Office Applications and Entertainment, Latin Squares

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12.0Latin Squares (12 x 12)

A Latin Square of order 12 is a 12 x 12 square filled with 12 different symbols, each occurring only once in each row and only once in each column.

12.1Latin Diagonal Squares (12 x 12)

Latin Diagonal Squares are Latin Squares for which the 12 different symbols occur also only once in each of the main diagonals.

12.2Magic Squares, Natural Numbers

12.2.2 Composed Magic Squares
Order 6 Semi Latin Sub Squares

Order 12 Magic Squares M composed of order 6 Magic Sub Squares can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B).

The example shown below is based on order 6 Semi-Latin Magic Squares, with Symmetrical Diagonals, as discussed in Section 6.2.2, and the symbols {ai, i = 1 ... 12} and {bj, j = 1 ... 12}.

A
a12 a2 a3 a10 a11 a1 a12 a2 a3 a10 a11 a1
a1 a11 a3 a10 a2 a12 a1 a11 a3 a10 a2 a12
a1 a2 a10 a3 a11 a12 a1 a2 a10 a3 a11 a12
a12 a2 a10 a3 a11 a1 a12 a2 a10 a3 a11 a1
a1 a11 a10 a3 a2 a12 a1 a11 a10 a3 a2 a12
a12 a11 a3 a10 a2 a1 a12 a11 a3 a10 a2 a1
a9 a5 a6 a7 a8 a4 a9 a5 a6 a7 a8 a4
a4 a8 a6 a7 a5 a9 a4 a8 a6 a7 a5 a9
a4 a5 a7 a6 a8 a9 a4 a5 a7 a6 a8 a9
a9 a5 a7 a6 a8 a4 a9 a5 a7 a6 a8 a4
a4 a8 a7 a6 a5 a9 a4 a8 a7 a6 a5 a9
a9 a8 a6 a7 a5 a4 a9 a8 a6 a7 a5 a4
B
b1 b12 b1 b12 b12 b1 b4 b9 b4 b9 b9 b4
b2 b2 b11 b11 b2 b11 b5 b5 b8 b8 b5 b8
b10 b3 b3 b3 b10 b10 b7 b6 b6 b6 b7 b7
b3 b10 b10 b10 b3 b3 b6 b7 b7 b7 b6 b6
b11 b11 b2 b2 b11 b2 b8 b8 b5 b5 b8 b5
b12 b1 b12 b1 b1 b12 b9 b4 b9 b4 b4 b9
b4 b9 b4 b9 b9 b4 b1 b12 b1 b12 b12 b1
b5 b5 b8 b8 b5 b8 b2 b2 b11 b11 b2 b11
b7 b6 b6 b6 b7 b7 b10 b3 b3 b3 b10 b10
b6 b7 b7 b7 b6 b6 b3 b10 b10 b10 b3 b3
b8 b8 b5 b5 b8 b5 b11 b11 b2 b2 b11 b2
b9 b4 b9 b4 b4 b9 b12 b1 b12 b1 b1 b12

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

A
11 1 2 9 10 0 11 1 2 9 10 0
0 10 2 9 1 11 0 10 2 9 1 11
0 1 9 2 10 11 0 1 9 2 10 11
11 1 9 2 10 0 11 1 9 2 10 0
0 10 9 2 1 11 0 10 9 2 1 11
11 10 2 9 1 0 11 10 2 9 1 0
8 4 5 6 7 3 8 4 5 6 7 3
3 7 5 6 4 8 3 7 5 6 4 8
3 4 6 5 7 8 3 4 6 5 7 8
8 4 6 5 7 3 8 4 6 5 7 3
3 7 6 5 4 8 3 7 6 5 4 8
8 7 5 6 4 3 8 7 5 6 4 3
B
0 11 0 11 11 0 3 8 3 8 8 3
1 1 10 10 1 10 4 4 7 7 4 7
9 2 2 2 9 9 6 5 5 5 6 6
2 9 9 9 2 2 5 6 6 6 5 5
10 10 1 1 10 1 7 7 4 4 7 4
11 0 11 0 0 11 8 3 8 3 3 8
3 8 3 8 8 3 0 11 0 11 11 0
4 4 7 7 4 7 1 1 10 10 1 10
6 5 5 5 6 6 9 2 2 2 9 9
5 6 6 6 5 5 2 9 9 9 2 2
7 7 4 4 7 4 10 10 1 1 10 1
8 3 8 3 3 8 11 0 11 0 0 11
M = A + 12 * B + 1
12 134 3 142 143 1
13 23 123 130 14 132
109 26 34 27 119 120
36 110 118 111 35 25
121 131 22 15 122 24
144 11 135 10 2 133
48 98 39 106 107 37
49 59 87 94 50 96
73 62 70 63 83 84
72 74 82 75 71 61
85 95 58 51 86 60
108 47 99 46 38 97
45 101 42 103 104 40
52 56 90 91 53 93
76 65 67 66 80 81
69 77 79 78 68 64
88 92 55 54 89 57
105 44 102 43 41 100
9 137 6 139 140 4
16 20 126 127 17 129
112 29 31 30 116 117
33 113 115 114 32 28
124 128 19 18 125 21
141 8 138 7 5 136

The balanced series {0, 1, 2, 3, 4 ... 11} have been split into two balanced sub series:

{0, 1, 2, 9, 10, 11} and {3, 4, 5, 6, 7, 8}

which have been used for the construction of four Magic Sub Squares with Symmetrical Diagonals.

Attachment 12.22.1 shows the unique sets (8 ea) of order 6 balanced lines for the integers 0 ... 11.

