Office Applications and Entertainment, Latin Squares

Vorige Pagina Volgende Pagina Index About the Author

14.0   Latin Squares (14 x 14)

A Latin Square of order 14 is a 14 x 14 square filled with 14 different symbols, each occurring only once in each row and only once in each column.

14.1   Latin Diagonal Squares (14 x 14)

Latin Diagonal Squares are Latin Squares for which the 14 different symbols occur also only once in each of the main diagonals.

14.2   Magic Squares, Natural Numbers

14.2.1 General

Although Euler postulated that sets of Greco Latin Squares did not exist for oddly even orders n ≡ 2 (mod 4), R. C. Bose and S. Shrinkhade proved the contrary for all n >= 14 (1959/1960).

The first pair of Orthogonal Latin Diagonal Squares of order 14 was constructed by L. ZHU (1982) and has been used as an example in Section 14.2.2 below.

14.2.2 Simple Magic Squares

An example of the construction of an order 14 Simple Magic Square M based on a pair of Orthogonal Latin Diagonal Squares (A, B), is shown below for the symbols {ai, i = 1 ... 14} and {bj, j = 1 ... 14).

A
a1 a4 a11 a5 a8 a2 a9 a3 a7 a14 a12 a13 a6 a10
a14 a2 a5 a11 a6 a3 a10 a4 a8 a12 a13 a7 a1 a9
a12 a14 a3 a6 a11 a4 a1 a5 a9 a13 a8 a2 a10 a7
a13 a12 a14 a4 a7 a5 a2 a6 a10 a9 a3 a1 a8 a11
a10 a13 a12 a14 a5 a6 a3 a7 a1 a4 a2 a9 a11 a8
a7 a8 a9 a10 a1 a11 a13 a14 a12 a6 a5 a4 a3 a2
a9 a10 a1 a2 a3 a14 a12 a11 a13 a8 a7 a6 a5 a4
a6 a7 a8 a9 a10 a12 a14 a13 a11 a5 a4 a3 a2 a1
a8 a9 a10 a1 a2 a13 a11 a12 a14 a7 a6 a5 a4 a3
a3 a11 a4 a7 a9 a1 a8 a2 a6 a10 a14 a12 a13 a5
a11 a3 a6 a8 a4 a10 a7 a1 a5 a2 a9 a14 a12 a13
a2 a5 a7 a3 a13 a9 a6 a10 a4 a11 a1 a8 a14 a12
a4 a6 a2 a13 a12 a8 a5 a9 a3 a1 a11 a10 a7 a14
a5 a1 a13 a12 a14 a7 a4 a8 a2 a3 a10 a11 a9 a6
B `
b1 b13 b10 b14 b11 b6 b7 b9 b8 b3 b4 b2 b12 b5
b4 b2 b13 b1 b14 b7 b8 b10 b9 b5 b3 b12 b6 b11
b6 b5 b3 b13 b2 b8 b9 b1 b10 b4 b12 b7 b11 b14
b5 b7 b6 b4 b13 b9 b10 b2 b1 b12 b8 b11 b14 b3
b12 b6 b8 b7 b5 b10 b1 b3 b2 b9 b11 b14 b4 b13
b9 b10 b1 b2 b3 b11 b14 b12 b13 b8 b7 b6 b5 b4
b2 b3 b4 b5 b6 b13 b12 b14 b11 b1 b10 b9 b8 b7
b3 b4 b5 b6 b7 b14 b11 b13 b12 b2 b1 b10 b9 b8
b7 b8 b9 b10 b1 b12 b13 b11 b14 b6 b5 b4 b3 b2
b13 b9 b14 b11 b4 b5 b6 b8 b7 b10 b2 b3 b1 b12
b8 b14 b11 b3 b12 b4 b5 b7 b6 b13 b9 b1 b2 b10
b14 b11 b2 b12 b9 b3 b4 b6 b5 b7 b13 b8 b10 b1
b11 b1 b12 b8 b10 b2 b3 b5 b4 b14 b6 b13 b7 b9
b10 b12 b7 b9 b8 b1 b2 b4 b3 b11 b14 b5 b13 b6

