Office Applications and Entertainment, Latin Squares

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16.0   Latin Squares (16 x 16)

A Latin Square of order 16 is a 16 x 16 square filled with 16 different symbols, each occurring only once in each row and only once in each column.

16.1   Latin Diagonal Squares (16 x 16)

Latin Diagonal Squares are Latin Squares for which the 16 different symbols occur also only once in each of the main diagonals.

16.2   Magic Squares, Natural Numbers

16.2.1 Compact, Complete Pan Magic Squares
       All 1/2 Rows and 1/2 Columns sum to s16/2

Order 16 Compact, Complete Pan Magic Squares, for which all 1/2 rows and 1/2 columns sum to s16/2, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 16 Pan Magic Squares is that:

  • The integers of each 2 × 2 sub square sum to s16/4 = 514 (Compact);
  • All pairs of integers distant n/2 along a (main) diagonal sum to s16/8 = 257 (Complete);
  • Consequently the main - and broken diagonals sum to s16 = 2056;
  • All 1/2 rows and 1/2 columns sum to s16/2 = 1028.

A numerical example is shown below:

A
13 12 11 10 7 6 1 0 3 2 5 4 9 8 15 14
2 3 4 5 8 9 14 15 12 13 10 11 6 7 0 1
13 12 11 10 7 6 1 0 3 2 5 4 9 8 15 14
2 3 4 5 8 9 14 15 12 13 10 11 6 7 0 1
13 12 11 10 7 6 1 0 3 2 5 4 9 8 15 14
2 3 4 5 8 9 14 15 12 13 10 11 6 7 0 1
13 12 11 10 7 6 1 0 3 2 5 4 9 8 15 14
2 3 4 5 8 9 14 15 12 13 10 11 6 7 0 1
12 13 10 11 6 7 0 1 2 3 4 5 8 9 14 15
3 2 5 4 9 8 15 14 13 12 11 10 7 6 1 0
12 13 10 11 6 7 0 1 2 3 4 5 8 9 14 15
3 2 5 4 9 8 15 14 13 12 11 10 7 6 1 0
12 13 10 11 6 7 0 1 2 3 4 5 8 9 14 15
3 2 5 4 9 8 15 14 13 12 11 10 7 6 1 0
12 13 10 11 6 7 0 1 2 3 4 5 8 9 14 15
3 2 5 4 9 8 15 14 13 12 11 10 7 6 1 0
B = T(A)
13 2 13 2 13 2 13 2 12 3 12 3 12 3 12 3
12 3 12 3 12 3 12 3 13 2 13 2 13 2 13 2
11 4 11 4 11 4 11 4 10 5 10 5 10 5 10 5
10 5 10 5 10 5 10 5 11 4 11 4 11 4 11 4
7 8 7 8 7 8 7 8 6 9 6 9 6 9 6 9
6 9 6 9 6 9 6 9 7 8 7 8 7 8 7 8
1 14 1 14 1 14 1 14 0 15 0 15 0 15 0 15
0 15 0 15 0 15 0 15 1 14 1 14 1 14 1 14
3 12 3 12 3 12 3 12 2 13 2 13 2 13 2 13
2 13 2 13 2 13 2 13 3 12 3 12 3 12 3 12
5 10 5 10 5 10 5 10 4 11 4 11 4 11 4 11
4 11 4 11 4 11 4 11 5 10 5 10 5 10 5 10
9 6 9 6 9 6 9 6 8 7 8 7 8 7 8 7
8 7 8 7 8 7 8 7 9 6 9 6 9 6 9 6
15 0 15 0 15 0 15 0 14 1 14 1 14 1 14 1
14 1 14 1 14 1 14 1 15 0 15 0 15 0 15 0
C = 16 * A + B + 1
222 195 190 163 126 99 30 3 61 36 93 68 157 132 253 228
45 52 77 84 141 148 237 244 206 211 174 179 110 115 14 19
220 197 188 165 124 101 28 5 59 38 91 70 155 134 251 230
43 54 75 86 139 150 235 246 204 213 172 181 108 117 12 21
216 201 184 169 120 105 24 9 55 42 87 74 151 138 247 234
39 58 71 90 135 154 231 250 200 217 168 185 104 121 8 25
210 207 178 175 114 111 18 15 49 48 81 80 145 144 241 240
33 64 65 96 129 160 225 256 194 223 162 191 98 127 2 31
196 221 164 189 100 125 4 29 35 62 67 94 131 158 227 254
51 46 83 78 147 142 243 238 212 205 180 173 116 109 20 13
198 219 166 187 102 123 6 27 37 60 69 92 133 156 229 252
53 44 85 76 149 140 245 236 214 203 182 171 118 107 22 11
202 215 170 183 106 119 10 23 41 56 73 88 137 152 233 248
57 40 89 72 153 136 249 232 218 199 186 167 122 103 26 7
208 209 176 177 112 113 16 17 47 50 79 82 143 146 239 242
63 34 95 66 159 130 255 226 224 193 192 161 128 97 32 1

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

With the 14 parameter procedure CompLat16a 474624 pairs of Orthogonal Semi-Latin Compact and Complete Squares (A, T(A)) could be generated within 1236 seconds.

Attachment 16.2.11 shows a few of the resulting order 16 Pan Magic, Compact and Complete Magic Squares.

