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18.0   Latin Squares (18 x 18)

A Latin Square of order 18 is a 18 x 18 square filled with 18 different symbols, each occurring only once in each row and only once in each column.

18.1   Latin Diagonal Squares (18 x 18)

Latin Diagonal Squares are Latin Squares for which the 18 different symbols occur also only once in each of the main diagonals.

18.2   Magic Squares, Natural Numbers

18.2.1 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 >= 10 (1959/1960).

Harry White provides on his Web Site miscellaneous (historical) construction methods for order 18 Self Orthogonal Latin Squares, of which one has been used as an example in Section 18.2.2 below.

18.2.2 Simple Magic Squares

An numerical example of the construction of an order 18 Simple Magic Square M based on a Self Orthogonal Latin Diagonal Square A, is shown below for the integers {0, 1, ... 17}:

A1
0 11 16 12 5 8 7 1 17 2 3 10 9 4 15 13 6 14
7 1 17 15 11 6 8 2 16 3 4 9 5 14 12 0 13 10
8 7 2 16 14 17 0 3 15 4 5 6 13 11 1 12 10 9
1 8 7 3 15 13 16 4 14 5 6 12 17 2 11 10 9 0
15 2 8 7 4 14 12 5 13 6 0 16 3 17 10 9 1 11
11 14 3 8 7 5 13 6 12 0 1 4 16 10 9 2 17 15
12 17 13 4 8 7 6 0 11 1 2 15 10 9 3 16 14 5
3 4 5 6 0 1 2 7 10 8 9 17 11 12 13 14 15 16
16 15 14 13 12 11 17 9 8 10 7 2 1 0 6 5 4 3
17 16 15 14 13 12 11 10 7 9 8 1 0 6 5 4 3 2
4 5 6 0 1 2 3 8 9 7 10 13 14 15 16 17 11 12
13 6 12 2 9 10 1 15 5 14 16 11 7 8 17 3 0 4
5 13 1 9 10 0 14 16 4 15 17 3 12 7 8 11 2 6
14 0 9 10 6 15 4 17 3 16 11 5 2 13 7 8 12 1
6 9 10 5 16 3 15 11 2 17 12 0 4 1 14 7 8 13
9 10 4 17 2 16 5 12 1 11 13 14 6 3 0 15 7 8
10 3 11 1 17 4 9 13 0 12 14 8 15 5 2 6 16 7
2 12 0 11 3 9 10 14 6 13 15 7 8 16 4 1 5 17
B = T(A)
0 7 8 1 15 11 12 3 16 17 4 13 5 14 6 9 10 2
11 1 7 8 2 14 17 4 15 16 5 6 13 0 9 10 3 12
16 17 2 7 8 3 13 5 14 15 6 12 1 9 10 4 11 0
12 15 16 3 7 8 4 6 13 14 0 2 9 10 5 17 1 11
5 11 14 15 4 7 8 0 12 13 1 9 10 6 16 2 17 3
8 6 17 13 14 5 7 1 11 12 2 10 0 15 3 16 4 9
7 8 0 16 12 13 6 2 17 11 3 1 14 4 15 5 9 10
1 2 3 4 5 6 0 7 9 10 8 15 16 17 11 12 13 14
17 16 15 14 13 12 11 10 8 7 9 5 4 3 2 1 0 6
2 3 4 5 6 0 1 8 10 9 7 14 15 16 17 11 12 13
3 4 5 6 0 1 2 9 7 8 10 16 17 11 12 13 14 15
10 9 6 12 16 4 15 17 2 1 13 11 3 5 0 14 8 7
9 5 13 17 3 16 10 11 1 0 14 7 12 2 4 6 15 8
4 14 11 2 17 10 9 12 0 6 15 8 7 13 1 3 5 16
15 12 1 11 10 9 3 13 6 5 16 17 8 7 14 0 2 4
13 0 12 10 9 2 16 14 5 4 17 3 11 8 7 15 6 1
6 13 10 9 1 17 14 15 4 3 11 0 2 12 8 7 16 5
14 10 9 0 11 15 5 16 3 2 12 4 6 1 13 8 7 17
M = A + 18 * B + 1
1 138 161 31 276 207 224 56 306 309 76 245 100 257 124 176 187 51
206 20 144 160 48 259 315 75 287 292 95 118 240 15 175 181 68 227
297 314 39 143 159 72 235 94 268 275 114 223 32 174 182 85 209 10
218 279 296 58 142 158 89 113 249 258 7 49 180 183 102 317 28 199
106 201 261 278 77 141 157 6 230 241 19 179 184 126 299 46 308 66
156 123 310 243 260 96 140 25 211 217 38 185 17 281 64 291 90 178
139 162 14 293 225 242 115 37 318 200 57 34 263 82 274 107 177 186
22 41 60 79 91 110 3 134 173 189 154 288 300 319 212 231 250 269
323 304 285 266 247 228 216 190 153 137 170 93 74 55 43 24 5 112
54 71 88 105 122 13 30 155 188 172 135 254 271 295 312 203 220 237
59 78 97 109 2 21 40 171 136 152 191 302 321 214 233 252 264 283
194 169 121 219 298 83 272 322 42 33 251 210 62 99 18 256 145 131
168 104 236 316 65 289 195 215 23 16 270 130 229 44 81 120 273 151
87 253 208 47 313 196 167 234 4 125 282 150 129 248 26 63 103 290
277 226 29 204 197 166 70 246 111 108 301 307 149 128 267 8 45 86
244 11 221 198 165 53 294 265 92 84 320 69 205 148 127 286 116 27
119 238 192 164 36 311 262 284 73 67 213 9 52 222 147 133 305 98
255 193 163 12 202 280 101 303 61 50 232 80 117 35 239 146 132 324