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

A
0 3 10 4 7 1 8 2 6 13 11 12 5 9
13 1 4 10 5 2 9 3 7 11 12 6 0 8
11 13 2 5 10 3 0 4 8 12 7 1 9 6
12 11 13 3 6 4 1 5 9 8 2 0 7 10
9 12 11 13 4 5 2 6 0 3 1 8 10 7
6 7 8 9 0 10 12 13 11 5 4 3 2 1
8 9 0 1 2 13 11 10 12 7 6 5 4 3
5 6 7 8 9 11 13 12 10 4 3 2 1 0
7 8 9 0 1 12 10 11 13 6 5 4 3 2
2 10 3 6 8 0 7 1 5 9 13 11 12 4
10 2 5 7 3 9 6 0 4 1 8 13 11 12
1 4 6 2 12 8 5 9 3 10 0 7 13 11
3 5 1 12 11 7 4 8 2 0 10 9 6 13
4 0 12 11 13 6 3 7 1 2 9 10 8 5
B
0 12 9 13 10 5 6 8 7 2 3 1 11 4
3 1 12 0 13 6 7 9 8 4 2 11 5 10
5 4 2 12 1 7 8 0 9 3 11 6 10 13
4 6 5 3 12 8 9 1 0 11 7 10 13 2
11 5 7 6 4 9 0 2 1 8 10 13 3 12
8 9 0 1 2 10 13 11 12 7 6 5 4 3
1 2 3 4 5 12 11 13 10 0 9 8 7 6
2 3 4 5 6 13 10 12 11 1 0 9 8 7
6 7 8 9 0 11 12 10 13 5 4 3 2 1
12 8 13 10 3 4 5 7 6 9 1 2 0 11
7 13 10 2 11 3 4 6 5 12 8 0 1 9
13 10 1 11 8 2 3 5 4 6 12 7 9 0
10 0 11 7 9 1 2 4 3 13 5 12 6 8
9 11 6 8 7 0 1 3 2 10 13 4 12 5
M = A + 14 * B + 1
1 172 137 187 148 72 93 115 105 42 54 27 160 66
56 16 173 11 188 87 108 130 120 68 41 161 71 149
82 70 31 174 25 102 113 5 135 55 162 86 150 189
69 96 84 46 175 117 128 20 10 163 101 141 190 39
164 83 110 98 61 132 3 35 15 116 142 191 53 176
119 134 9 24 29 151 195 168 180 104 89 74 59 44
23 38 43 58 73 182 166 193 153 8 133 118 103 88
34 49 64 79 94 194 154 181 165 19 4 129 114 99
92 107 122 127 2 167 179 152 196 77 62 47 32 17
171 123 186 147 51 57 78 100 90 136 28 40 13 159
109 185 146 36 158 52 63 85 75 170 121 14 26 139
184 145 21 157 125 37 48 80 60 95 169 106 140 12
144 6 156 111 138 22 33 65 45 183 81 178 91 126
131 155 97 124 112 7 18 50 30 143 192 67 177 76

Each orthogonal set (A, B) corresponds with 322560 transformations, as described below.

  • Any line n can be interchanged with line (15 - n). The possible number of transformations is 27 = 128.
    It should be noted that for each square the 180o rotated aspect is included in this collection.

  • Any permutation can be applied to the lines 1, 2, ... 7, provided that the same permutation is applied to the lines 14, 13, ... 8. The possible number of transformations is 7! = 5040.

The resulting number of transformations, excluding the 180o rotated aspects, is 128/2 * 5040 = 322560.

Attachment 14.2.2 shows alternatively the construction of a Simple Magic Square M based on a Self Orthogonal Diagonal Latin Square A.

14.2.3 Simple Magic Squares

Symmetrical Diagonals

An example of the construction of an order 14 Simple Magic Square M - with Symmetrical Diagonals - based on pairs of Orthogonal Semi-Latin Squares (A, B), is shown below for the symbols {ai, i = 1 ... 14} and {bj, j = 1 ... 14}.

