70
62 74 86 98 110 111 2 14 26 38 50
42 54 66 67 79 91 103 115 6 18 30
22 23 35 47 59 71 83 95 107 119 10
112 3 15 27 39 51 63 75 87 99 100
92 104 116 7 19 31 43 55 56 68 80
72 84 96 108 120 11 12 24 36 48 60
52 64 76 88 89 101 113 4 16 28 40
32 44 45 57 69 81 93 105 117 8 20

Each type Latin Diagonal Square described above, corresponds with 10! = 3628800 Latin Diagonal Squares.

The possible combinations of square A and B described above will result in 28 * 36288002 / 4 = 92.177.326.080.000 unique solutions.

Each unique Pan Magic Square results in a Class Cn and finally in 8 * 121 * 92.177.326.080.000 = 89.227.651.645.440.000 possible Pan Magic Squares of the 11th order.

Attachment 11.2.2 shows one Pan Magic Square for each valid type combination (A, B) as defined above.

A Self Orthogonal Latin Square is a Latin Square that is Orthogonal to its Transposed. If the main diagonal contains the integers 0 to 10 in natural order the Self Orthogonal Latin Square is called Idempotent.

Attachment 11.2.3 shows the construction of a Pan Magic Square M based on an Idempotent Latin Square A.

11.2.2 Ultra Magic Squaress

An example of the construction of an order 11 Ultra Magic Square M based on pairs of Orthogonal Latin Diagonal Squares (A, B), is shown below for the symbols {ai, i = 1 ... 11} and {bj, j = 1 ... 11}.

A
a11 a1 a10 a9 a8 a7 a6 a5 a4 a3 a2
a3 a2 a11 a1 a10 a9 a8 a7 a6 a5 a4
a5 a4 a3 a2 a11 a1 a10 a9 a8 a7 a6
a7 a6 a5 a4 a3 a2 a11 a1 a10 a9 a8
a9 a8 a7 a6 a5 a4 a3 a2 a11 a1 a10
a1 a10 a9 a8 a7 a6 a5 a4 a3 a2 a11
a2 a11 a1 a10 a9 a8 a7 a6 a5 a4 a3
a4 a3 a2 a11 a1 a10 a9 a8 a7 a6 a5
a6 a5 a4 a3 a2 a11 a1 a10 a9 a8 a7
a8 a7 a6 a5 a4 a3 a2 a11 a1 a10 a9
a10 a9 a8 a7 a6 a5 a4 a3 a2 a11 a1
B = T(A)
b11 b3 b5 b7 b9 b1 b2 b4 b6 b8 b10
b1 b2 b4 b6 b8 b10 b11 b3 b5 b7 b9
b10 b11 b3 b5 b7 b9 b1 b2 b4 b6 b8
b9 b1 b2 b4 b6 b8 b10 b11 b3 b5 b7
b8 b10 b11 b3 b5 b7 b9 b1 b2 b4 b6
b7 b9 b1 b2 b4 b6 b8 b10 b11 b3 b5
b6 b8 b10 b11 b3 b5 b7 b9 b1 b2 b4
b5 b7 b9 b1 b2 b4 b6 b8 b10 b11 b3
b4 b6 b8 b10 b11 b3 b5 b7 b9 b1 b2
b3 b5 b7 b9 b1 b2 b4 b6 b8 b10 b11
b2 b4 b6 b8 b10 b11 b3 b5 b7 b9 b1

The Latin Square B is the transposed square of A (rows and columns exchanged).