Attachment 12.22.2 shows the resulting order 12 Magic Squares composed of Magic Sub Squares with Symmetrical Diagonals.

Based on the construction method described above at least 2 * (6 * 128)2 = 1179648 squares can be constructed for each set of order 6 balanced lines shown in Attachment 12.22.1.

Each square shown in Attachment 12.22.2 corresponds at least with (4!) * (6 * 128)4 = 8,35 1012 squares, which can be obtained by permutation of the sub squares and (or) the rows and columns within the sub squares.

12.2.3 Composed Magic Squares
Order 4 Latin Diagonal Sub Squares

Order 12 (Pan) Magic Squares M composed of order 4 (Pan) Magic Sub Squares can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B).

The example shown below is based on order 4 Latin Diagonal Pan Magic Squares, as discussed in Section 4.2.2, and the symbols {ai, i = 1 ... 12} and {bj, j = 1 ... 12).

A
a1 a2 a11 a12 a3 a4 a9 a10 a5 a6 a7 a8
a12 a11 a2 a1 a10 a9 a4 a3 a8 a7 a6 a5
a2 a1 a12 a11 a4 a3 a10 a9 a6 a5 a8 a7
a11 a12 a1 a2 a9 a10 a3 a4 a7 a8 a5 a6
a1 a2 a11 a12 a3 a4 a9 a10 a5 a6 a7 a8
a12 a11 a2 a1 a10 a9 a4 a3 a8 a7 a6 a5
a2 a1 a12 a11 a4 a3 a10 a9 a6 a5 a8 a7
a11 a12 a1 a2 a9 a10 a3 a4 a7 a8 a5 a6
a1 a2 a11 a12 a3 a4 a9 a10 a5 a6 a7 a8
a12 a11 a2 a1 a10 a9 a4 a3 a8 a7 a6 a5
a2 a1 a12 a11 a4 a3 a10 a9 a6 a5 a8 a7
a11 a12 a1 a2 a9 a10 a3 a4 a7 a8 a5 a6
B
b1 b12 b2 b11 b1 b12 b2 b11 b1 b12 b2 b11
b2 b11 b1 b12 b2 b11 b1 b12 b2 b11 b1 b12
b11 b2 b12 b1 b11 b2 b12 b1 b11 b2 b12 b1
b12 b1 b11 b2 b12 b1 b11 b2 b12 b1 b11 b2
b3 b10 b4 b9 b3 b10 b4 b9 b3 b10 b4 b9
b4 b9 b3 b10 b4 b9 b3 b10 b4 b9 b3 b10
b9 b4 b10 b3 b9 b4 b10 b3 b9 b4 b10 b3
b10 b3 b9 b4 b10 b3 b9 b4 b10 b3 b9 b4
b5 b8 b6 b7 b5 b8 b6 b7 b5 b8 b6 b7
b6 b7 b5 b8 b6 b7 b5 b8 b6 b7 b5 b8
b7 b6 b8 b5 b7 b6 b8 b5 b7 b6 b8 b5
b8 b5 b7 b6 b8 b5 b7 b6 b8 b5 b7 b6

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

A
0 1 10 11 2 3 8 9 4 5 6 7
11 10 1 0 9 8 3 2 7 6 5 4
1 0 11 10 3 2 9 8 5 4 7 6
10 11 0 1 8 9 2 3 6 7 4 5
0 1 10 11 2 3 8 9 4 5 6 7
11 10 1 0 9 8 3 2 7 6 5 4
1 0 11 10 3 2 9 8 5 4 7 6
10 11 0 1 8 9 2 3 6 7 4 5
0 1 10 11 2 3 8 9 4 5 6 7
11 10 1 0 9 8 3 2 7 6 5 4
1 0 11 10 3 2 9 8 5 4 7 6
10 11 0 1 8 9 2 3 6 7 4 5
B = T(A)
0 11 1 10 0 11 1 10 0 11 1 10
1 10 0 11 1 10 0 11 1 10 0 11
10 1 11 0 10 1 11 0 10 1 11 0
11 0 10 1 11 0 10 1 11 0 10 1
2 9 3 8 2 9 3 8 2 9 3 8
3 8 2 9 3 8 2 9 3 8 2 9
8 3 9 2 8 3 9 2 8 3 9 2
9 2 8 3 9 2 8 3 9 2 8 3
4 7 5 6 4 7 5 6 4 7 5 6
5 6 4 7 5 6 4 7 5 6 4 7
6 5 7 4 6 5 7 4 6 5 7 4
7 4 6 5 7 4 6 5 7 4 6 5
M = A + 12 * B + 1
1 134 23 132 3 136 21 130 5 138 19 128
24 131 2 133 22 129 4 135 20 127 6 137
122 13 144 11 124 15 142 9 126 17 140 7
143 12 121 14 141 10 123 16 139 8 125 18
25 110 47 108 27 112 45 106 29 114 43 104
48 107 26 109 46 105 28 111 44 103 30 113
98 37 120 35 100 39 118 33 102 41 116 31
119 36 97 38 117 34 99 40 115 32 101 42
49 86 71 84 51 88 69 82 53 90 67 80
72 83 50 85 70 81 52 87 68 79 54 89
74 61 96 59 76 63 94 57 78 65 92 55
95 60 73 62 93 58 75 64 91 56 77 66

The balanced series {0, 1, 2, 3, 4 ... 11} have been split into three balanced sub series:

{0, 1, 10. 11}, {2, 3, 8, 9} and {4, 5, 6, 7}

which have been used for the construction of nine Pan Magic Sub Squares.

Attachment 12.23.1 shows the unique sets (15 ea) of order 4 balanced lines for the integers 0 ... 11.