A
a1 a1 a1 a1 a14 a14 a14 a14 a14 a1 a14 a1 a14 a1
a2 a2 a13 a2 a13 a13 a13 a13 a13 a2 a13 a2 a2 a2
a12 a12 a3 a12 a3 a3 a3 a3 a3 a12 a12 a3 a12 a12
a4 a4 a4 a4 a11 a11 a11 a11 a11 a11 a4 a11 a4 a4
a10 a10 a10 a10 a5 a5 a5 a5 a5 a5 a5 a10 a10 a10
a9 a9 a6 a9 a6 a6 a6 a6 a6 a9 a6 a9 a9 a9
a8 a8 a8 a7 a8 a7 a7 a7 a7 a7 a8 a7 a8 a8
a7 a7 a7 a8 a7 a8 a8 a8 a8 a8 a7 a8 a7 a7
a6 a6 a9 a6 a9 a9 a9 a9 a9 a6 a9 a6 a6 a6
a5 a5 a5 a5 a10 a10 a10 a10 a10 a10 a10 a5 a5 a5
a11 a11 a11 a11 a4 a4 a4 a4 a4 a4 a11 a4 a11 a11
a3 a3 a12 a3 a12 a12 a12 a12 a12 a3 a3 a12 a3 a3
a13 a13 a2 a13 a2 a2 a2 a2 a2 a13 a2 a13 a13 a13
a14 a14 a14 a14 a1 a1 a1 a1 a1 a14 a1 a14 a1 a14
B = T(A)
b1 b2 b12 b4 b10 b9 b8 b7 b6 b5 b11 b3 b13 b14
b1 b2 b12 b4 b10 b9 b8 b7 b6 b5 b11 b3 b13 b14
b1 b13 b3 b4 b10 b6 b8 b7 b9 b5 b11 b12 b2 b14
b1 b2 b12 b4 b10 b9 b7 b8 b6 b5 b11 b3 b13 b14
b14 b13 b3 b11 b5 b6 b8 b7 b9 b10 b4 b12 b2 b1
b14 b13 b3 b11 b5 b6 b7 b8 b9 b10 b4 b12 b2 b1
b14 b13 b3 b11 b5 b6 b7 b8 b9 b10 b4 b12 b2 b1
b14 b13 b3 b11 b5 b6 b7 b8 b9 b10 b4 b12 b2 b1
b14 b13 b3 b11 b5 b6 b7 b8 b9 b10 b4 b12 b2 b1
b1 b2 b12 b11 b5 b9 b7 b8 b6 b10 b4 b3 b13 b14
b14 b13 b12 b4 b5 b6 b8 b7 b9 b10 b11 b3 b2 b1
b1 b2 b3 b11 b10 b9 b7 b8 b6 b5 b4 b12 b13 b14
b14 b2 b12 b4 b10 b9 b8 b7 b6 b5 b11 b3 b13 b1
b1 b2 b12 b4 b10 b9 b8 b7 b6 b5 b11 b3 b13 b14

The Semi-Latin Square A has Semi-Latin Rows, Latin Columns and Latin Diagonals (Symmetrical).
The Semi-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
0 0 0 0 13 13 13 13 13 0 13 0 13 0
1 1 12 1 12 12 12 12 12 1 12 1 1 1
11 11 2 11 2 2 2 2 2 11 11 2 11 11
3 3 3 3 10 10 10 10 10 10 3 10 3 3
9 9 9 9 4 4 4 4 4 4 4 9 9 9
8 8 5 8 5 5 5 5 5 8 5 8 8 8
7 7 7 6 7 6 6 6 6 6 7 6 7 7
6 6 6 7 6 7 7 7 7 7 6 7 6 6
5 5 8 5 8 8 8 8 8 5 8 5 5 5
4 4 4 4 9 9 9 9 9 9 9 4 4 4
10 10 10 10 3 3 3 3 3 3 10 3 10 10
2 2 11 2 11 11 11 11 11 2 2 11 2 2
12 12 1 12 1 1 1 1 1 12 1 12 12 12
13 13 13 13 0 0 0 0 0 13 0 13 0 13
B = T(A)
0 1 11 3 9 8 7 6 5 4 10 2 12 13
0 1 11 3 9 8 7 6 5 4 10 2 12 13
0 12 2 3 9 5 7 6 8 4 10 11 1 13
0 1 11 3 9 8 6 7 5 4 10 2 12 13
13 12 2 10 4 5 7 6 8 9 3 11 1 0
13 12 2 10 4 5 6 7 8 9 3 11 1 0
13 12 2 10 4 5 6 7 8 9 3 11 1 0
13 12 2 10 4 5 6 7 8 9 3 11 1 0
13 12 2 10 4 5 6 7 8 9 3 11 1 0
0 1 11 10 4 8 6 7 5 9 3 2 12 13
13 12 11 3 4 5 7 6 8 9 10 2 1 0
0 1 2 10 9 8 6 7 5 4 3 11 12 13
13 1 11 3 9 8 7 6 5 4 10 2 12 0
0 1 11 3 9 8 7 6 5 4 10 2 12 13
M = A + 14 * B + 1
1 15 155 43 140 126 112 98 84 57 154 29 182 183
2 16 167 44 139 125 111 97 83 58 153 30 170 184
12 180 31 54 129 73 101 87 115 68 152 157 26 194
4 18 158 46 137 123 95 109 81 67 144 39 172 186
192 178 38 150 61 75 103 89 117 131 47 164 24 10
191 177 34 149 62 76 90 104 118 135 48 163 23 9
190 176 36 147 64 77 91 105 119 133 50 161 22 8
189 175 35 148 63 78 92 106 120 134 49 162 21 7
188 174 37 146 65 79 93 107 121 132 51 160 20 6
5 19 159 145 66 122 94 108 80 136 52 33 173 187
193 179 165 53 60 74 102 88 116 130 151 32 25 11
3 17 40 143 138 124 96 110 82 59 45 166 171 185
195 27 156 55 128 114 100 86 72 69 142 41 181 13
14 28 168 56 127 113 99 85 71 70 141 42 169 196