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
10 0 9 8 7 6 5 4 3 2 1
2 1 10 0 9 8 7 6 5 4 3
4 3 2 1 10 0 9 8 7 6 5
6 5 4 3 2 1 10 0 9 8 7
8 7 6 5 4 3 2 1 10 0 9
0 9 8 7 6 5 4 3 2 1 10
1 10 0 9 8 7 6 5 4 3 2
3 2 1 10 0 9 8 7 6 5 4
5 4 3 2 1 10 0 9 8 7 6
7 6 5 4 3 2 1 10 0 9 8
9 8 7 6 5 4 3 2 1 10 0
B = T(A)
10 2 4 6 8 0 1 3 5 7 9
0 1 3 5 7 9 10 2 4 6 8
9 10 2 4 6 8 0 1 3 5 7
8 0 1 3 5 7 9 10 2 4 6
7 9 10 2 4 6 8 0 1 3 5
6 8 0 1 3 5 7 9 10 2 4
5 7 9 10 2 4 6 8 0 1 3
4 6 8 0 1 3 5 7 9 10 2
3 5 7 9 10 2 4 6 8 0 1
2 4 6 8 0 1 3 5 7 9 10
1 3 5 7 9 10 2 4 6 8 0
M = A + 11 * B + 1
121 23 54 75 96 7 17 38 59 80 101
3 13 44 56 87 108 118 29 50 71 92
104 114 25 46 77 89 10 20 41 62 83
95 6 16 37 58 79 110 111 32 53 74
86 107 117 28 49 70 91 2 22 34 65
67 98 9 19 40 61 82 103 113 24 55
57 88 100 120 31 52 73 94 5 15 36
48 69 90 11 12 43 64 85 106 116 27
39 60 81 102 112 33 45 76 97 8 18
30 51 72 93 4 14 35 66 78 109 119
21 42 63 84 105 115 26 47 68 99 1

The Latin Diagonal Squares A can be determined based on the defining equations of the top and bottom row, as provided below for Latin Diagonal Squares type R2:

a(120) = 2 * s1 / 11 - a(121)
a(115) = 3 * s1 / 11 - a(120) - a(121)
a(114) = 2 * s1 / 11 - a(116)
a(113) = 2 * s1 / 11 - a(117)
a(112) = 2 * s1 / 11 - a(118)
a(111) = 2 * s1 / 11 - a(119)

a(7) = 3 * s1 / 11 - a(120) - a(121)
a(6) = 2 * s1 / 11 - a(116)
a(5) = 2 * s1 / 11 - a(117)
a(4) = 2 * s1 / 11 - a(118)
a(3) = 2 * s1 / 11 - a(119)

a(11) = a(119)
a(10) = a(118)
a( 9) = a(117)
a( 8) = a(116)
a( 2) = a(121)
a( 1) = a(120)

The solutions can be obtained by guessing the 5 parameters:

      a(i) for i = 121, 119, 118, 117 and 116

filling out these guesses in the abovementioned equations, and completing the square by copying the first row into the following rows, while shifting 2 columns to the center.

With an appropriate guessing routine (UltraLat11) 3840 Ultra Magic Squares can be constructed based on Diagonal Latin Squares type R2.

Comparable results can be obtained for Diagonal Latin Squares type L2, L3, R3, L4, R4, L5 and R5.

11.2.3 Concentric Magic Squares

Order 11 Concentric Magic Squares M can be constructed based on pairs of Orthogonal Concentric Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 11} and {bj, j = 1 ... 11).

A
a6 a2 a3 a4 a5 a7 a8 a9 a10 a11 a1
a1 a10 a3 a4 a2 a5 a7 a8 a9 a6 a11
a11 a2 a9 a3 a4 a5 a7 a8 a6 a10 a1
a11 a2 a9 a8 a4 a5 a7 a6 a3 a10 a1
a1 a10 a3 a4 a7 a5 a6 a8 a9 a2 a11
a1 a10 a3 a8 a5 a6 a7 a4 a9 a2 a11
a1 a10 a9 a4 a6 a7 a5 a8 a3 a2 a11
a1 a2 a3 a6 a8 a7 a5 a4 a9 a10 a11
a11 a2 a6 a9 a8 a7 a5 a4 a3 a10 a1
a11 a6 a9 a8 a10 a7 a5 a4 a3 a2 a1
a11 a10 a9 a8 a7 a5 a4 a3 a2 a1 a6
B
b1 b1 b1 b11 b11 b11 b11 b1 b1 b11 b6
b11 b6 b10 b10 b2 b2 b2 b10 b10 b2 b1
b10 b9 b6 b9 b3 b9 b9 b3 b3 b3 b2
b9 b8 b8 b6 b4 b8 b8 b4 b4 b4 b3
b8 b7 b7 b5 b6 b5 b7 b7 b5 b5 b4
b7 b5 b5 b4 b7 b6 b5 b8 b7 b7 b5
b5 b2 b4 b7 b5 b7 b6 b5 b8 b10 b7
b4 b4 b3 b8 b8 b4 b4 b6 b9 b8 b8
b3 b3 b9 b3 b9 b3 b3 b9 b6 b9 b9
b2 b10 b2 b2 b10 b10 b10 b2 b2 b6 b10
b6 b11 b11 b1 b1 b1 b1 b11 b11 b1 b11