Attachment 12.23.2 shows the resulting order 12 Magic Squares composed of Pan Magic Sub Squares.

It can be noticed that the constructed square shown in the example above is Pan Magic and that the integers of every 2 2 sub square sum to s12/3 = 290 (Compact).

Based on the construction method described above 6 * (16)3 = 24576 squares can be constructed for each set of order 4 balanced lines shown in Attachment 12.23.1.

Each square shown in Attachment 12.23.2 corresponds with (9!) * (384)4 = 6,6 1028 squares, which can be obtained by permutation of the sub squares and (or) selecting other aspects of the sub squares.

12.2.4 Composed Magic Squares

Barink Restrictions

Order 12 Composed Pan Magic Barink Squares, as discussed in detail in Section 12.1.1, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 12 Pan Magic Sqaure, composed of order 4 Pan Magic Sub Squares, is that

  • The integers of every 2 2 sub square sum to s12/3 = 290 (Compact)
  • Any 4 consecutive integers starting on any odd place in a row or column sum to s12/3 = 290

A numerical example based on order 4 Latin Diagonal Pan Magic Sub Squares, as discussed in Section 4.2.2, is shown below.

A
1 4 11 6 3 2 9 8 5 0 7 10
10 7 0 5 8 9 2 3 6 11 4 1
0 5 10 7 2 3 8 9 4 1 6 11
11 6 1 4 9 8 3 2 7 10 5 0
1 4 11 6 3 2 9 8 5 0 7 10
10 7 0 5 8 9 2 3 6 11 4 1
0 5 10 7 2 3 8 9 4 1 6 11
11 6 1 4 9 8 3 2 7 10 5 0
1 4 11 6 3 2 9 8 5 0 7 10
10 7 0 5 8 9 2 3 6 11 4 1
0 5 10 7 2 3 8 9 4 1 6 11
11 6 1 4 9 8 3 2 7 10 5 0
B = T(A)
1 10 0 11 1 10 0 11 1 10 0 11
4 7 5 6 4 7 5 6 4 7 5 6
11 0 10 1 11 0 10 1 11 0 10 1
6 5 7 4 6 5 7 4 6 5 7 4
3 8 2 9 3 8 2 9 3 8 2 9
2 9 3 8 2 9 3 8 2 9 3 8
9 2 8 3 9 2 8 3 9 2 8 3
8 3 9 2 8 3 9 2 8 3 9 2
5 6 4 7 5 6 4 7 5 6 4 7
0 11 1 10 0 11 1 10 0 11 1 10
7 4 6 5 7 4 6 5 7 4 6 5
10 1 11 0 10 1 11 0 10 1 11 0
M = A + 12 * B + 1
14 125 12 139 16 123 10 141 18 121 8 143
59 92 61 78 57 94 63 76 55 96 65 74
133 6 131 20 135 4 129 22 137 2 127 24
84 67 86 53 82 69 88 51 80 71 90 49
38 101 36 115 40 99 34 117 42 97 32 119
35 116 37 102 33 118 39 100 31 120 41 98
109 30 107 44 111 28 105 46 113 26 103 48
108 43 110 29 106 45 112 27 104 47 114 25
62 77 60 91 64 75 58 93 66 73 56 95
11 140 13 126 9 142 15 124 7 144 17 122
85 54 83 68 87 52 81 70 89 50 79 72
132 19 134 5 130 21 136 3 128 23 138 1

Attachment 12.24.1 shows 960 ea order 12 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat12a).

Attachment 12.24.2 shows 960 ea order 12 Barink Squares based on M = A + 12 * T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged).

Based on the total collection {Ai, T(Aj)}, 645120 (= 2 * 322560) order 12 Composed Pan Magic Squares can be generated (ref. CnstrSqrs12a).

12.2.5 Composed Magic Squares

John Hendricks

Order 12 Composed Pan Magic Hendricks Squares, as discussed in detail in Section 12.1.4, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 12 Pan Magic Sqaure, composed of order 4 Pan Magic Sub Squares, is that

  • In addition to the nine order 4 Pan Magic Squares, the square contains 16 order 4 Embedded Simple Magic Squares.

A numerical example based on order 4 Latin Diagonal Pan Magic Sub Squares, as discussed in Section 4.2.2, is shown below.

A
5 0 11 6 4 1 10 7 3 2 9 8
8 9 2 3 7 10 1 4 6 11 0 5
0 5 6 11 1 4 7 10 2 3 8 9
9 8 3 2 10 7 4 1 11 6 5 0
5 0 11 6 4 1 10 7 3 2 9 8
8 9 2 3 7 10 1 4 6 11 0 5
0 5 6 11 1 4 7 10 2 3 8 9
9 8 3 2 10 7 4 1 11 6 5 0
5 0 11 6 4 1 10 7 3 2 9 8
8 9 2 3 7 10 1 4 6 11 0 5
0 5 6 11 1 4 7 10 2 3 8 9
9 8 3 2 10 7 4 1 11 6 5 0
B = T(A)
5 8 0 9 5 8 0 9 5 8 0 9
0 9 5 8 0 9 5 8 0 9 5 8
11 2 6 3 11 2 6 3 11 2 6 3
6 3 11 2 6 3 11 2 6 3 11 2
4 7 1 10 4 7 1 10 4 7 1 10
1 10 4 7 1 10 4 7 1 10 4 7
10 1 7 4 10 1 7 4 10 1 7 4
7 4 10 1 7 4 10 1 7 4 10 1
3 6 2 11 3 6 2 11 3 6 2 11
2 11 3 6 2 11 3 6 2 11 3 6
9 0 8 5 9 0 8 5 9 0 8 5
8 5 9 0 8 5 9 0 8 5 9 0
M = A + 12 * B + 1
66 97 12 115 65 98 11 116 64 99 10 117
9 118 63 100 8 119 62 101 7 120 61 102
133 30 79 48 134 29 80 47 135 28 81 46
82 45 136 27 83 44 137 26 84 43 138 25
54 85 24 127 53 86 23 128 52 87 22 129
21 130 51 88 20 131 50 89 19 132 49 90
121 18 91 60 122 17 92 59 123 16 93 58
94 57 124 15 95 56 125 14 96 55 126 13
42 73 36 139 41 74 35 140 40 75 34 141
33 142 39 76 32 143 38 77 31 144 37 78
109 6 103 72 110 5 104 71 111 4 105 70
106 69 112 3 107 68 113 2 108 67 114 1