The Self Orthogonal Latin Diagonal Square A shown above corresponds with 18! = 6,40 10i5 Self Orthogonal Latin Diagonal Squares, which can be obtained by permutation of the integers {ai, i = 1 ... 18}.

The magic constant of the order 4 centre square - around which the Self Orthogonal Latin Diagonal Square A has been constructed - will change For the majority of these permutations.

In addition to the permutations described above, each Self Orthogonal Latin Diagonal Square A corresponds with 92.897.280 transformations, as described below.

  • Any line n can be interchanged with line (19 - n). The possible number of transformations is 29 = 512
    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, ... 9, provided that the same permutation is applied to the lines 18, 17, ... 10. The possible number of transformations is 9! = 362.880.

The resulting number of transformations, excluding the 180o rotated aspects, is 512/2 * 362.880 = 92.897.280.

For the majority of these transformations the order 4 centre square will change into a hardly recognisable order 4 inlaid magic square.

18.2.4 Almost Associated Magic Squares

Order 18 Almost Associated Magic Squares composed out of

  • An order 18 Associated Border and
  • An order 14 Almost Associated Center Square (ref. Section 14.2.4)

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

A
17 16 15 14 13 11 10 9 8 7 6 5 4 12 3 2 1 0
0 1 2 3 11 6 5 4 8 9 7 10 13 12 14 15 16 17
17 16 15 14 15 14 14 2 3 2 2 15 15 3 3 2 1 0
0 1 2 3 3 15 3 14 2 3 14 2 14 15 14 15 16 17
17 1 13 11 13 11 13 5 4 4 13 5 12 5 5 5 0 16
16 17 11 6 4 12 4 13 12 12 5 12 4 7 4 13 1 0
16 1 10 5 5 10 11 6 11 8 6 9 10 9 7 12 17 0
0 16 9 4 6 9 7 10 7 7 10 10 13 6 10 11 1 17
1 0 8 8 9 6 6 9 8 11 9 8 9 10 8 10 16 17
0 1 7 9 7 8 9 8 6 9 8 11 11 8 9 9 17 16
0 16 6 7 11 4 10 7 10 10 7 7 8 11 13 8 1 17
17 0 5 10 8 7 8 11 9 6 11 6 7 12 12 7 16 1
17 16 4 13 10 13 5 12 5 5 4 13 5 13 11 6 0 1
1 17 12 12 12 5 12 4 13 13 12 4 6 4 6 4 16 0
0 1 2 3 2 3 15 3 14 15 3 14 2 14 14 15 16 17
17 16 15 14 14 2 2 15 15 14 15 3 3 2 3 2 1 0
0 1 2 3 5 4 7 10 8 9 13 12 11 6 14 15 16 17
17 16 15 14 5 13 12 11 10 9 8 7 6 4 3 2 1 0
B
17 0 17 0 17 16 16 0 1 0 0 17 17 1 0 17 0 17
16 1 16 1 1 17 1 16 0 1 16 0 16 17 1 16 1 16
15 2 15 2 13 11 10 9 8 7 6 5 4 12 2 15 2 15
14 3 14 3 11 6 5 4 8 9 7 10 13 12 3 14 3 14
13 11 15 3 13 4 5 6 9 7 11 8 10 12 2 14 5 5
11 6 14 15 11 12 10 9 6 8 4 7 13 5 3 2 4 13