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
5 1 2 3 4 6 7 8 9 10 0
0 9 2 3 1 4 6 7 8 5 10
10 1 8 2 3 4 6 7 5 9 0
10 1 8 7 3 4 6 5 2 9 0
0 9 2 3 6 4 5 7 8 1 10
0 9 2 7 4 5 6 3 8 1 10
0 9 8 3 5 6 4 7 2 1 10
0 1 2 5 7 6 4 3 8 9 10
10 1 5 8 7 6 4 3 2 9 0
10 5 8 7 9 6 4 3 2 1 0
10 9 8 7 6 4 3 2 1 0 5
B
0 0 0 10 10 10 10 0 0 10 5
10 5 9 9 1 1 1 9 9 1 0
9 8 5 8 2 8 8 2 2 2 1
8 7 7 5 3 7 7 3 3 3 2
7 6 6 4 5 4 6 6 4 4 3
6 4 4 3 6 5 4 7 6 6 4
4 1 3 6 4 6 5 4 7 9 6
3 3 2 7 7 3 3 5 8 7 7
2 2 8 2 8 2 2 8 5 8 8
1 9 1 1 9 9 9 1 1 5 9
5 10 10 0 0 0 0 10 10 0 10
M = A + 11 * B + 1
6 2 3 114 115 117 118 9 10 121 56
111 65 102 103 13 16 18 107 108 17 11
110 90 64 91 26 93 95 30 28 32 12
99 79 86 63 37 82 84 39 36 43 23
78 76 69 48 62 49 72 74 53 46 44
67 54 47 41 71 61 51 81 75 68 55
45 21 42 70 50 73 60 52 80 101 77
34 35 25 83 85 40 38 59 97 87 88
33 24 94 31 96 29 27 92 58 98 89
22 105 20 19 109 106 104 15 14 57 100
66 120 119 8 7 5 4 113 112 1 116

A pair of order 11 Orthogonal Semi-Latin Borders can be constructed for each pair of order 9 Orthogonal Concentric Semi-Latin Squares (A9, B9), as found in Section 9.2.2.

Each pair of order 11 Orthogonal Semi-Latin Borders corresponds with 8 * (9!)2 = 1.053.455.155.200 pairs.

Consequently 2,8410 1028 Concentric Magic Squares can be constructed based on the method described above.

11.2.4 Bordered Magic Squares

Pan Magic Sub Squares Order 4 and 5

Order 11 Bordered Magic Squares M can be constructed based on pairs of Orthogonal Bordered Semi-Latin Squares (A, B) for miscellaneous types of Center Squares.

The example shown below is based on Center Squares with Pan Magic Sub Squares as discussed in Section 9.2.6 and the symbols {ai, i = 1 ... 11} and {bj, j = 1 ... 11).

A
a6 a2 a3 a4 a5 a7 a8 a9 a10 a11 a1
a1 a5 a2 a10 a7 a9 a8 a3 a6 a4 a11
a11 a7 a10 a2 a5 a3 a6 a4 a9 a8 a1
a11 a2 a5 a7 a10 a4 a9 a8 a3 a6 a1
a1 a10 a7 a5 a2 a8 a3 a6 a4 a9 a11
a1 a5 a2 a10 a7 a6 a4 a9 a8 a3 a11
a1 a7 a6 a5 a2 a10 a4 a3 a9 a8 a11
a1 a2 a10 a7 a6 a5 a8 a9 a3 a4 a11
a11 a6 a5 a2 a10 a7 a3 a4 a8 a9 a1
a11 a10 a7 a6 a5 a2 a9 a8 a4 a3 a1
a11 a10 a9 a8 a7 a5 a4 a3 a2 a1 a6
B
b1 b1 b1 b11 b11 b11 b11 b1 b1 b11 b6
b11 b5 b7 b2 b10 b7 b2 b6 b10 b5 b1
b10 b2 b10 b5 b7 b10 b5 b7 b2 b6 b2
b9 b10 b2 b7 b5 b2 b6 b10 b5 b7 b3
b8 b7 b5 b10 b2 b5 b7 b2 b6 b10 b4
b7 b9 b3 b4 b8 b6 b10 b5 b7 b2 b5
b5 b4 b8 b6 b9 b3 b4 b8 b3 b9 b7
b4 b6 b9 b3 b4 b8 b3 b9 b4 b8 b8
b3 b3 b4 b8 b6 b9 b9 b3 b8 b4 b9
b2 b8 b6 b9 b3 b4 b8 b4 b9 b3 b10
b6 b11 b11 b1 b1 b1 b1 b11 b11 b1 b11

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example.