Attachment 12.25.1 shows 768 ea order 12 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat12b).

Attachment 12.25.2 shows 768 ea order 12 Hendricks Squares based on M = A + 12 * T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged).

Based on the total collection {Ai, T(Aj)}, 405504 (= 2 * 202752) order 12 Composed Pan Magic Squares can be generated (ref. CnstrSqrs12a).

12.2.6 Most Perfect Pan Magic Squares

Bent Diagonals

Order 12 Most Perfect Pan Magic Squares, as discussed in detail in Section 12.2.3, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 12 Most Perfect Pan Magic Squares is that

  • The integers of each 2 2 sub square sum to s12/3 = 290 (Compact)
  • All pairs of integers distant n/2 along a (main) diagonal sum to s12/6 = 145 (Complete)
  • The main bent diagonals and all the bent diagonals parallel to it sum to s12 = 870

A numerical example is shown below:

A
4 6 3 2 1 0 7 5 8 9 10 11
7 5 8 9 10 11 4 6 3 2 1 0
4 6 3 2 1 0 7 5 8 9 10 11
7 5 8 9 10 11 4 6 3 2 1 0
4 6 3 2 1 0 7 5 8 9 10 11
7 5 8 9 10 11 4 6 3 2 1 0
4 6 3 2 1 0 7 5 8 9 10 11
7 5 8 9 10 11 4 6 3 2 1 0
4 6 3 2 1 0 7 5 8 9 10 11
7 5 8 9 10 11 4 6 3 2 1 0
4 6 3 2 1 0 7 5 8 9 10 11
7 5 8 9 10 11 4 6 3 2 1 0
B = T(A)
4 7 4 7 4 7 4 7 4 7 4 7
6 5 6 5 6 5 6 5 6 5 6 5
3 8 3 8 3 8 3 8 3 8 3 8
2 9 2 9 2 9 2 9 2 9 2 9
1 10 1 10 1 10 1 10 1 10 1 10
0 11 0 11 0 11 0 11 0 11 0 11
7 4 7 4 7 4 7 4 7 4 7 4
5 6 5 6 5 6 5 6 5 6 5 6
8 3 8 3 8 3 8 3 8 3 8 3
9 2 9 2 9 2 9 2 9 2 9 2
10 1 10 1 10 1 10 1 10 1 10 1
11 0 11 0 11 0 11 0 11 0 11 0
M = A + 12 * B + 1
53 91 52 87 50 85 56 90 57 94 59 96
80 66 81 70 83 72 77 67 76 63 74 61
41 103 40 99 38 97 44 102 45 106 47 108
32 114 33 118 35 120 29 115 28 111 26 109
17 127 16 123 14 121 20 126 21 130 23 132
8 138 9 142 11 144 5 139 4 135 2 133
89 55 88 51 86 49 92 54 93 58 95 60
68 78 69 82 71 84 65 79 64 75 62 73
101 43 100 39 98 37 104 42 105 46 107 48
116 30 117 34 119 36 113 31 112 27 110 25
125 19 124 15 122 13 128 18 129 22 131 24
140 6 141 10 143 12 137 7 136 3 134 1

Attachment 12.26.1 shows 1440 ea order 12 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat12c).

Attachment 12.26.2 shows 1440 ea order 12 Most Perfect Pan Magic Squares based on M = A + 12 * T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged).

Based on the total collection {Ai, T(Aj)}, 4.147.200 (= 2 * 14402) order 12 Most Perfect Pan Magic Squares can be generated (ref. CnstrSqrs12a).

All 1/3 Rows and 1/3 Columns Sum to s12/3

Also order 12 Compact Complete Pan Magic Squares, with Every 1/3 Row and 1/3 Column summing to s12/3, as discussed in detail in Section 12.2.5, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

A numerical example is shown below:

A
4 1 11 6 5 0 10 7 3 2 9 8
7 10 0 5 6 11 1 4 8 9 2 3
1 4 8 9 2 3 7 10 0 5 6 11
10 7 3 2 9 8 4 1 11 6 5 0
4 1 11 6 5 0 10 7 3 2 9 8
7 10 0 5 6 11 1 4 8 9 2 3
1 4 8 9 2 3 7 10 0 5 6 11
10 7 3 2 9 8 4 1 11 6 5 0
4 1 11 6 5 0 10 7 3 2 9 8
7 10 0 5 6 11 1 4 8 9 2 3
1 4 8 9 2 3 7 10 0 5 6 11
10 7 3 2 9 8 4 1 11 6 5 0
B = T(A)
4 7 1 10 4 7 1 10 4 7 1 10
1 10 4 7 1 10 4 7 1 10 4 7
11 0 8 3 11 0 8 3 11 0 8 3
6 5 9 2 6 5 9 2 6 5 9 2
5 6 2 9 5 6 2 9 5 6 2 9
0 11 3 8 0 11 3 8 0 11 3 8
10 1 7 4 10 1 7 4 10 1 7 4
7 4 10 1 7 4 10 1 7 4 10 1
3 8 0 11 3 8 0 11 3 8 0 11
2 9 5 6 2 9 5 6 2 9 5 6
9 2 6 5 9 2 6 5 9 2 6 5
8 3 11 0 8 3 11 0 8 3 11 0
M = A + 12 * B + 1
53 86 24 127 54 85 23 128 52 87 22 129
20 131 49 90 19 132 50 89 21 130 51 88
134 5 105 46 135 4 104 47 133 6 103 48
83 68 112 27 82 69 113 26 84 67 114 25
65 74 36 115 66 73 35 116 64 75 34 117
8 143 37 102 7 144 38 101 9 142 39 100
122 17 93 58 123 16 92 59 121 18 91 60
95 56 124 15 94 57 125 14 96 55 126 13
41 98 12 139 42 97 11 140 40 99 10 141
32 119 61 78 31 120 62 77 33 118 63 76
110 29 81 70 111 28 80 71 109 30 79 72
107 44 136 3 106 45 137 2 108 43 138 1

Attachment 12.26.3 shows 960 ea order 12 Semi-Latin Pan Magic Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat12d).

Attachment 12.26.4 shows 960 ea order 12 Compact Complete Pan Magic Squares based on M = A + 12 * T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged).

Based on the total collection {Ai, T(Aj)}, 645.120 (= 2 * 322.560) order 12 Compact Complete Pan Magic Squares can be generated (ref. CnstrSqrs12a).

All 1/2 Rows and 1/2 Columns Sum to s12/2

Also order 12 Compact Complete Pan Magic Squares, with Every 1/2 Row and 1/2 Column summing to s12/2, as discussed in detail in Section 12.2.7, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

A numerical example is shown below:

A
10 8 7 6 2 0 1 3 4 5 9 11
1 3 4 5 9 11 10 8 7 6 2 0
10 8 7 6 2 0 1 3 4 5 9 11
1 3 4 5 9 11 10 8 7 6 2 0
10 8 7 6 2 0 1 3 4 5 9 11
1 3 4 5 9 11 10 8 7 6 2 0
10 8 7 6 2 0 1 3 4 5 9 11
1 3 4 5 9 11 10 8 7 6 2 0
10 8 7 6 2 0 1 3 4 5 9 11
1 3 4 5 9 11 10 8 7 6 2 0
10 8 7 6 2 0 1 3 4 5 9 11
1 3 4 5 9 11 10 8 7 6 2 0
B = T(A)
10 1 10 1 10 1 10 1 10 1 10 1
8 3 8 3 8 3 8 3 8 3 8 3
7 4 7 4 7 4 7 4 7 4 7 4
6 5 6 5 6 5 6 5 6 5 6 5
2 9 2 9 2 9 2 9 2 9 2 9
0 11 0 11 0 11 0 11 0 11 0 11
1 10 1 10 1 10 1 10 1 10 1 10
3 8 3 8 3 8 3 8 3 8 3 8
4 7 4 7 4 7 4 7 4 7 4 7
5 6 5 6 5 6 5 6 5 6 5 6
9 2 9 2 9 2 9 2 9 2 9 2
11 0 11 0 11 0 11 0 11 0 11 0
C = 12 * A + B + 1
131 98 95 74 35 2 23 38 59 62 119 134
21 40 57 64 117 136 129 100 93 76 33 4
128 101 92 77 32 5 20 41 56 65 116 137
19 42 55 66 115 138 127 102 91 78 31 6
123 106 87 82 27 10 15 46 51 70 111 142
13 48 49 72 109 144 121 108 85 84 25 12
122 107 86 83 26 11 14 47 50 71 110 143
16 45 52 69 112 141 124 105 88 81 28 9
125 104 89 80 29 8 17 44 53 68 113 140
18 43 54 67 114 139 126 103 90 79 30 7
130 99 94 75 34 3 22 39 58 63 118 135
24 37 60 61 120 133 132 97 96 73 36 1

Attachment 12.28.5 shows 1440 ea order 12 Semi-Latin Pan Magic Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat12f).

Attachment 12.28.6 shows 1440 ea order 12 Pan Magic, Compact, Complete Squares based on M = 12 * A + T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged).

Based on the total collection {Ai, T(Aj)}, 4.147.200 (= 2 * 14402) order 12 Pan Magic, Compact, Complete Squares can be generated (ref. CnstrSqrs12a).

12.2.7 Pan Magic Squares
Franklin Like Properties

Order 12 Pan Magic Squares with Franklin like Properties, as discussed in detail in Section 12.2.1, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 12 Pan Magic Squares is that

  • The integers of each 2 2 sub square sum to s12/3 = 290 (Compact)
  • All 1/3 rows and 1/3 columns sum to s12/3 = 290
  • The main bent diagonals and all the bent diagonals parallel to it sum to s12 = 870

A numerical example is shown below:

A
3 4 5 10 6 0 9 7 2 1 8 11
8 7 6 1 5 11 2 4 9 10 3 0
3 4 5 10 6 0 9 7 2 1 8 11
8 7 6 1 5 11 2 4 9 10 3 0
3 4 5 10 6 0 9 7 2 1 8 11
8 7 6 1 5 11 2 4 9 10 3 0
3 4 5 10 6 0 9 7 2 1 8 11
8 7 6 1 5 11 2 4 9 10 3 0
3 4 5 10 6 0 9 7 2 1 8 11
8 7 6 1 5 11 2 4 9 10 3 0
3 4 5 10 6 0 9 7 2 1 8 11
8 7 6 1 5 11 2 4 9 10 3 0
B = T(A)
3 8 3 8 3 8 3 8 3 8 3 8
4 7 4 7 4 7 4 7 4 7 4 7
5 6 5 6 5 6 5 6 5 6 5 6
10 1 10 1 10 1 10 1 10 1 10 1
6 5 6 5 6 5 6 5 6 5 6 5
0 11 0 11 0 11 0 11 0 11 0 11
9 2 9 2 9 2 9 2 9 2 9 2
7 4 7 4 7 4 7 4 7 4 7 4
2 9 2 9 2 9 2 9 2 9 2 9
1 10 1 10 1 10 1 10 1 10 1 10
8 3 8 3 8 3 8 3 8 3 8 3
11 0 11 0 11 0 11 0 11 0 11 0
M = A + 12 * B + 1
40 101 42 107 43 97 46 104 39 98 45 108
57 92 55 86 54 96 51 89 58 95 52 85
64 77 66 83 67 73 70 80 63 74 69 84
129 20 127 14 126 24 123 17 130 23 124 13
76 65 78 71 79 61 82 68 75 62 81 72
9 140 7 134 6 144 3 137 10 143 4 133
112 29 114 35 115 25 118 32 111 26 117 36
93 56 91 50 90 60 87 53 94 59 88 49
28 113 30 119 31 109 34 116 27 110 33 120
21 128 19 122 18 132 15 125 22 131 16 121
100 41 102 47 103 37 106 44 99 38 105 48
141 8 139 2 138 12 135 5 142 11 136 1

Attachment 12.27.1 shows 1408 ea order 12 Semi-Latin Pan Magic Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat12e).

Attachment 12.27.2 shows 1280 ea order 12 Franklin like Pan Magic Squares based on M = A + 12 * T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged).

Based on the total collection {Ai, T(Aj)}, 2.662.400 (= 2 * 1.331.200) order 12 Franklin like Pan Magic Squares can be generated (ref. CnstrSqrs12a).

12.2.8 Bordered Magic Squares

Order 12 Borders M can be constructed based on pairs of Semi-Latin Orthogonal Borders (A, B).

A pair of order 12 Orthogonal Semi-Latin Borders can be combined with any pair of order 10 Orthogonal Latin or Semi-Latin Squares (A10, B10), of which a few types have been discussed in Section 10.2.

A numerical example of the construction of a Bordered Magic Square based on Simple Magic Latin Center Squares (ref. Section 10.2.2) is shown below:

A
0 1 2 3 4 5 6 7 8 9 10 11
11 10 8 3 1 7 4 2 9 5 6 0
0 8 2 6 9 10 3 5 7 1 4 11
11 1 10 8 2 9 7 6 5 4 3 0
0 3 9 1 4 5 8 10 6 7 2 11
11 5 7 10 8 6 2 1 4 3 9 0
11 4 3 7 6 8 5 9 1 2 10 0
0 9 1 2 3 4 6 7 8 10 5 11
11 2 6 9 5 1 10 4 3 8 7 0
0 6 4 5 7 2 1 3 10 9 8 11
11 7 5 4 10 3 9 8 2 6 1 0
0 10 9 8 7 6 5 4 3 2 1 11
B
0 11 0 11 0 11 11 0 11 0 11 0
1 10 6 5 8 1 3 4 2 7 9 10
2 7 2 4 3 9 10 1 8 5 6 9
3 6 5 8 1 10 9 3 4 2 7 8
4 9 7 2 4 3 1 8 5 6 10 7
5 2 4 3 9 6 7 10 1 8 5 6
6 8 1 10 7 2 5 6 9 3 4 5
7 4 3 9 6 5 2 7 10 1 8 4
8 5 8 1 10 7 6 9 3 4 2 3
9 1 10 6 5 8 4 2 7 9 3 2
10 3 9 7 2 4 8 5 6 10 1 1
11 0 11 0 11 0 0 11 0 11 0 11
M = A + 12 * B + 1
1 134 3 136 5 138 139 8 141 10 143 12
24 131 81 64 98 20 41 51 34 90 115 121
25 93 27 55 46 119 124 18 104 62 77 120
48 74 71 105 15 130 116 43 54 29 88 97
49 112 94 26 53 42 21 107 67 80 123 96
72 30 56 47 117 79 87 122 17 100 70 73
84 101 16 128 91 33 66 82 110 39 59 61
85 58 38 111 76 65 31 92 129 23 102 60
108 63 103 22 126 86 83 113 40 57 32 37
109 19 125 78 68 99 50 28 95 118 45 36
132 44 114 89 35 52 106 69 75 127 14 13
133 11 142 9 140 7 6 137 4 135 2 144

Each pair of order 12 Orthogonal Semi-Latin Borders corresponds with 8 * (10!)2 = 29.030.400 pairs.

Consequently 8 * 1920 * 8 * (10!)2 = 1,61811 1018 Bordered Magic Squares with Simple Magic Center Squares can be constructed, based on the pair of order 12 Orthogonal Non-Latin Borders shown above.

12.2.9 Composed Magic Squares
Associated Border

Order 12 Magic Squares M, composed of order 5 (Pan) Magic Sub Squares and an Associated Border, can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B).

The example shown below is based on order 5 Latin Diagonal Pan Magic Squares, as discussed in Section 5.2.2, for the symbols {ai, i = 1 ... 12} and {bj, j = 1 ... 12}.