The amount of Semi-Latin Squares with Symmetrical Diagonals is so substantial, that the example shown above is based on following (restricting) properties:

  • Row 1 and 14 contain the integers 0 and 13
  • Row 2 and 13 contain the integers 1 and 12
  • Row 3 and 12 contain the integers 2 and 11
  • Row 4 and 11 contain the integers 3 and 10
  • Row 5 and 10 contain the integers 4 and 9
  • Row 6 and  9 contain the integers 5 and 8
  • Row 7 and  8 contain the integers 6 and 7

The number of Orthogonal Sets (A, B) which can be generated under these conditions, with both diagonals and five top rows constant, is 450864 (ref. SemiLat14).

Composed Border (1)

With the 'Check Outer Border' option activated it is possible to filter Orthogonal Semi-Latin Squares (A, B) with Composed Borders from the collection described above, as illustrated by following numerical example (ref. SemiLat14):

A
0 0 0 0 13 13 13 13 13 0 13 0 13 0
1 1 12 1 12 12 12 12 12 1 12 1 1 1
2 11 2 2 2 2 11 11 11 11 11 2 11 2
3 10 3 3 3 10 10 10 10 10 3 3 10 3
4 9 4 9 4 4 9 9 9 4 9 4 9 4
5 5 5 5 8 5 8 8 5 8 8 5 8 8
6 6 6 6 6 7 6 6 7 7 7 7 7 7
7 7 7 7 7 6 7 7 6 6 6 6 6 6
8 8 8 8 5 8 5 5 8 5 5 8 5 5
9 4 9 4 9 9 4 4 4 9 4 9 4 9
10 3 10 10 10 3 3 3 3 3 10 10 3 10
11 2 11 11 11 11 2 2 2 2 2 11 2 11
12 12 1 12 1 1 1 1 1 12 1 12 12 12
13 13 13 13 0 0 0 0 0 13 0 13 0 13
B = T(A)
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 11 10 9 5 6 7 8 4 3 2 12 13
0 12 2 3 4 5 6 7 8 9 10 11 1 13
0 1 2 3 9 5 6 7 8 4 10 11 12 13
13 12 2 3 4 8 6 7 5 9 10 11 1 0
13 12 2 10 4 5 7 6 8 9 3 11 1 0
13 12 11 10 9 8 6 7 5 4 3 2 1 0
13 12 11 10 9 8 6 7 5 4 3 2 1 0
13 12 11 10 9 5 7 6 8 4 3 2 1 0
0 1 11 10 4 8 7 6 5 9 3 2 12 13
13 12 11 3 9 8 7 6 5 4 10 2 1 0
0 1 2 3 4 5 7 6 8 9 10 11 12 13
13 1 11 10 9 8 7 6 5 4 3 2 12 0
0 1 2 3 4 8 7 6 5 9 10 11 12 13
M = A + 14 * B + 1
1 15 29 43 70 84 98 112 126 127 154 155 182 183
2 16 167 142 139 83 97 111 125 58 55 30 170 184
3 180 31 45 59 73 96 110 124 138 152 157 26 185
4 25 32 46 130 81 95 109 123 67 144 158 179 186
187 178 33 52 61 117 94 108 80 131 150 159 24 5
188 174 34 146 65 76 107 93 118 135 51 160 23 9
189 175 161 147 133 120 91 105 78 64 50 36 22 8
190 176 162 148 134 119 92 106 77 63 49 35 21 7
191 177 163 149 132 79 104 90 121 62 48 37 20 6
10 19 164 145 66 122 103 89 75 136 47 38 173 192
193 172 165 53 137 116 102 88 74 60 151 39 18 11
12 17 40 54 68 82 101 87 115 129 143 166 171 194
195 27 156 153 128 114 100 86 72 69 44 41 181 13
14 28 42 56 57 113 99 85 71 140 141 168 169 196

The number of Orthogonal Sets (A, B), which can be filtered from the collection with both diagonals and five top rows constant, is 17712 out of 450864.