A
5 1 2 3 4 6 7 8 9 10 0
0 4 1 9 6 8 7 2 5 3 10
10 6 9 1 4 2 5 3 8 7 0
10 1 4 6 9 3 8 7 2 5 0
0 9 6 4 1 7 2 5 3 8 10
0 4 1 9 6 5 3 8 7 2 10
0 6 5 4 1 9 3 2 8 7 10
0 1 9 6 5 4 7 8 2 3 10
10 5 4 1 9 6 2 3 7 8 0
10 9 6 5 4 1 8 7 3 2 0
10 9 8 7 6 4 3 2 1 0 5
B
0 0 0 10 10 10 10 0 0 10 5
10 4 6 1 9 6 1 5 9 4 0
9 1 9 4 6 9 4 6 1 5 1
8 9 1 6 4 1 5 9 4 6 2
7 6 4 9 1 4 6 1 5 9 3
6 8 2 3 7 5 9 4 6 1 4
4 3 7 5 8 2 3 7 2 8 6
3 5 8 2 3 7 2 8 3 7 7
2 2 3 7 5 8 8 2 7 3 8
1 7 5 8 2 3 7 3 8 2 9
5 10 10 0 0 0 0 10 10 0 10
M = A + 11 * B + 1
6 2 3 114 115 117 118 9 10 121 56
111 49 68 21 106 75 19 58 105 48 11
110 18 109 46 71 102 50 70 20 63 12
99 101 16 73 54 15 64 107 47 72 23
78 76 51 104 13 52 69 17 59 108 44
67 93 24 43 84 61 103 53 74 14 55
45 40 83 60 90 32 37 80 31 96 77
34 57 98 29 39 82 30 97 36 81 88
33 28 38 79 65 95 91 26 85 42 89
22 87 62 94 27 35 86 41 92 25 100
66 120 119 8 7 5 4 113 112 1 116

A pair of order 11 Orthogonal Semi-Latin Borders can be combined with any pair of order 9 Orthogonal Diagonal-Latin or Semi-Latin Squares (A9, B9), of which a few types have been discussed in Section 9.2.

Each pair of order 11 Orthogonal Semi-Latin Borders corresponds with 8 * (9!)2 = 1.053.455.155.200 pairs.

Consequently 1282 * 11522 * 8 * (9!)2 = 2,29056 1022 Bordered Magic Squares with Center Squares composed of Pan Magic Sub Squares can be constructed, based on the pair of order 11 Orthogonal Semi-Latin Borders shown above.

Diamond Inlays Order 4 and 5

The example shown below is based on Center Squares with order 4 and 5 Diamond Inlays as discussed in Section 9.2.8 and the symbols {ai, i = 1 ... 11} and {bj, j = 1 ... 11).