Attachment 16.2.12 shows the corresponding Associated (Partly Compact) Magic Squares, with the semi diagonals summing to s16, which can be obtained by Eulers transformation.

16.2.2 Compact, Associated Pan Magic Squares
       All 1/2 Rows and 1/2 Columns sum to s16/2

Order 16 Compact, Associated Pan Magic Squares, for which all 1/2 rows and 1/2 columns sum to s16/2, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 16 Pan Magic Squares is that:

  • The integers of each 2 × 2 sub square sum to s16/4 = 514 (Compact);
  • All pairs of integers, which can be connected with a straight line through the center, sum to s16/8 = 257
    (Associated).
  • All main - and broken diagonals sum to s16 = 2056;
  • All 1/2 rows and 1/2 columns sum to s16/2 = 1028.

A numerical example is shown below:

A
15 12 11 8 7 4 3 0 2 1 6 5 10 9 14 13
2 1 6 5 10 9 14 13 15 12 11 8 7 4 3 0
13 14 9 10 5 6 1 2 0 3 4 7 8 11 12 15
0 3 4 7 8 11 12 15 13 14 9 10 5 6 1 2
13 14 9 10 5 6 1 2 0 3 4 7 8 11 12 15
0 3 4 7 8 11 12 15 13 14 9 10 5 6 1 2
15 12 11 8 7 4 3 0 2 1 6 5 10 9 14 13
2 1 6 5 10 9 14 13 15 12 11 8 7 4 3 0
15 12 11 8 7 4 3 0 2 1 6 5 10 9 14 13
2 1 6 5 10 9 14 13 15 12 11 8 7 4 3 0
13 14 9 10 5 6 1 2 0 3 4 7 8 11 12 15
0 3 4 7 8 11 12 15 13 14 9 10 5 6 1 2
13 14 9 10 5 6 1 2 0 3 4 7 8 11 12 15
0 3 4 7 8 11 12 15 13 14 9 10 5 6 1 2
15 12 11 8 7 4 3 0 2 1 6 5 10 9 14 13
2 1 6 5 10 9 14 13 15 12 11 8 7 4 3 0
B = T(A)
15 2 13 0 13 0 15 2 15 2 13 0 13 0 15 2
12 1 14 3 14 3 12 1 12 1 14 3 14 3 12 1
11 6 9 4 9 4 11 6 11 6 9 4 9 4 11 6
8 5 10 7 10 7 8 5 8 5 10 7 10 7 8 5
7 10 5 8 5 8 7 10 7 10 5 8 5 8 7 10
4 9 6 11 6 11 4 9 4 9 6 11 6 11 4 9
3 14 1 12 1 12 3 14 3 14 1 12 1 12 3 14
0 13 2 15 2 15 0 13 0 13 2 15 2 15 0 13
2 15 0 13 0 13 2 15 2 15 0 13 0 13 2 15
1 12 3 14 3 14 1 12 1 12 3 14 3 14 1 12
6 11 4 9 4 9 6 11 6 11 4 9 4 9 6 11
5 8 7 10 7 10 5 8 5 8 7 10 7 10 5 8
10 7 8 5 8 5 10 7 10 7 8 5 8 5 10 7
9 4 11 6 11 6 9 4 9 4 11 6 11 6 9 4
14 3 12 1 12 1 14 3 14 3 12 1 12 1 14 3
13 0 15 2 15 2 13 0 13 0 15 2 15 2 13 0
C = 16 * A + B + 1
256 195 190 129 126 65 64 3 48 19 110 81 174 145 240 211
45 18 111 84 175 148 237 210 253 194 191 132 127 68 61 2
220 231 154 165 90 101 28 39 12 55 74 117 138 181 204 247
9 54 75 120 139 184 201 246 217 230 155 168 91 104 25 38
216 235 150 169 86 105 24 43 8 59 70 121 134 185 200 251
5 58 71 124 135 188 197 250 213 234 151 172 87 108 21 42
244 207 178 141 114 77 52 15 36 31 98 93 162 157 228 223
33 30 99 96 163 160 225 222 241 206 179 144 115 80 49 14
243 208 177 142 113 78 51 16 35 32 97 94 161 158 227 224
34 29 100 95 164 159 226 221 242 205 180 143 116 79 50 13
215 236 149 170 85 106 23 44 7 60 69 122 133 186 199 252
6 57 72 123 136 187 198 249 214 233 152 171 88 107 22 41
219 232 153 166 89 102 27 40 11 56 73 118 137 182 203 248
10 53 76 119 140 183 202 245 218 229 156 167 92 103 26 37
255 196 189 130 125 66 63 4 47 20 109 82 173 146 239 212
46 17 112 83 176 147 238 209 254 193 192 131 128 67 62 1

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

With the 14 parameter procedure AssLat16a 967680 pairs of Orthogonal Semi-Latin Compact, Associated Pan Magic Squares (A, T(A)) could be generated within 2793 seconds.

Attachment 16.2.2 shows a few of the resulting order 16 Compact, Associated Pan Magic Squares.

16.2.3 Composed Magic Squares
       Order 4 Latin Diagonal Sub Squares

Order 16 (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).