A
a1 a7 a8 a9 a11 a3 a3 a11 a9 a8 a7 a1
a12 a2 a4 a5 a6 a10 a2 a4 a5 a6 a10 a12
a12 a5 a6 a10 a2 a4 a5 a6 a10 a2 a4 a12
a12 a10 a2 a4 a5 a6 a10 a2 a4 a5 a6 a12
a12 a4 a5 a6 a10 a2 a4 a5 a6 a10 a2 a12
a12 a6 a10 a2 a4 a5 a6 a10 a2 a4 a5 a12
a1 a11 a9 a8 a7 a3 a11 a9 a8 a7 a3 a1
a1 a8 a7 a3 a11 a9 a8 a7 a3 a11 a9 a1
a1 a3 a11 a9 a8 a7 a3 a11 a9 a8 a7 a1
a1 a9 a8 a7 a3 a11 a9 a8 a7 a3 a11 a1
a1 a7 a3 a11 a9 a8 a7 a3 a11 a9 a8 a1
a12 a6 a5 a4 a2 a10 a10 a2 a4 a5 a6 a12
B
b1 b12 b12 b12 b12 b12 b1 b1 b1 b1 b1 b12
b7 b2 b4 b5 b6 b10 b11 b9 b8 b7 b3 b6
b8 b6 b10 b2 b4 b5 b7 b3 b11 b9 b8 b5
b9 b4 b5 b6 b10 b2 b9 b8 b7 b3 b11 b4
b11 b10 b2 b4 b5 b6 b3 b11 b9 b8 b7 b2
b3 b5 b6 b10 b2 b4 b8 b7 b3 b11 b9 b10
b3 b2 b4 b5 b6 b10 b11 b9 b8 b7 b3 b10
b11 b6 b10 b2 b4 b5 b7 b3 b11 b9 b8 b2
b9 b4 b5 b6 b10 b2 b9 b8 b7 b3 b11 b4
b8 b10 b2 b4 b5 b6 b3 b11 b9 b8 b7 b5
b7 b5 b6 b10 b2 b4 b8 b7 b3 b11 b9 b6
b1 b12 b12 b12 b12 b12 b1 b1 b1 b1 b1 b12

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

A
0 6 7 8 10 2 2 10 8 7 6 0
11 1 3 4 5 9 1 3 4 5 9 11
11 4 5 9 1 3 4 5 9 1 3 11
11 9 1 3 4 5 9 1 3 4 5 11
11 3 4 5 9 1 3 4 5 9 1 11
11 5 9 1 3 4 5 9 1 3 4 11
0 10 8 7 6 2 10 8 7 6 2 0
0 7 6 2 10 8 7 6 2 10 8 0
0 2 10 8 7 6 2 10 8 7 6 0
0 8 7 6 2 10 8 7 6 2 10 0
0 6 2 10 8 7 6 2 10 8 7 0
11 5 4 3 1 9 9 1 3 4 5 11
Sa
22 22
33 33
B
0 11 11 11 11 11 0 0 0 0 0 11
6 1 3 4 5 9 10 8 7 6 2 5
7 5 9 1 3 4 6 2 10 8 7 4
8 3 4 5 9 1 8 7 6 2 10 3
10 9 1 3 4 5 2 10 8 7 6 1
2 4 5 9 1 3 7 6 2 10 8 9
2 1 3 4 5 9 10 8 7 6 2 9
10 5 9 1 3 4 6 2 10 8 7 1
8 3 4 5 9 1 8 7 6 2 10 3
7 9 1 3 4 5 2 10 8 7 6 4
6 4 5 9 1 3 7 6 2 10 8 5
0 11 11 11 11 11 0 0 0 0 0 11
Sb
22 33
22 33
M = A + 12 * B + 1
1 139 140 141 143 135 3 11 9 8 7 133
84 14 40 53 66 118 122 100 89 78 34 72
96 65 114 22 38 52 77 30 130 98 88 60
108 46 50 64 113 18 106 86 76 29 126 48
132 112 17 42 58 62 28 125 102 94 74 24
36 54 70 110 16 41 90 82 26 124 101 120
25 23 45 56 67 111 131 105 92 79 27 109
121 68 115 15 47 57 80 31 123 107 93 13
97 39 59 69 116 19 99 95 81 32 127 37
85 117 20 43 51 71 33 128 103 87 83 49
73 55 63 119 21 44 91 75 35 129 104 61
12 138 137 136 134 142 10 2 4 5 6 144
Sm
291 423
302 434

The balanced series {0, 1, 2, 3, 4 ... 11} have been split into two unbalanced sub series:

{1, 3, 4, 5, 9}, {2, 6, 7, 8, 10} and a pair {0, 11}

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

Attachment 12.28.1 shows a few suitable unique sets (78 ea) of order 5 sub series for the integers 0 ... 11.

Attachment 12.28.2 shows the resulting order 12 Magic Squares composed of Pan Magic Sub Squares.

Based on the construction method described above 2402 * (5!)2 = 8,2944 108 squares can be constructed for each set of order 5 sub series shown in Attachment 12.28.1.

Each square shown in Attachment 12.28.2 corresponds with 288004 * (5!)2 = 9,90678 1021 squares, which can be obtained by permutations within the border and/or selecting other aspects of the Pan Magic Sub Squares.

12.2.10 Associated Magic Squares
Pan Magic Square Inlays

Comparable with Section 12.2.9 above, order 12 Associated Magic Squares M, with order 5 Pan Magic Square Inlays, can be constructed based on pairs of Orthogonal Inlaid Semi-Latin Squares (A, B).

Attachment 12.28.3 shows a few suitable unique sets (6 ea) of order 5 sub series for the integers 0 ... 11.