A
a6 a2 a3 a4 a5 a7 a8 a9 a10 a11 a1
a1 a3 a6 a8 a2 a9 a7 a5 a4 a10 a11
a11 a10 a8 a6 a9 a3 a2 a4 a5 a7 a1
a11 a7 a6 a2 a4 a5 a8 a9 a10 a3 a1
a1 a3 a2 a8 a7 a9 a4 a10 a6 a5 a11
a1 a8 a9 a10 a5 a6 a7 a2 a3 a4 a11
a1 a7 a6 a2 a8 a3 a5 a4 a10 a9 a11
a1 a9 a2 a3 a4 a7 a8 a10 a6 a5 a11
a11 a5 a7 a8 a10 a9 a3 a6 a4 a2 a1
a11 a2 a8 a7 a5 a3 a10 a4 a6 a9 a1
a11 a10 a9 a8 a7 a5 a4 a3 a2 a1 a6
B
b1 b1 b1 b11 b11 b11 b11 b1 b1 b11 b6
b11 b2 b5 b9 b7 b8 b3 b7 b10 b3 b1
b10 b8 b7 b2 b6 b9 b2 b6 b8 b6 b2
b9 b7 b8 b3 b2 b10 b8 b2 b6 b8 b3
b8 b5 b10 b4 b8 b5 b7 b4 b9 b2 b4
b7 b3 b9 b7 b3 b6 b9 b5 b3 b9 b5
b5 b10 b3 b8 b5 b7 b4 b8 b2 b7 b7
b4 b4 b6 b10 b4 b2 b10 b9 b4 b5 b8
b3 b6 b4 b6 b10 b3 b6 b10 b5 b4 b9
b2 b9 b2 b5 b9 b4 b5 b3 b7 b10 b10
b6 b11 b11 b1 b1 b1 b1 b11 b11 b1 b11

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
5 1 2 3 4 6 7 8 9 10 0
0 2 5 7 1 8 6 4 3 9 10
10 9 7 5 8 2 1 3 4 6 0
10 6 5 1 3 4 7 8 9 2 0
0 2 1 7 6 8 3 9 5 4 10
0 7 8 9 4 5 6 1 2 3 10
0 6 5 1 7 2 4 3 9 8 10
0 8 1 2 3 6 7 9 5 4 10
10 4 6 7 9 8 2 5 3 1 0
10 1 7 6 4 2 9 3 5 8 0
10 9 8 7 6 4 3 2 1 0 5
B
0 0 0 10 10 10 10 0 0 10 5
10 1 4 8 6 7 2 6 9 2 0
9 7 6 1 5 8 1 5 7 5 1
8 6 7 2 1 9 7 1 5 7 2
7 4 9 3 7 4 6 3 8 1 3
6 2 8 6 2 5 8 4 2 8 4
4 9 2 7 4 6 3 7 1 6 6
3 3 5 9 3 1 9 8 3 4 7
2 5 3 5 9 2 5 9 4 3 8
1 8 1 4 8 3 4 2 6 9 9
5 10 10 0 0 0 0 10 10 0 10
M = A + 11 * B + 1
6 2 3 114 115 117 118 9 10 121 56
111 14 50 96 68 86 29 71 103 32 11
110 87 74 17 64 91 13 59 82 62 12
99 73 83 24 15 104 85 20 65 80 23
78 47 101 41 84 53 70 43 94 16 44
67 30 97 76 27 61 95 46 25 92 55
45 106 28 79 52 69 38 81 21 75 77
34 42 57 102 37 18 107 98 39 49 88
33 60 40 63 109 31 58 105 48 35 89
22 90 19 51 93 36 54 26 72 108 100
66 120 119 8 7 5 4 113 112 1 116

A pair of order 11 Orthogonal Semi-Latin Borders can be constructed for each pair of order 9 Orthogonal Inlaid Semi-Latin Squares (A9, B9), as found in Section 9.2.8.

Each pair of order 11 Orthogonal Semi-Latin Borders corresponds 8 * (9!)2 = 1.053.455.155.200 pairs.

Consequently 2 * 1090 * 8 * (9!)2 = 2,29653 1015 Bordered Magic Squares with order 4 and 5 Diamond Inlays can be constructed based on the method described above.

11.2.5 Associated Magic Squares
       Square Inlays Order 4 and 5 (Overlapping)

Order 11 Associated Magic Squares M, with order 4 and 5 Magic Square Inlays - as discussed in Section 11.3.3 - can be constructed based on pairs of Orthogonal Inlaid Semi-Latin Squares (A, B).

The example shown below is based on order 4 and 5 Latin Diagonal (Pan) Magic Sub Squares - as discussed in Section 4.2.1 and Section 5.2.2 - for the symbols {ai, i = 1 ... 11} and {bj, j = 1 ... 11}.