A numerical example is shown below:

A
0 1 14 15 2 3 12 13 4 5 10 11 6 7 8 9
15 14 1 0 13 12 3 2 11 10 5 4 9 8 7 6
1 0 15 14 3 2 13 12 5 4 11 10 7 6 9 8
14 15 0 1 12 13 2 3 10 11 4 5 8 9 6 7
0 1 14 15 2 3 12 13 4 5 10 11 6 7 8 9
15 14 1 0 13 12 3 2 11 10 5 4 9 8 7 6
1 0 15 14 3 2 13 12 5 4 11 10 7 6 9 8
14 15 0 1 12 13 2 3 10 11 4 5 8 9 6 7
0 1 14 15 2 3 12 13 4 5 10 11 6 7 8 9
15 14 1 0 13 12 3 2 11 10 5 4 9 8 7 6
1 0 15 14 3 2 13 12 5 4 11 10 7 6 9 8
14 15 0 1 12 13 2 3 10 11 4 5 8 9 6 7
0 1 14 15 2 3 12 13 4 5 10 11 6 7 8 9
15 14 1 0 13 12 3 2 11 10 5 4 9 8 7 6
1 0 15 14 3 2 13 12 5 4 11 10 7 6 9 8
14 15 0 1 12 13 2 3 10 11 4 5 8 9 6 7
B = T(A)
0 15 1 14 0 15 1 14 0 15 1 14 0 15 1 14
1 14 0 15 1 14 0 15 1 14 0 15 1 14 0 15
14 1 15 0 14 1 15 0 14 1 15 0 14 1 15 0
15 0 14 1 15 0 14 1 15 0 14 1 15 0 14 1
2 13 3 12 2 13 3 12 2 13 3 12 2 13 3 12
3 12 2 13 3 12 2 13 3 12 2 13 3 12 2 13
12 3 13 2 12 3 13 2 12 3 13 2 12 3 13 2
13 2 12 3 13 2 12 3 13 2 12 3 13 2 12 3
4 11 5 10 4 11 5 10 4 11 5 10 4 11 5 10
5 10 4 11 5 10 4 11 5 10 4 11 5 10 4 11
10 5 11 4 10 5 11 4 10 5 11 4 10 5 11 4
11 4 10 5 11 4 10 5 11 4 10 5 11 4 10 5
6 9 7 8 6 9 7 8 6 9 7 8 6 9 7 8
7 8 6 9 7 8 6 9 7 8 6 9 7 8 6 9
8 7 9 6 8 7 9 6 8 7 9 6 8 7 9 6
9 6 8 7 9 6 8 7 9 6 8 7 9 6 8 7
C = 16 * A + B + 1
1 32 226 255 33 64 194 223 65 96 162 191 97 128 130 159
242 239 17 16 210 207 49 48 178 175 81 80 146 143 113 112
31 2 256 225 63 34 224 193 95 66 192 161 127 98 160 129
240 241 15 18 208 209 47 50 176 177 79 82 144 145 111 114
3 30 228 253 35 62 196 221 67 94 164 189 99 126 132 157
244 237 19 14 212 205 51 46 180 173 83 78 148 141 115 110
29 4 254 227 61 36 222 195 93 68 190 163 125 100 158 131
238 243 13 20 206 211 45 52 174 179 77 84 142 147 109 116
5 28 230 251 37 60 198 219 69 92 166 187 101 124 134 155
246 235 21 12 214 203 53 44 182 171 85 76 150 139 117 108
27 6 252 229 59 38 220 197 91 70 188 165 123 102 156 133
236 245 11 22 204 213 43 54 172 181 75 86 140 149 107 118
7 26 232 249 39 58 200 217 71 90 168 185 103 122 136 153
248 233 23 10 216 201 55 42 184 169 87 74 152 137 119 106
25 8 250 231 57 40 218 199 89 72 186 167 121 104 154 135
234 247 9 24 202 215 41 56 170 183 73 88 138 151 105 120

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

The balanced series {0, 1, 2, 3, 4 ... 15} have been split into three balanced sub series:

       {0, 1, 14, 15}, {2, 3, 12, 13}, (4, 5, 10, 11) and {6, 7, 8, 9}

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

Attachment 16.23.1 shows the unique sets (105 ea) of order 4 balanced lines for the integers 0 ... 15.

Attachment 16.23.2 shows the resulting order 16 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 s16/4 = 514 (Compact).

Based on the construction method described above 4! * (16)4 = 1572864 squares can be constructed for each set of order 4 balanced lines shown in Attachment 16.23.1.

Each square shown in Attachment 16.23.2 corresponds with (16!) * (384)4 = 4,7 1054 squares, which can be obtained by permutation of the sub squares and (or) selecting other aspects of the sub squares.

16.2.4 Franklin Pan Magic Squares

Order 16 Franklin (Pan Magic) Squares, as discussed in detail in Section 12.3.1, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 16 Franklin Squares is that:

  • All half-rows and half-columns sum to s16/2 = 1028;
  • The main bent diagonals and all the bent diagonals parallel to it sum to s16 = 2056;
  • All 2 × 2 sub squares sum to s16/4 = 514;
  • The 4 middle rows and columns of each 8 x 8 sub square sum to s1/4 = 514.