Composed Border (2)

With also the 'Check Inner Border' option activated it is possible to filter Orthogonal Semi-Latin Squares (A, B) with Double Composed Borders from the collection described above, as illustrated below (ref. SemiLat14):

A
0 0 0 0 13 13 13 13 13 0 13 0 13 0
1 1 12 1 12 12 12 12 12 1 12 1 1 1
2 11 2 2 2 2 11 11 11 11 11 2 11 2
3 10 3 3 3 10 10 10 10 10 3 3 10 3
4 9 4 9 4 4 9 9 9 4 9 4 9 4
5 5 5 8 8 5 5 8 5 8 8 5 8 8
6 6 6 6 6 7 6 6 7 7 7 7 7 7
7 7 7 7 7 6 7 7 6 6 6 6 6 6
8 8 8 5 5 8 8 5 8 5 5 8 5 5
9 4 9 4 9 9 4 4 4 9 4 9 4 9
10 3 10 10 10 3 3 3 3 3 10 10 3 10
11 2 11 11 11 11 2 2 2 2 2 11 2 11
12 12 1 12 1 1 1 1 1 12 1 12 12 12
13 13 13 13 0 0 0 0 0 13 0 13 0 13
B = T(A)
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 11 10 9 5 6 7 8 4 3 2 12 13
0 12 2 3 4 5 6 7 8 9 10 11 1 13
0 1 2 3 9 8 6 7 5 4 10 11 12 13
13 12 2 3 4 8 6 7 5 9 10 11 1 0
13 12 2 10 4 5 7 6 8 9 3 11 1 0
13 12 11 10 9 5 6 7 8 4 3 2 1 0
13 12 11 10 9 8 6 7 5 4 3 2 1 0
13 12 11 10 9 5 7 6 8 4 3 2 1 0
0 1 11 10 4 8 7 6 5 9 3 2 12 13
13 12 11 3 9 8 7 6 5 4 10 2 1 0
0 1 2 3 4 5 7 6 8 9 10 11 12 13
13 1 11 10 9 8 7 6 5 4 3 2 12 0
0 1 2 3 4 8 7 6 5 9 10 11 12 13
M = A + 14 * B + 1
1 15 29 43 70 84 98 112 126 127 154 155 182 183
2 16 167 142 139 83 97 111 125 58 55 30 170 184
3 180 31 45 59 73 96 110 124 138 152 157 26 185
4 25 32 46 130 123 95 109 81 67 144 158 179 186
187 178 33 52 61 117 94 108 80 131 150 159 24 5
188 174 34 149 65 76 104 93 118 135 51 160 23 9
189 175 161 147 133 78 91 105 120 64 50 36 22 8
190 176 162 148 134 119 92 106 77 63 49 35 21 7
191 177 163 146 132 79 107 90 121 62 48 37 20 6
10 19 164 145 66 122 103 89 75 136 47 38 173 192
193 172 165 53 137 116 102 88 74 60 151 39 18 11
12 17 40 54 68 82 101 87 115 129 143 166 171 194
195 27 156 153 128 114 100 86 72 69 44 41 181 13
14 28 42 56 57 113 99 85 71 140 141 168 169 196

The number of Orthogonal Sets (A, B), which can be filtered from the collection with Composed Outer Borders is 1728 out of 17712.

14.2.4 Almost Associated Magic Squares

Order 14 Almost Associated Magic Squares composed out of

  • An order 14 Associated Border and
  • An order 10 Almost Associated Center Square (ref. Section 10.2.4)

can be constructed based on Orthogonal Semi-Latin Squares (A, B) as illustrted by following numerical example:

A
13 12 13 12 12 0 1 0 0 13 13 1 1 0
0 1 1 13 1 12 0 1 12 0 12 13 12 13
11 9 11 9 11 3 2 2 11 3 10 3 3 3
9 4 2 10 2 11 10 10 3 10 2 5 2 11
8 3 3 8 9 4 9 6 4 7 8 7 5 10
7 2 4 7 5 8 5 5 8 8 11 4 8 9
6 6 7 4 4 7 6 9 7 6 7 8 6 8
5 7 5 6 7 6 4 7 6 9 9 6 7 7
4 5 9 2 8 5 8 8 5 5 6 9 11 6
3 8 6 5 6 9 7 4 9 4 5 10 10 5
2 11 8 11 3 10 3 3 2 11 3 11 9 4
10 10 10 3 10 2 11 11 10 2 4 2 4 2
0 1 0 1 13 1 12 13 1 12 0 12 12 13
13 12 12 0 0 13 13 12 13 1 1 0 1 0
B = T(A)
13 0 11 9 8 7 6 5 4 3 2 10 0 13
12 1 9 4 3 2 6 7 5 8 11 10 1 12
13 1 11 2 3 4 7 5 9 6 8 10 0 12
12 13 9 10 8 7 4 6 2 5 11 3 1 0
12 1 11 2 9 5 7 4 8 6 3 10 13 0
0 12 3 11 6 8 4 9 5 7 10 2 1 13
1 0 2 10 7 5 6 4 8 9 3 11 12 13
0 1 2 10 4 5 9 7 8 6 3 11 13 12
0 12 11 3 6 8 4 9 5 7 2 10 1 13
13 0 3 10 7 8 9 6 5 4 11 2 12 1
13 12 10 2 8 11 7 9 6 5 3 4 0 1
1 13 3 5 7 4 8 6 9 10 11 2 12 0
1 12 3 2 5 8 6 7 11 10 9 4 12 1
0 13 3 11 10 9 8 7 6 5 4 2 13 0
M = A + 14 * B + 1
196 13 168 139 125 99 86 71 57 56 42 142 2 183
169 16 128 70 44 41 85 100 83 113 167 154 27 182
194 24 166 38 54 60 101 73 138 88 123 144 4 172
178 187 129 151 115 110 67 95 32 81 157 48 17 12
177 18 158 37 136 75 108 63 117 92 51 148 188 11
8 171 47 162 90 121 62 132 79 107 152 33 23 192
21 7 36 145 103 78 91 66 120 133 50 163 175 191
6 22 34 147 64 77 131 106 119 94 52 161 190 176
5 174 164 45 93 118 65 135 76 104 35 150 26 189
186 9 49 146 105 122 134 89 80 61 160 39 179 20
185 180 149 40 116 165 102 130 87 82 46 68 10 19
25 193 53 74 109 59 124 96 137 143 159 31 173 3
15 170 43 30 84 114 97 112 156 153 127 69 181 28
14 195 55 155 141 140 126 111 98 72 58 29 184 1

Based on the order 10 Almost Associated Magic Squares as found in Section 10.2.4 the order 14 Associated Border shown above result in 18.874.368 order 14 Almost Magic Squares.

14.2.5 Bordered Magic Squares

The order 12 Orthogonal Latin Diagonal or Semi-Latin Squares (A12, B12), as discussed in Section 12.2, have been used to construct collections of Simple Magic Squares based on the Balanced Series {0 ... 11}.

From the Balanced Series {0 ... 13} seven suitable Balanced Sub Series can be selected to construct Center Squares for order 14 Bordered Magic Squares e.g. {1 ... 12}.

Suitable Borders can be constructed for each of these Center Squares, based on pairs of Non Latin but Orthogonal Borders (A, B).

A numerical example of the construction of a Bordered Magic Square with a Center Square composed of order 4 Pan Magic Sub Squares is shown below:

A
0 8 6 5 1 13 13 13 13 9 4 3 2 1
0 1 2 11 12 3 4 9 10 5 6 7 8 13
0 12 11 2 1 10 9 4 3 8 7 6 5 13
0 2 1 12 11 4 3 10 9 6 5 8 7 13
13 11 12 1 2 9 10 3 4 7 8 5 6 0
0 1 2 11 12 3 4 9 10 5 6 7 8 13
13 12 11 2 1 10 9 4 3 8 7 6 5 0
0 2 1 12 11 4 3 10 9 6 5 8 7 13
13 11 12 1 2 9 10 3 4 7 8 5 6 0
13 1 2 11 12 3 4 9 10 5 6 7 8 0
11 12 11 2 1 10 9 4 3 8 7 6 5 2
10 2 1 12 11 4 3 10 9 6 5 8 7 3
6 11 12 1 2 9 10 3 4 7 8 5 6 7
12 5 7 8 12 0 0 0 0 4 9 10 11 13
B
0 13 13 13 13 12 11 10 6 0 0 0 0 0
13 1 12 2 11 1 12 2 11 1 12 2 11 0
12 2 11 1 12 2 11 1 12 2 11 1 12 1
10 11 2 12 1 11 2 12 1 11 2 12 1 3
9 12 1 11 2 12 1 11 2 12 1 11 2 4
9 3 10 4 9 3 10 4 9 3 10 4 9 4
8 4 9 3 10 4 9 3 10 4 9 3 10 5
8 9 4 10 3 9 4 10 3 9 4 10 3 5
7 10 3 9 4 10 3 9 4 10 3 9 4 6
2 5 8 6 7 5 8 6 7 5 8 6 7 11
0 6 7 5 8 6 7 5 8 6 7 5 8 13
0 7 6 8 5 7 6 8 5 7 6 8 5 13
0 8 5 7 6 8 5 7 6 8 5 7 6 13
13 0 0 0 0 1 2 3 7 13 13 13 13 13
M = A + 14 * B + 1
1 191 189 188 184 182 168 154 98 10 5 4 3 2
183 16 171 40 167 18 173 38 165 20 175 36 163 14
169 41 166 17 170 39 164 19 172 37 162 21 174 28
141 157 30 181 26 159 32 179 24 161 34 177 22 56
140 180 27 156 31 178 25 158 33 176 23 160 35 57
127 44 143 68 139 46 145 66 137 48 147 64 135 70
126 69 138 45 142 67 136 47 144 65 134 49 146 71
113 129 58 153 54 131 60 151 52 133 62 149 50 84
112 152 55 128 59 150 53 130 61 148 51 132 63 85
42 72 115 96 111 74 117 94 109 76 119 92 107 155
12 97 110 73 114 95 108 75 116 93 106 77 118 185
11 101 86 125 82 103 88 123 80 105 90 121 78 186
7 124 83 100 87 122 81 102 89 120 79 104 91 190
195 6 8 9 13 15 29 43 99 187 192 193 194 196