A
a8 a4 a8 a4 a8 a4 a8 a4 a8 a4 a6
a10 a7 a10 a6 a1 a3 a1 a3 a7 a10 a8
a11 a6 a1 a3 a7 a10 a10 a7 a3 a1 a7
a9 a3 a7 a10 a6 a1 a3 a1 a10 a7 a9
a7 a10 a6 a1 a3 a7 a7 a10 a1 a3 a11
a2 a1 a3 a7 a10 a6 a2 a5 a9 a11 a10
a1 a9 a11 a2 a5 a5 a9 a11 a6 a2 a5
a3 a5 a2 a11 a9 a11 a6 a2 a5 a9 a3
a5 a11 a9 a5 a2 a2 a5 a9 a11 a6 a1
a4 a2 a5 a9 a11 a9 a11 a6 a2 a5 a2
a6 a8 a4 a8 a4 a8 a4 a8 a4 a8 a4
B
b6 b8 b7 b9 b11 b10 b5 b3 b1 b2 b4
b4 b7 b6 b3 b10 b1 b9 b5 b11 b2 b8
b8 b10 b1 b7 b6 b3 b11 b2 b9 b5 b4
b4 b6 b3 b10 b1 b7 b2 b11 b5 b9 b8
b8 b1 b7 b6 b3 b10 b5 b9 b2 b11 b4
b4 b3 b10 b1 b7 b6 b5 b11 b2 b9 b8
b8 b1 b10 b3 b7 b2 b9 b6 b5 b11 b4
b4 b3 b7 b1 b10 b5 b11 b2 b9 b6 b8
b8 b7 b3 b10 b1 b9 b6 b5 b11 b2 b4
b4 b10 b1 b7 b3 b11 b2 b9 b6 b5 b8
b8 b10 b11 b9 b7 b2 b1 b3 b5 b4 b6

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
7 3 7 3 7 3 7 3 7 3 5
9 6 9 5 0 2 0 2 6 9 7
10 5 0 2 6 9 9 6 2 0 6
8 2 6 9 5 0 2 0 9 6 8
6 9 5 0 2 6 6 9 0 2 10
1 0 2 6 9 5 1 4 8 10 9
0 8 10 1 4 4 8 10 5 1 4
2 4 1 10 8 10 5 1 4 8 2
4 10 8 4 1 1 4 8 10 5 0
3 1 4 8 10 8 10 5 1 4 1
5 7 3 7 3 7 3 7 3 7 3
Sa
22 17
23 28
B
5 7 6 8 10 9 4 2 0 1 3
3 6 5 2 9 0 8 4 10 1 7
7 9 0 6 5 2 10 1 8 4 3
3 5 2 9 0 6 1 10 4 8 7
7 0 6 5 2 9 4 8 1 10 3
3 2 9 0 6 5 4 10 1 8 7
7 0 9 2 6 1 8 5 4 10 3
3 2 6 0 9 4 10 1 8 5 7
7 6 2 9 0 8 5 4 10 1 3
3 9 0 6 2 10 1 8 5 4 7
7 9 10 8 6 1 0 2 4 3 5
Sb
22 23
17 28
M = A + 11 * B + 1
63 81 74 92 118 103 52 26 8 15 39
43 73 65 28 100 3 89 47 117 21 85
88 105 1 69 62 32 120 18 91 45 40
42 58 29 109 6 67 14 111 54 95 86
84 10 72 56 25 106 51 98 12 113 44
35 23 102 7 76 61 46 115 20 99 87
78 9 110 24 71 16 97 66 50 112 38
36 27 68 11 108 55 116 13 93 64 80
82 77 31 104 2 90 60 53 121 17 34
37 101 5 75 33 119 22 94 57 49 79
83 107 114 96 70 19 4 30 48 41 59
Sm
269 274
214 341

The balanced series {0, 1, 2, 3, 4 ... 10} have been split into two unbalanced sub series:

     {0, 2, 6, 9}, {5, 1, 4, 8, 10} and a pair {3, 7}

which have been used for the construction of four Magic Sub Squares and the associated border.

Attachment 11.25.1 shows a few suitable sets (12 ea) of order 4/5 sub series for the integers 0 ... 10.

Attachment 11.25.2 shows the resulting order 11 Associated Magic Squares with Ocerlapping Square Inlays.

Based on the construction method described above 48 * 48 * (4!)2 = 1.327.104 squares can be constructed for each set of order 4/5 series shown in Attachment 11.25.1.

Each square shown in Attachment 11.25.2 corresponds with 1152 * 1152 * (4!)2 = 764.411.904 squares, which can be obtained by permutations within the border and/or selecting other aspects of the order 4 and 5 Sub Squares.