A numerical example is shown below:

A
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
5 4 7 6 8 9 10 11 2 1 0 3 15 12 13 14
10 11 8 9 7 6 5 4 13 14 15 12 0 3 2 1
B = T(A)
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
4 11 4 11 4 11 4 11 4 11 4 11 4 11 4 11
7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8
6 9 6 9 6 9 6 9 6 9 6 9 6 9 6 9
8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7
9 6 9 6 9 6 9 6 9 6 9 6 9 6 9 6
10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5
11 4 11 4 11 4 11 4 11 4 11 4 11 4 11 4
2 13 2 13 2 13 2 13 2 13 2 13 2 13 2 13
1 14 1 14 1 14 1 14 1 14 1 14 1 14 1 14
0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15
3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12
15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0
12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3
13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2
14 1 14 1 14 1 14 1 14 1 14 1 14 1 14 1
C = 16 * A + B + 1
86 75 118 107 134 155 166 187 38 27 6 59 246 203 214 235
165 188 133 156 117 108 85 76 213 236 245 204 5 60 37 28
88 73 120 105 136 153 168 185 40 25 8 57 248 201 216 233
167 186 135 154 119 106 87 74 215 234 247 202 7 58 39 26
89 72 121 104 137 152 169 184 41 24 9 56 249 200 217 232
170 183 138 151 122 103 90 71 218 231 250 199 10 55 42 23
91 70 123 102 139 150 171 182 43 22 11 54 251 198 219 230
172 181 140 149 124 101 92 69 220 229 252 197 12 53 44 21
83 78 115 110 131 158 163 190 35 30 3 62 243 206 211 238
162 191 130 159 114 111 82 79 210 239 242 207 2 63 34 31
81 80 113 112 129 160 161 192 33 32 1 64 241 208 209 240
164 189 132 157 116 109 84 77 212 237 244 205 4 61 36 29
96 65 128 97 144 145 176 177 48 17 16 49 256 193 224 225
173 180 141 148 125 100 93 68 221 228 253 196 13 52 45 20
94 67 126 99 142 147 174 179 46 19 14 51 254 195 222 227
175 178 143 146 127 98 95 66 223 226 255 194 15 50 47 18

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

With the 21 parameter procedure Franklin16a and a(256) = 1, 97696 pairs of Orthogonal Semi-Latin Franklin Pan Magic Squares (A, T(A)) could be generated within 682 seconds.

Attachment 16.2.4 shows a few of the resulting order 16 Franklin Pan Magic Squares.

16.2.5 Franklin Squares, Pan Magic and Complete

Also order 16 Pan Magic and Complete Franklin Squares, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

A numerical example is shown below:

A
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
11 12 13 10 0 7 6 1 4 3 2 5 15 8 9 14
4 3 2 5 15 8 9 14 11 12 13 10 0 7 6 1
B = T(A)
11 4 11 4 11 4 11 4 11 4 11 4 11 4 11 4
12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3
13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2
10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5
0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15
7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8
6 9 6 9 6 9 6 9 6 9 6 9 6 9 6 9
1 14 1 14 1 14 1 14 1 14 1 14 1 14 1 14
4 11 4 11 4 11 4 11 4 11 4 11 4 11 4 11
3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12
2 13 2 13 2 13 2 13 2 13 2 13 2 13 2 13
5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0
8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7
9 6 9 6 9 6 9 6 9 6 9 6 9 6 9 6
14 1 14 1 14 1 14 1 14 1 14 1 14 1 14 1
C = 16 * A + B + 1
188 197 220 165 12 117 108 21 76 53 44 85 252 133 156 229
77 52 45 84 253 132 157 228 189 196 221 164 13 116 109 20
190 195 222 163 14 115 110 19 78 51 46 83 254 131 158 227
75 54 43 86 251 134 155 230 187 198 219 166 11 118 107 22
177 208 209 176 1 128 97 32 65 64 33 96 241 144 145 240
72 57 40 89 248 137 152 233 184 201 216 169 8 121 104 25
183 202 215 170 7 122 103 26 71 58 39 90 247 138 151 234
66 63 34 95 242 143 146 239 178 207 210 175 2 127 98 31
181 204 213 172 5 124 101 28 69 60 37 92 245 140 149 236
68 61 36 93 244 141 148 237 180 205 212 173 4 125 100 29
179 206 211 174 3 126 99 30 67 62 35 94 243 142 147 238
70 59 38 91 246 139 150 235 182 203 214 171 6 123 102 27
192 193 224 161 16 113 112 17 80 49 48 81 256 129 160 225
73 56 41 88 249 136 153 232 185 200 217 168 9 120 105 24
186 199 218 167 10 119 106 23 74 55 42 87 250 135 154 231
79 50 47 82 255 130 159 226 191 194 223 162 15 114 111 18

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

With the 10 parameter procedure Franklin16b 7680 pairs of Orthogonal Semi-Latin Pan Magic and Complete Franklin Squares (A, T(A)) could be generated within 25 seconds.

Attachment 16.2.5 shows a few of the resulting order 16 Pan Magic and Complete Franklin Squares.