Attachment 12.28.4 shows the resulting order 12 Associated Magic Squares with Pan Magic Square Inlays.

Based on the construction method described above 240 * (5!)2 = 3,456 106 squares can be constructed for each set of order 5 sub series shown in Attachment 12.28.3.

Each square shown in Attachment 12.28.4 corresponds with 288002 * (5!)2 = 1,1944 1013 squares, which can be obtained by permutations within the border and/or selecting other aspects of two of the Pan Magic Sub Squares.

12.3Magic Squares, Prime Numbers

12.3.1 Simple Magic Squares

When the elements {ai, i = 1 ... 12} and {bj, j = 1 ... 12) of a valid pair of Orthogonal Diagonal Latin Squares (A, B) comply with following condition:

  • mij = ai + bj = prime for i = 1 ... 12 and j = 1 ... 12 (correlated)

the resulting square M = A + B will be an order 12 Prime Number Simple Magic Square.

Sa = 76198
1 3 43 73 211 381 463 1393 4333 9621 21903 37773
37773 21903 9621 4333 1393 463 381 211 73 43 3 1
43 73 211 381 1 3 21903 37773 463 1393 4333 9621
9621 4333 1393 463 37773 21903 3 1 381 211 73 43
211 381 1 3 43 73 4333 9621 21903 37773 463 1393
1393 463 37773 21903 9621 4333 73 43 3 1 381 211
73 43 381 211 3 1 37773 21903 1393 463 9621 4333
4333 9621 463 1393 21903 37773 1 3 211 381 43 73
3 1 73 43 381 211 1393 463 9621 4333 37773 21903
21903 37773 4333 9621 463 1393 211 381 43 73 1 3
381 211 3 1 73 43 9621 4333 37773 21903 1393 463
463 1393 21903 37773 4333 9621 43 73 1 3 211 381

Sb = 160692
40 58 190 1018 1150 1930 1996 2266 6448 37336 51340 56920
190 1018 1150 1930 40 58 51340 56920 1996 2266 6448 37336
1930 1150 58 40 1018 190 37336 6448 56920 51340 2266 1996
51340 56920 6448 37336 1996 2266 1150 1930 190 1018 40 58
1996 2266 51340 56920 6448 37336 190 1018 40 58 1150 1930
56920 51340 37336 6448 2266 1996 1930 1150 1018 190 58 40
58 40 1018 190 1930 1150 2266 1996 37336 6448 56920 51340
1018 190 1930 1150 58 40 56920 51340 2266 1996 37336 6448
37336 6448 2266 1996 56920 51340 58 40 1930 1150 1018 190
1150 1930 40 58 190 1018 6448 37336 51340 56920 1996 2266
6448 37336 1996 2266 51340 56920 40 58 1150 1930 190 1018
2266 1996 56920 51340 37336 6448 1018 190 58 40 1930 1150

Sm = 236890
41 61 233 1091 1361 2311 2459 3659 10781 46957 73243 94693
37963 22921 10771 6263 1433 521 51721 57131 2069 2309 6451 37337
1973 1223 269 421 1019 193 59239 44221 57383 52733 6599 11617
60961 61253 7841 37799 39769 24169 1153 1931 571 1229 113 101
2207 2647 51341 56923 6491 37409 4523 10639 21943 37831 1613 3323
58313 51803 75109 28351 11887 6329 2003 1193 1021 191 439 251
131 83 1399 401 1933 1151 40039 23899 38729 6911 66541 55673
5351 9811 2393 2543 21961 37813 56921 51343 2477 2377 37379 6521
37339 6449 2339 2039 57301 51551 1451 503 11551 5483 38791 22093
23053 39703 4373 9679 653 2411 6659 37717 51383 56993 1997 2269
6829 37547 1999 2267 51413 56963 9661 4391 38923 23833 1583 1481
2729 3389 78823 89113 41669 16069 1061 263 59 43 2141 1531

Attachment 12.3 contains miscellaneous correlated series {ai, i = 1 ... 12} and {bj, j = 1 ... 12).

Attachment 12.3.1 contains the resulting Prime Number Pan Magic Squares for miscellaneous Magic Sums (Sm).

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

12.3.2 Symmetric Magic Squares

Order 12 Correlated Balanced Magic Series, suitable for Prime Number Symmetric Magic Squares, have not yet been found.

12.4Summary

The obtained results regarding the order 12 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Subject

Subroutine

Attachment 12.22.2

Composed Magic Squares

CnstrSqrs12a

Attachment 12.23.2

Composed Magic Squares

Attachment 12.28.6

Pan Magic Squares, Compact, Complete

-

-

-

Attachment 12.24.1

Composed Pan Magic Semi-Latin Squares

CompLat12a

Attachment 12.25.1

Composed Pan Magic Semi-Latin Squares

CompLat12b

Attachment 12.26.1

Most Perfect Pan Magic Semi-Latin Squares (1)

CompLat12c

Attachment 12.26.3

Most Perfect Pan Magic Semi-Latin Squares (2)

CompLat12d

Attachment 12.27.1

Franklin Like Semi-Latin Pan Magic Squares

CompLat12e

Attachment 12.28.5

Semi-Latin Pan Magic Squares, Compact, Complete

CompLat12f

-

-

-

Attachment 12.28.2

Composed Magic Squares, Associated Border

-

Attachment 12.28.4

Composed Magic Squares, Associated

-

-

-

-

Attachment 12.3

Correlated Series

-

Attachment 12.3.1

Prime Number Pan Magic Squares

CnstrSqrs12b

-

-

-

Comparable methods as described above, can be used to construct order 13 (Semi) Latin - and related (Pan) Magic Squares, which will be described in following sections.


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