11.2.6 Associated Magic Squares

Diamond Inlays order 5 and 6

Order 11 Associated Magic Squares M, with order 5 and 6 Diamond Inlays, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 11} and {bj, j = 1 ... 11).

A
a9 a11 a5 a4 a8 a2 a6 a1 a3 a7 a10
a5 a2 a1 a8 a4 a11 a9 a3 a6 a7 a10
a6 a2 a11 a1 a5 a9 a10 a8 a3 a4 a7
a7 a11 a8 a4 a9 a2 a3 a1 a5 a10 a6
a3 a10 a1 a6 a7 a9 a4 a8 a5 a11 a2
a11 a9 a2 a4 a5 a6 a7 a8 a10 a3 a1
a10 a1 a7 a4 a8 a3 a5 a6 a11 a2 a9
a6 a2 a7 a11 a9 a10 a3 a8 a4 a1 a5
a5 a8 a9 a4 a2 a3 a7 a11 a1 a10 a6
a2 a5 a6 a9 a3 a1 a8 a4 a11 a10 a7
a2 a5 a9 a11 a6 a10 a4 a8 a7 a1 a3
B = T(A)
b9 b5 b6 b7 b3 b11 b10 b6 b5 b2 b2
b11 b2 b2 b11 b10 b9 b1 b2 b8 b5 b5
b5 b1 b11 b8 b1 b2 b7 b7 b9 b6 b9
b4 b8 b1 b4 b6 b4 b4 b11 b4 b9 b11
b8 b4 b5 b9 b7 b5 b8 b9 b2 b3 b6
b2 b11 b9 b2 b9 b6 b3 b10 b3 b1 b10
b6 b9 b10 b3 b4 b7 b5 b3 b7 b8 b4
b1 b3 b8 b1 b8 b8 b6 b8 b11 b4 b8
b3 b6 b3 b5 b5 b10 b11 b4 b1 b11 b7
b7 b7 b4 b10 b11 b3 b2 b1 b10 b10 b1
b10 b10 b7 b6 b2 b1 b9 b5 b6 b7 b3

The Latin Square B is the transposed square of A (rows and columns exchanged).

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
8 10 4 3 7 1 5 0 2 6 9
4 1 0 7 3 10 8 2 5 6 9
5 1 10 0 4 8 9 7 2 3 6
6 10 7 3 8 1 2 0 4 9 5
2 9 0 5 6 8 3 7 4 10 1
10 8 1 3 4 5 6 7 9 2 0
9 0 6 3 7 2 4 5 10 1 8
5 1 6 10 8 9 2 7 3 0 4
4 7 8 3 1 2 6 10 0 9 5
1 4 5 8 2 0 7 3 10 9 6
1 4 8 10 5 9 3 7 6 0 2
B = T(A)
8 4 5 6 2 10 9 5 4 1 1
10 1 1 10 9 8 0 1 7 4 4
4 0 10 7 0 1 6 6 8 5 8
3 7 0 3 5 3 3 10 3 8 10
7 3 4 8 6 4 7 8 1 2 5
1 10 8 1 8 5 2 9 2 0 9
5 8 9 2 3 6 4 2 6 7 3
0 2 7 0 7 7 5 7 10 3 7
2 5 2 4 4 9 10 3 0 10 6
6 6 3 9 10 2 1 0 9 9 0
9 9 6 5 1 0 8 4 5 6 2
M = 11 * A + B + 1
97 115 50 40 80 22 65 6 27 68 101
55 13 2 88 43 119 89 24 63 71 104
60 12 121 8 45 90 106 84 31 39 75
70 118 78 37 94 15 26 11 48 108 66
30 103 5 64 73 93 41 86 46 113 17
112 99 20 35 53 61 69 87 102 23 10
105 9 76 36 81 29 49 58 117 19 92
56 14 74 111 96 107 28 85 44 4 52
47 83 91 38 16 32 77 114 1 110 62
18 51 59 98 33 3 79 34 120 109 67
21 54 95 116 57 100 42 82 72 7 25

With procedure SemiLat11 numerous pairs of Orthogonal Semi-Latin Associated Squares (A, T(A)) with order 5 and 6 Diamond Inlays can be found, of which a few are shown in Attachment 11.2.6.