10 5 14 3 13 4 11 7 9 6 10 8 5 12 15 2 7 12
9 4 2 14 5 13 8 10 6 11 7 9 12 4 3 15 10 11
8 8 3 2 4 12 9 7 8 6 10 11 5 13 14 15 8 10
7 9 2 3 4 12 6 7 11 9 10 8 5 13 15 14 9 9
6 7 2 14 13 5 8 10 6 11 7 9 4 12 3 15 13 8
5 10 15 2 5 12 9 10 11 8 7 6 13 4 14 3 12 7
4 13 15 14 12 4 10 13 9 11 8 7 5 6 2 3 11 6
12 12 3 15 5 7 9 6 10 8 11 12 13 4 14 2 6 4
3 14 3 14 5 4 7 10 8 9 13 12 11 6 14 3 14 3
2 15 2 15 5 13 12 11 10 9 8 7 6 4 15 2 15 2
1 16 1 16 0 1 17 1 16 17 1 16 0 16 16 1 16 1
0 17 0 17 16 0 0 17 17 16 17 1 1 0 17 0 17 0
M = A + 18 * B + 1
324 17 322 15 320 300 299 10 27 8 7 312 311 31 4 309 2 307
289 20 291 22 30 313 24 293 9 28 296 11 302 319 33 304 35 306
288 53 286 51 250 213 195 165 148 129 111 106 88 220 40 273 38 271
253 56 255 58 202 124 94 87 147 166 141 183 249 232 69 268 71 270
252 200 284 66 248 84 104 114 167 131 212 150 193 222 42 258 91 107
215 126 264 277 203 229 185 176 121 157 78 139 239 98 59 50 74 235
197 92 263 60 240 83 210 133 174 117 187 154 101 226 278 49 144 217
163 89 46 257 97 244 152 191 116 206 137 173 230 79 65 282 182 216
146 145 63 45 82 223 169 136 153 120 190 207 100 245 261 281 161 198
127 164 44 64 80 225 118 135 205 172 189 156 102 243 280 262 180 179
109 143 43 260 246 95 155 188 119 209 134 170 81 228 68 279 236 162
108 181 276 47 99 224 171 192 208 151 138 115 242 85 265 62 233 128
90 251 275 266 227 86 186 247 168 204 149 140 96 122 48 61 199 110
218 234 67 283 103 132 175 113 194 158 211 221 241 77 259 41 125 73
55 254 57 256 93 76 142 184 159 178 238 231 201 123 267 70 269 72
54 287 52 285 105 237 219 214 196 177 160 130 112 75 274 39 272 37
19 290 21 292 6 23 314 29 297 316 32 301 12 295 303 34 305 36
18 323 16 321 294 14 13 318 317 298 315 26 25 5 310 3 308 1

The Mutual Orthogonal Semi-Latin Squares (A, B) shown above, are based on the order 14 Almost Associated Magic Squares shown in Section 14.2.4.

Due to the application of an orde 6 Almost Associated Centre Square as discussed in Section 6.2.4, only the Associated Border of the Semi-Latin Square A is Self-Orthogonal.

18.2.5 Bordered Magic Squares

The order 16 Orthogonal Latin Diagonal or Semi-Latin Squares (A16, B16), as discussed in e.g. Section 16.2.7, have been used to construct collections of Magic Squares based on the Balanced Series {0 ... 15}.

From the Balanced Series {0 ... 17} nine suitable Balanced Sub Series can be selected to construct Center Squares for order 18 Bordered Magic Squares e.g. {1 ... 16}.