Each pair of order 14 Orthogonal Non-Latin Borders corresponds with 8 * (12!)2 = 1,835 1018 pairs.

Consequently (9!) * (384)4 * 8 * (12!)2 = 3,20 1029 Bordered Magic Squares with Composed Magic Center Squares can be constructed, based on the pair of order 14 Orthogonal Non-Latin Borders shown above.

14.2.6 Composed Magic Squares

Magic Center Cross (2 x 14)

Pairs of Non-Latin but Orthogonal Borders (A', B'), can be transformed to pairs of Non-Latin but Orthogonal Center Crosses (A, B), which can be completed with pairs of order 6 Orthogonal Semi-Latin Squares (A6, B6).

Following numerical example is based on Simple Magic Sub Squares and the balanced Sub Series {1 ... 12}:

A
12 2 3 10 11 1 0 13 12 2 3 10 11 1
1 11 3 10 2 12 0 13 1 11 3 10 2 12
1 2 10 3 11 12 0 13 1 2 10 3 11 12
12 2 10 3 11 1 13 0 12 2 10 3 11 1
1 11 10 3 2 12 0 13 1 11 10 3 2 12
12 11 3 10 2 1 13 0 12 11 3 10 2 1
8 6 5 1 13 13 0 1 13 13 9 4 3 2
5 7 8 12 0 0 12 13 0 0 4 9 10 11
9 5 6 7 8 4 0 13 9 5 6 7 8 4
4 8 6 7 5 9 13 0 4 8 6 7 5 9
4 5 7 6 8 9 13 0 4 5 7 6 8 9
9 5 7 6 8 4 11 2 9 5 7 6 8 4
4 8 7 6 5 9 10 3 4 8 7 6 5 9
9 8 6 7 5 4 6 7 9 8 6 7 5 4
B
1 12 1 12 12 1 13 0 4 9 4 9 9 4
2 2 11 11 2 11 12 1 5 5 8 8 5 8
10 3 3 3 10 10 10 3 7 6 6 6 7 7
3 10 10 10 3 3 9 4 6 7 7 7 6 6
11 11 2 2 11 2 9 4 8 8 5 5 8 5
12 1 12 1 1 12 8 5 9 4 9 4 4 9
13 13 13 13 12 11 0 0 10 6 0 0 0 0
0 0 0 0 1 2 13 13 3 7 13 13 13 13
4 9 4 9 9 4 8 5 1 12 1 12 12 1
5 5 8 8 5 8 7 6 2 2 11 11 2 11
7 6 6 6 7 7 2 11 10 3 3 3 10 10
6 7 7 7 6 6 0 13 3 10 10 10 3 3
8 8 5 5 8 5 0 13 11 11 2 2 11 2
9 4 9 4 4 9 0 13 12 1 12 1 1 12
M = A + 14 * B + 1
27 171 18 179 180 16 183 14 69 129 60 137 138 58
30 40 158 165 31 167 169 28 72 82 116 123 73 125
142 45 53 46 152 153 141 56 100 87 95 88 110 111
55 143 151 144 54 44 140 57 97 101 109 102 96 86
156 166 39 32 157 41 127 70 114 124 81 74 115 83
181 26 172 25 17 170 126 71 139 68 130 67 59 128
191 189 188 184 182 168 1 2 154 98 10 5 4 3
6 8 9 13 15 29 195 196 43 99 187 192 193 194
66 132 63 134 135 61 113 84 24 174 21 176 177 19
75 79 119 120 76 122 112 85 33 37 161 162 34 164
103 90 92 91 107 108 42 155 145 48 50 49 149 150
94 104 106 105 93 89 12 185 52 146 148 147 51 47
117 121 78 77 118 80 11 186 159 163 36 35 160 38
136 65 133 64 62 131 7 190 178 23 175 22 20 173

Each pair of order 14 Orthogonal Non-Latin Center Crosses corresponds with 8 * (12!)2 = 1,835 1018 pairs.