Attachment 11.2.7 shows the resulting order 11 Associated Magic Squares with order 5 and 6 Diamond Inlays.

11.3   Magic Squares, Prime Numbers

11.3.1 Pan Magic Squares

When the elements {ai, i = 1 ... 11} and {bj, j = 1 ... 11) of a valid pair of Orthogonal Diagonal Latin Squares (A, B) - as applied in Section 11.2.1 above - comply with following condition:

  • mij = ai + bj = prime for i = 1 ... 11 and j = 1 ... 11 (correlated)

the resulting square M = A + B will be an order 11 Prime Number Pan Magic Square.

Sa = 20407
1 8351 5 71 145 551 665 841 1471 2281 6025
2281 6025 1 8351 5 71 145 551 665 841 1471
841 1471 2281 6025 1 8351 5 71 145 551 665
551 665 841 1471 2281 6025 1 8351 5 71 145
71 145 551 665 841 1471 2281 6025 1 8351 5
8351 5 71 145 551 665 841 1471 2281 6025 1
6025 1 8351 5 71 145 551 665 841 1471 2281
1471 2281 6025 1 8351 5 71 145 551 665 841
665 841 1471 2281 6025 1 8351 5 71 145 551
145 551 665 841 1471 2281 6025 1 8351 5 71
5 71 145 551 665 841 1471 2281 6025 1 8351
Sb = 22860
12 2238 396 276 192 14418 2706 1992 312 222 96
14418 2706 1992 312 222 96 12 2238 396 276 192
96 12 2238 396 276 192 14418 2706 1992 312 222
192 14418 2706 1992 312 222 96 12 2238 396 276
222 96 12 2238 396 276 192 14418 2706 1992 312
276 192 14418 2706 1992 312 222 96 12 2238 396
312 222 96 12 2238 396 276 192 14418 2706 1992
396 276 192 14418 2706 1992 312 222 96 12 2238
1992 312 222 96 12 2238 396 276 192 14418 2706
2238 396 276 192 14418 2706 1992 312 222 96 12
2706 1992 312 222 96 12 2238 396 276 192 14418
Sm = 43267
13 10589 401 347 337 14969 3371 2833 1783 2503 6121
16699 8731 1993 8663 227 167 157 2789 1061 1117 1663
937 1483 4519 6421 277 8543 14423 2777 2137 863 887
743 15083 3547 3463 2593 6247 97 8363 2243 467 421
293 241 563 2903 1237 1747 2473 20443 2707 10343 317
8627 197 14489 2851 2543 977 1063 1567 2293 8263 397
6337 223 8447 17 2309 541 827 857 15259 4177 4273
1867 2557 6217 14419 11057 1997 383 367 647 677 3079
2657 1153 1693 2377 6037 2239 8747 281 263 14563 3257
2383 947 941 1033 15889 4987 8017 313 8573 101 83
2711 2063 457 773 761 853 3709 2677 6301 193 22769

Attachment 11.3 contains miscellaneous correlated series {ai, i = 1 ... 11} and {bj, j = 1 ... 11).

Attachment 11.3.1 contains the resulting Prime Number Pan Magic Squares for miscellaneous Magic Sums (Sm).

Each square shown corresponds with numerous Prime Number Pan Magic Squares.

11.3.2 Symmetric Magic Squares

Order 11 Correlated Balanced Magic Series, suitable for Prime Number Symmetric Magic Squares, have not yet been found.

11.4   Summary

The obtained results regarding the order 11 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Subject

Subroutine

Attachment 11.2.1

Latin Diagonal Squares

-

Attachment 11.2.2

Pan Magic Squares

CnstrSqrs11a

Attachment 11.2.3

Pan Magic Square (Idempotent Square A)

Attachment 11.25.2

Associated Magic Squares, Composed

Attachment 11.2.7

Associated Magic Squares, Diamond Inlays

SemiLat11

-

-

-

Attachment 11.3

Correlated Series

-

Attachment 11.3.1

Prime Number Pan Magic Squares

CnstrSqrs11b

-

-

-

Comparable methods as described above, can be used to construct order 12 (Semi) Latin - and related (Pan) Magic Squares, which will be described in following sections.

Vorige Pagina Volgende Pagina Index About the Author