16.2.6 Franklin Squares, Pan Magic and Associated

Also order 16 Pan Magic and Associated Franklin Squares (Ultra Magic), can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

A numerical example is shown below:

A
13 11 8 14 5 3 0 6 4 2 1 7 12 10 9 15
2 4 7 1 10 12 15 9 11 13 14 8 3 5 6 0
15 9 10 12 7 1 2 4 6 0 3 5 14 8 11 13
0 6 5 3 8 14 13 11 9 15 12 10 1 7 4 2
15 9 10 12 7 1 2 4 6 0 3 5 14 8 11 13
0 6 5 3 8 14 13 11 9 15 12 10 1 7 4 2
13 11 8 14 5 3 0 6 4 2 1 7 12 10 9 15
2 4 7 1 10 12 15 9 11 13 14 8 3 5 6 0
15 9 10 12 7 1 2 4 6 0 3 5 14 8 11 13
0 6 5 3 8 14 13 11 9 15 12 10 1 7 4 2
13 11 8 14 5 3 0 6 4 2 1 7 12 10 9 15
2 4 7 1 10 12 15 9 11 13 14 8 3 5 6 0
13 11 8 14 5 3 0 6 4 2 1 7 12 10 9 15
2 4 7 1 10 12 15 9 11 13 14 8 3 5 6 0
15 9 10 12 7 1 2 4 6 0 3 5 14 8 11 13
0 6 5 3 8 14 13 11 9 15 12 10 1 7 4 2
B = T(A)
13 2 15 0 15 0 13 2 15 0 13 2 13 2 15 0
11 4 9 6 9 6 11 4 9 6 11 4 11 4 9 6
8 7 10 5 10 5 8 7 10 5 8 7 8 7 10 5
14 1 12 3 12 3 14 1 12 3 14 1 14 1 12 3
5 10 7 8 7 8 5 10 7 8 5 10 5 10 7 8
3 12 1 14 1 14 3 12 1 14 3 12 3 12 1 14
0 15 2 13 2 13 0 15 2 13 0 15 0 15 2 13
6 9 4 11 4 11 6 9 4 11 6 9 6 9 4 11
4 11 6 9 6 9 4 11 6 9 4 11 4 11 6 9
2 13 0 15 0 15 2 13 0 15 2 13 2 13 0 15
1 14 3 12 3 12 1 14 3 12 1 14 1 14 3 12
7 8 5 10 5 10 7 8 5 10 7 8 7 8 5 10
12 3 14 1 14 1 12 3 14 1 12 3 12 3 14 1
10 5 8 7 8 7 10 5 8 7 10 5 10 5 8 7
9 6 11 4 11 4 9 6 11 4 9 6 9 6 11 4
15 0 13 2 13 2 15 0 13 2 15 0 15 0 13 2
C = 16 * A + B + 1
222 179 144 225 96 49 14 99 80 33 30 115 206 163 160 241
44 69 122 23 170 199 252 149 186 215 236 133 60 85 106 7
249 152 171 198 123 22 41 72 107 6 57 88 233 136 187 214
15 98 93 52 141 228 223 178 157 244 207 162 31 114 77 36
246 155 168 201 120 25 38 75 104 9 54 91 230 139 184 217
4 109 82 63 130 239 212 189 146 255 196 173 20 125 66 47
209 192 131 238 83 62 1 112 67 46 17 128 193 176 147 254
39 74 117 28 165 204 247 154 181 220 231 138 55 90 101 12
245 156 167 202 119 26 37 76 103 10 53 92 229 140 183 218
3 110 81 64 129 240 211 190 145 256 195 174 19 126 65 48
210 191 132 237 84 61 2 111 68 45 18 127 194 175 148 253
40 73 118 27 166 203 248 153 182 219 232 137 56 89 102 11
221 180 143 226 95 50 13 100 79 34 29 116 205 164 159 242
43 70 121 24 169 200 251 150 185 216 235 134 59 86 105 8
250 151 172 197 124 21 42 71 108 5 58 87 234 135 188 213
16 97 94 51 142 227 224 177 158 243 208 161 32 113 78 35

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

With the 10 parameter procedure Franklin16c 9216 pairs of Orthogonal Semi-Latin Ultra Magic Franklin Squares (A, T(A)) could be generated within 30 seconds.

Attachment 16.2.6 shows a few of the resulting order 16 Ultra Magic Franklin Squares.

16.2.7 Most Perfect Franklin Pan Magic Squares

Order 16 Most Perfect Franklin Pan Magic Squares, as discussed in detail in Section 12.4.1, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 16 Most Perfect Franklin Pan Magic Squares is that:

  • The 4 x 4 subsquares (16 ea) are Pan Magic.
    Consequently the rows, columns and main diagonals sum to s16 = 2056.
  • The main bent diagonals and all the bent diagonals parallel to it sum to s16 = 2056;
  • The main - and broken diagonals sum to s16 = 2056;
  • All 2 × 2 sub squares sum to s16/4 = 514.

A numerical example is shown below:

A
7 6 9 8 5 4 11 10 3 2 13 12 1 0 15 14
8 9 6 7 10 11 4 5 12 13 2 3 14 15 0 1
6 7 8 9 4 5 10 11 2 3 12 13 0 1 14 15
9 8 7 6 11 10 5 4 13 12 3 2 15 14 1 0
7 6 9 8 5 4 11 10 3 2 13 12 1 0 15 14
8 9 6 7 10 11 4 5 12 13 2 3 14 15 0 1
6 7 8 9 4 5 10 11 2 3 12 13 0 1 14 15
9 8 7 6 11 10 5 4 13 12 3 2 15 14 1 0
7 6 9 8 5 4 11 10 3 2 13 12 1 0 15 14
8 9 6 7 10 11 4 5 12 13 2 3 14 15 0 1
6 7 8 9 4 5 10 11 2 3 12 13 0 1 14 15
9 8 7 6 11 10 5 4 13 12 3 2 15 14 1 0
7 6 9 8 5 4 11 10 3 2 13 12 1 0 15 14
8 9 6 7 10 11 4 5 12 13 2 3 14 15 0 1
6 7 8 9 4 5 10 11 2 3 12 13 0 1 14 15
9 8 7 6 11 10 5 4 13 12 3 2 15 14 1 0
B=T(A)
7 8 6 9 7 8 6 9 7 8 6 9 7 8 6 9
6 9 7 8 6 9 7 8 6 9 7 8 6 9 7 8
9 6 8 7 9 6 8 7 9 6 8 7 9 6 8 7
8 7 9 6 8 7 9 6 8 7 9 6 8 7 9 6
5 10 4 11 5 10 4 11 5 10 4 11 5 10 4 11
4 11 5 10 4 11 5 10 4 11 5 10 4 11 5 10
11 4 10 5 11 4 10 5 11 4 10 5 11 4 10 5
10 5 11 4 10 5 11 4 10 5 11 4 10 5 11 4
3 12 2 13 3 12 2 13 3 12 2 13 3 12 2 13
2 13 3 12 2 13 3 12 2 13 3 12 2 13 3 12
13 2 12 3 13 2 12 3 13 2 12 3 13 2 12 3
12 3 13 2 12 3 13 2 12 3 13 2 12 3 13 2
1 14 0 15 1 14 0 15 1 14 0 15 1 14 0 15
0 15 1 14 0 15 1 14 0 15 1 14 0 15 1 14
15 0 14 1 15 0 14 1 15 0 14 1 15 0 14 1
14 1 15 0 14 1 15 0 14 1 15 0 14 1 15 0
C = 16 * A + B + 1
120 105 151 138 88 73 183 170 56 41 215 202 24 9 247 234
135 154 104 121 167 186 72 89 199 218 40 57 231 250 8 25
106 119 137 152 74 87 169 184 42 55 201 216 10 23 233 248
153 136 122 103 185 168 90 71 217 200 58 39 249 232 26 7
118 107 149 140 86 75 181 172 54 43 213 204 22 11 245 236
133 156 102 123 165 188 70 91 197 220 38 59 229 252 6 27
108 117 139 150 76 85 171 182 44 53 203 214 12 21 235 246
155 134 124 101 187 166 92 69 219 198 60 37 251 230 28 5
116 109 147 142 84 77 179 174 52 45 211 206 20 13 243 238
131 158 100 125 163 190 68 93 195 222 36 61 227 254 4 29
110 115 141 148 78 83 173 180 46 51 205 212 14 19 237 244
157 132 126 99 189 164 94 67 221 196 62 35 253 228 30 3
114 111 145 144 82 79 177 176 50 47 209 208 18 15 241 240
129 160 98 127 161 192 66 95 193 224 34 63 225 256 2 31
112 113 143 146 80 81 175 178 48 49 207 210 16 17 239 242
159 130 128 97 191 162 96 65 223 194 64 33 255 226 32 1

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

With the 16 parameter procedure MstPrfct16a and a(256) = 0, 829440 pairs of Orthogonal Semi-Latin Most Perfect Franklin Pan Magic Squares (A, T(A)) could be generated within 30 minutes.

Attachment 16.2.7 shows a few of the resulting order 16 Most Perfect Franklin Pan Magic Squares.

16.2.8 Most Perfect Franklin Pan Magic Squares
       Barink Restrictions

Order 16 Most Perfect Franklin Pan Magic Squares (Barink Restrictions), as discussed in detail in Section 12.5.1, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this type order 16 Most Perfect Franklin Pan Magic Squares (Barink Restrictions) is that:

  • The 4 x 4 subsquares (16 ea) are Pan Magic.
    Consequently the rows, columns and main diagonals sum to the s16 = 2056.
  • The main bent diagonals and all the bent diagonals parallel to it sum to s16 = 2056;
  • The main - and broken diagonals sum to s16 = 2056;
  • All 2 × 2 sub squares sum to s16/4 = 514;
  • Any 4 consecutive numbers, starting on any odd place in a row or column, sum to s16/4 = 514.

A numerical example is shown below:

A
4 3 15 8 5 2 14 9 6 1 13 10 7 0 12 11
11 12 0 7 10 13 1 6 9 14 2 5 8 15 3 4
0 7 11 12 1 6 10 13 2 5 9 14 3 4 8 15
15 8 4 3 14 9 5 2 13 10 6 1 12 11 7 0
4 3 15 8 5 2 14 9 6 1 13 10 7 0 12 11
11 12 0 7 10 13 1 6 9 14 2 5 8 15 3 4
0 7 11 12 1 6 10 13 2 5 9 14 3 4 8 15
15 8 4 3 14 9 5 2 13 10 6 1 12 11 7 0
4 3 15 8 5 2 14 9 6 1 13 10 7 0 12 11
11 12 0 7 10 13 1 6 9 14 2 5 8 15 3 4
0 7 11 12 1 6 10 13 2 5 9 14 3 4 8 15
15 8 4 3 14 9 5 2 13 10 6 1 12 11 7 0
4 3 15 8 5 2 14 9 6 1 13 10 7 0 12 11
11 12 0 7 10 13 1 6 9 14 2 5 8 15 3 4
0 7 11 12 1 6 10 13 2 5 9 14 3 4 8 15
15 8 4 3 14 9 5 2 13 10 6 1 12 11 7 0
B=T(A)
4 11 0 15 4 11 0 15 4 11 0 15 4 11 0 15
3 12 7 8 3 12 7 8 3 12 7 8 3 12 7 8
15 0 11 4 15 0 11 4 15 0 11 4 15 0 11 4
8 7 12 3 8 7 12 3 8 7 12 3 8 7 12 3
5 10 1 14 5 10 1 14 5 10 1 14 5 10 1 14
2 13 6 9 2 13 6 9 2 13 6 9 2 13 6 9
14 1 10 5 14 1 10 5 14 1 10 5 14 1 10 5
9 6 13 2 9 6 13 2 9 6 13 2 9 6 13 2
6 9 2 13 6 9 2 13 6 9 2 13 6 9 2 13
1 14 5 10 1 14 5 10 1 14 5 10 1 14 5 10
13 2 9 6 13 2 9 6 13 2 9 6 13 2 9 6
10 5 14 1 10 5 14 1 10 5 14 1 10 5 14 1
7 8 3 12 7 8 3 12 7 8 3 12 7 8 3 12
0 15 4 11 0 15 4 11 0 15 4 11 0 15 4 11
12 3 8 7 12 3 8 7 12 3 8 7 12 3 8 7
11 4 15 0 11 4 15 0 11 4 15 0 11 4 15 0
C = 16 * A + B + 1
69 60 241 144 85 44 225 160 101 28 209 176 117 12 193 192
180 205 8 121 164 221 24 105 148 237 40 89 132 253 56 73
16 113 188 197 32 97 172 213 48 81 156 229 64 65 140 245
249 136 77 52 233 152 93 36 217 168 109 20 201 184 125 4
70 59 242 143 86 43 226 159 102 27 210 175 118 11 194 191
179 206 7 122 163 222 23 106 147 238 39 90 131 254 55 74
15 114 187 198 31 98 171 214 47 82 155 230 63 66 139 246
250 135 78 51 234 151 94 35 218 167 110 19 202 183 126 3
71 58 243 142 87 42 227 158 103 26 211 174 119 10 195 190
178 207 6 123 162 223 22 107 146 239 38 91 130 255 54 75
14 115 186 199 30 99 170 215 46 83 154 231 62 67 138 247
251 134 79 50 235 150 95 34 219 166 111 18 203 182 127 2
72 57 244 141 88 41 228 157 104 25 212 173 120 9 196 189
177 208 5 124 161 224 21 108 145 240 37 92 129 256 53 76
13 116 185 200 29 100 169 216 45 84 153 232 61 68 137 248
252 133 80 49 236 149 96 33 220 165 112 17 204 181 128 1

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

With the 10 parameter procedure MstPrfct16b 8448 pairs of Orthogonal Semi-Latin Most Perfect Franklin Pan Magic Squares with Barink Restrictions (A, T(A)) could be generated within 16.3 seconds.

Attachment 16.2.8 shows a few of the resulting order 16 Most Perfect Franklin Pan Magic Squares with Barink Restrictions.

16.2.9 Composed Magic Squares
       Associated Border

Order 16 Magic Squares M, composed of order 7 (Pan) Magic Sub Squares and an Associated Border, can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B).

The numerical example shown below is based on order 7 Latin Diagonal Pan Magic Squares, as discussed in Section 7.2.1,