Consequently (4!) * (6 * 128)4 * 8 * (12!)2 = 1,53 1031 Composed Magic Squares can be constructed, based on the pair of order 14 Orthogonal Non-Latin Center Crosses shown above.

14.2.7 Simple Magic Squares

A nice numerical example of the construction of an order 14 Simple Magic Square M, based on a pair of Orthogonal Squares (A, B) with a Non-Latin square A and a Latin square B, is shown below:

A (Non-Latin)
6 1 5 9 13 4 12 1 4 14 14 8 13 1
4 11 10 8 14 7 2 13 7 5 3 1 6 14
7 2 1 12 13 3 5 14 2 1 6 14 11 14
12 13 13 10 4 6 8 13 5 4 6 7 1 3
9 1 8 5 7 9 14 8 7 13 11 3 8 2
7 9 7 8 10 13 4 4 13 4 12 3 2 9
4 6 9 11 1 5 10 8 12 9 2 10 10 8
8 10 6 6 5 9 3 4 11 7 2 12 8 14
14 5 9 1 4 2 2 12 8 3 13 11 9 12
1 13 1 3 9 12 3 6 4 14 12 10 11 6
10 14 5 8 6 5 10 5 1 3 11 5 11 11
11 9 13 10 7 7 12 2 4 12 2 3 7 6
10 7 11 2 9 12 11 1 13 6 5 10 5 3
2 4 7 12 3 11 9 14 14 10 6 8 3 2
B (Latin)
14 13 12 11 10 9 8 7 6 5 4 3 2 1
13 3 11 5 6 7 1 14 8 9 10 4 12 2
12 4 5 6 7 1 2 13 14 8 9 10 11 3
11 5 6 7 1 2 12 3 13 14 8 9 10 4
5 6 7 1 2 12 11 4 3 13 14 8 9 10
6 7 1 2 12 11 10 5 4 3 13 14 8 9
7 1 2 12 11 10 9 6 5 4 3 13 14 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14
8 14 13 3 4 5 6 9 10 11 12 2 1 7
9 8 14 13 3 4 5 10 11 12 2 1 7 6
10 9 8 14 13 3 4 11 12 2 1 7 6 5
4 10 9 8 14 13 3 12 2 1 7 6 5 11
3 11 10 9 8 14 13 2 1 7 6 5 4 12
2 12 4 10 9 8 14 1 7 6 5 11 3 13
M = A + (B – 1) * 14
188 169 159 149 139 116 110 85 74 70 56 36 27 1
172 39 150 64 84 91 2 195 105 117 129 43 160 28
161 44 57 82 97 3 19 182 184 99 118 140 151 42
152 69 83 94 4 20 162 41 173 186 104 119 127 45
65 71 92 5 21 163 154 50 35 181 193 101 120 128
77 93 7 22 164 153 130 60 55 32 180 185 100 121
88 6 23 165 141 131 122 78 68 51 30 178 192 106
8 24 34 48 61 79 87 102 123 133 142 166 176 196
112 187 177 29 46 58 72 124 134 143 167 25 9 96
113 111 183 171 37 54 59 132 144 168 26 10 95 76
136 126 103 190 174 33 52 145 155 17 11 89 81 67
53 135 125 108 189 175 40 156 18 12 86 73 63 146
38 147 137 114 107 194 179 15 13 90 75 66 47 157
16 158 49 138 115 109 191 14 98 80 62 148 31 170

Square B (A. Breedijk) contains fourteen patterns - 1 (grey) thru 14 (blue) - each populated with fourteen times the same number.

To enable the construction of a (correct) Simple Magic Square M, the comparable patterns in square A should be populated with the numbers 1 thru 14.

14.4   Summary

The obtained results regarding the order 14 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, can be summarized as folows:

  • The construction of order 14 Simple Magic Squares based on Orthogonal Latin Diagonal Squares (Historical).
  • The construction of order 14 Simple Magic Squares with Symmetrical Diagonals based on Orthogonal Semi Latin Diagonal Squares.
  • The construction of order 14 Almost Associated Magic Squares based on Orthogonal Semi Latin Diagonal Squares.
  • The construction of order 14 Concentric Borders based on Non Latin but Orthogonal Borders.
  • The construction of order 14 Concentric Center Crosses based on Non Latin but Orthogonal Center Crosses.
  • The construction of order 14 Simple Magic Squares based on Orthogonal Sets of one Non-Latin and one Latin Diagonal Square.

Comparable methods as described in previous sections, can be used to construct order 16 (Semi) Latin - and related (Pan) Magic Squares, which will be described in following sections.

Vorige Pagina Volgende Pagina Index About the Author