A
0 3 5 6 8 11 13 14 14 13 11 8 6 5 3 0
15 1 2 3 4 8 13 14 1 2 3 4 8 13 14 15
15 8 13 14 1 2 3 4 8 13 14 1 2 3 4 15
15 2 3 4 8 13 14 1 2 3 4 8 13 14 1 15
15 13 14 1 2 3 4 8 13 14 1 2 3 4 8 15
15 3 4 8 13 14 1 2 3 4 8 13 14 1 2 15
15 14 1 2 3 4 8 13 14 1 2 3 4 8 13 15
15 4 8 13 14 1 2 3 4 8 13 14 1 2 3 15
0 5 6 7 9 10 11 12 5 6 7 9 10 11 12 0
0 10 11 12 5 6 7 9 10 11 12 5 6 7 9 0
0 6 7 9 10 11 12 5 6 7 9 10 11 12 5 0
0 11 12 5 6 7 9 10 11 12 5 6 7 9 10 0
0 7 9 10 11 12 5 6 7 9 10 11 12 5 6 0
0 12 5 6 7 9 10 11 12 5 6 7 9 10 11 0
0 9 10 11 12 5 6 7 9 10 11 12 5 6 7 0
15 12 10 9 7 4 2 1 1 2 4 7 9 10 12 15
Sa
45 45
60 60
B = T(A)
0 15 15 15 15 15 15 15 0 0 0 0 0 0 0 15
3 1 8 2 13 3 14 4 5 10 6 11 7 12 9 12
5 2 13 3 14 4 1 8 6 11 7 12 9 5 10 10
6 3 14 4 1 8 2 13 7 12 9 5 10 6 11 9
8 4 1 8 2 13 3 14 9 5 10 6 11 7 12 7
11 8 2 13 3 14 4 1 10 6 11 7 12 9 5 4
13 13 3 14 4 1 8 2 11 7 12 9 5 10 6 2
14 14 4 1 8 2 13 3 12 9 5 10 6 11 7 1
14 1 8 2 13 3 14 4 5 10 6 11 7 12 9 1
13 2 13 3 14 4 1 8 6 11 7 12 9 5 10 2
11 3 14 4 1 8 2 13 7 12 9 5 10 6 11 4
8 4 1 8 2 13 3 14 9 5 10 6 11 7 12 7
6 8 2 13 3 14 4 1 10 6 11 7 12 9 5 9
5 13 3 14 4 1 8 2 11 7 12 9 5 10 6 10
3 14 4 1 8 2 13 3 12 9 5 10 6 11 7 12
0 15 15 15 15 15 15 15 0 0 0 0 0 0 0 15
Sb
45 60
45 60
C = 16 * A + B + 1
1 64 96 112 144 192 224 240 225 209 177 129 97 81 49 16
244 18 41 51 78 132 223 229 22 43 55 76 136 221 234 253
246 131 222 228 31 37 50 73 135 220 232 29 42 54 75 251
247 36 63 69 130 217 227 30 40 61 74 134 219 231 28 250
249 213 226 25 35 62 68 143 218 230 27 39 60 72 141 248
252 57 67 142 212 239 21 34 59 71 140 216 237 26 38 245
254 238 20 47 53 66 137 211 236 24 45 58 70 139 215 243
255 79 133 210 233 19 46 52 77 138 214 235 23 44 56 242
15 82 105 115 158 164 191 197 86 107 119 156 168 189 202 2
14 163 190 196 95 101 114 153 167 188 200 93 106 118 155 3
12 100 127 149 162 185 195 94 104 125 154 166 187 199 92 5
9 181 194 89 99 126 148 175 186 198 91 103 124 152 173 8
7 121 147 174 180 207 85 98 123 151 172 184 205 90 102 10
6 206 84 111 117 146 169 179 204 88 109 122 150 171 183 11
4 159 165 178 201 83 110 116 157 170 182 203 87 108 120 13
241 208 176 160 128 80 48 32 17 33 65 113 145 161 193 256
Sc
772 787
1012 1027

The balanced series {0, 1, 2, 3, 4 ... 15} has been split into two unbalanced sub series:

       {1, 2, 3, 4, 8, 13, 14}, {5, 6, 7, 9, 10, 11, 12} and a pair {0, 15}

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

Attachment 16.28.1 shows a few suitable unique sets (16 ea) of order 7 sub series for the integers 0 ... 15.

Attachment 16.28.2 shows the resulting order 16 Magic Squares composed of Pan Magic Sub Squares.

Based on the construction method described above 28802 * (7!)2 = 2,11 1014 squares can be constructed for each set of order 7 sub series shown in Attachment 16.28.1.

Each square shown in Attachment 16.28.2 corresponds with (304.819.200)4 * (7!)2 = 2,19 1041 squares, which can be obtained by permutations within the border and/or selecting other aspects of the Pan Magic Sub Squares.

16.2.10 Associated Magic Squares
        Pan Magic Square Inlays

Comparable with Section 16.2.9 above, order 16 Associated Magic Squares M, with order 7 Pan Magic Square Inlays, can be constructed based on pairs of Orthogonal Inlaid Semi-Latin Squares (A, B).

Attachment 16.28.3 shows a few suitable unique sets (8 ea) of order 7 sub series for the integers 0 ... 15.

Attachment 16.28.4 shows the resulting order 16 Associated Magic Squares with Pan Magic Square Inlays.

Based on the construction method described above 2880 * (7!)2 = 7,32 1010 squares can be constructed for each set of order 7 sub series shown in Attachment 16.28.3.

Each square shown in Attachment 16.28.4 corresponds with (304.819.200)2 * (7!)2 = 2,36 1024 squares, which can be obtained by permutations within the border and/or selecting other aspects of two of the Pan Magic Sub Squares.

16.3   Summary

The obtained results regarding the order 16 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Type

Characteristics

Subroutine

Attachment 16.2.11

Pan Magic

Compact, Complete Pan Magic Squares
All 1/2 Rows and 1/2 Columns sum to s16/2

CompLat16a

Attachment 16.2.12

Associated

Partly Compact, Semi-Diagonals sum to s16
All 1/2 Rows and 1/2 Columns sum to s16/2

-

Attachment 16.2.2

Ultra Magic

Compact, Associated Pan Magic Squares
All 1/2 Rows and 1/2 Columns sum to s16/2

AssLat16a

Attachment 16.23.2

Composed

Composed of Latin Diagonal Sub Squares

-

-

-

-

-

Attachment 16.2.4

Franklin

Pan Magic Squares

Franklin16a

Attachment 16.2.5

Pan Magic and Complete Squares

Franklin16b

Attachment 16.2.6

Pan Magic and Associated Squares

Franklin16c

Attachment 16.2.7

Most Perfect Franklin Pan Magic Squares

MstPrfct16a

Attachment 16.2.8

Most Perfect Franklin Pan Magic Squares
Barink Restrictions

MstPrfct16b

-

-

-

-

Attachment 16.28.2

Composed

Associated Border

-

Attachment 16.28.4

Composed

Associated

-

-

-

-

-

Comparable methods as described above, can be used to construct order 17 (Semi) Latin - and related (Pan) Magic Squares, which will be described in following sections.

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