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 7 6 5 4 3 1 16 17 0 17 17 17 17 0 3 15
15 8 7 10 9 6 5 12 11 4 3 14 13 2 1 16 15 2
13 9 10 7 8 11 12 5 6 13 14 3 4 15 16 1 2 4
12 7 8 9 10 5 6 11 12 3 4 13 14 1 2 15 16 5
11 10 9 8 7 12 11 6 5 14 13 4 3 16 15 2 1 6
17 8 7 10 9 6 5 12 11 4 3 14 13 2 1 16 15 0
0 9 10 7 8 11 12 5 6 13 14 3 4 15 16 1 2 17
7 7 8 9 10 5 6 11 12 3 4 13 14 1 2 15 16 10
8 10 9 8 7 12 11 6 5 14 13 4 3 16 15 2 1 9
17 8 7 10 9 6 5 12 11 4 3 14 13 2 1 16 15 0
17 9 10 7 8 11 12 5 6 13 14 3 4 15 16 1 2 0
0 7 8 9 10 5 6 11 12 3 4 13 14 1 2 15 16 17
17 10 9 8 7 12 11 6 5 14 13 4 3 16 15 2 1 0
0 8 7 10 9 6 5 12 11 4 3 14 13 2 1 16 15 17
17 9 10 7 8 11 12 5 6 13 14 3 4 15 16 1 2 0
0 7 8 9 10 5 6 11 12 3 4 13 14 1 2 15 16 17
0 10 9 8 7 12 11 6 5 14 13 4 3 16 15 2 1 17
2 9 10 11 12 13 14 16 1 0 17 0 0 0 0 17 14 17
B
0 17 17 17 17 17 17 0 0 0 1 1 7 13 14 15 0 0
17 8 9 7 10 8 9 7 10 8 9 7 10 8 9 7 10 0
17 7 10 8 9 7 10 8 9 7 10 8 9 7 10 8 9 0
17 10 7 9 8 10 7 9 8 10 7 9 8 10 7 9 8 0
17 9 8 10 7 9 8 10 7 9 8 10 7 9 8 10 7 0
15 6 11 5 12 6 11 5 12 6 11 5 12 6 11 5 12 2
9 5 12 6 11 5 12 6 11 5 12 6 11 5 12 6 11 8
0 12 5 11 6 12 5 11 6 12 5 11 6 12 5 11 6 17
0 11 6 12 5 11 6 12 5 11 6 12 5 11 6 12 5 17
3 4 13 3 14 4 13 3 14 4 13 3 14 4 13 3 14 14
4 3 14 4 13 3 14 4 13 3 14 4 13 3 14 4 13 13
5 14 3 13 4 14 3 13 4 14 3 13 4 14 3 13 4 12
5 13 4 14 3 13 4 14 3 13 4 14 3 13 4 14 3 12
6 2 15 1 16 2 15 1 16 2 15 1 16 2 15 1 16 11
6 1 16 2 15 1 16 2 15 1 16 2 15 1 16 2 15 11
7 16 1 15 2 16 1 15 2 16 1 15 2 16 1 15 2 10
8 15 2 16 1 15 2 16 1 15 2 16 1 15 2 16 1 9
17 0 0 0 0 0 0 17 17 17 16 16 10 4 3 2 17 17
M = A + 18 * B + 1
1 315 314 313 312 311 310 2 17 18 19 36 144 252 270 271 4 16
322 153 170 137 190 151 168 139 192 149 166 141 194 147 164 143 196 3
320 136 191 152 171 138 193 150 169 140 195 148 167 142 197 146 165 5
319 188 135 172 155 186 133 174 157 184 131 176 159 182 129 178 161 6
318 173 154 189 134 175 156 187 132 177 158 185 130 179 160 183 128 7
288 117 206 101 226 115 204 103 228 113 202 105 230 111 200 107 232 37
163 100 227 116 207 102 229 114 205 104 231 112 203 106 233 110 201 162
8 224 99 208 119 222 97 210 121 220 95 212 123 218 93 214 125 317
9 209 118 225 98 211 120 223 96 213 122 221 94 215 124 219 92 316
72 81 242 65 262 79 240 67 264 77 238 69 266 75 236 71 268 253
90 64 263 80 243 66 265 78 241 68 267 76 239 70 269 74 237 235
91 260 63 244 83 258 61 246 85 256 59 248 87 254 57 250 89 234
108 245 82 261 62 247 84 259 60 249 86 257 58 251 88 255 56 217
109 45 278 29 298 43 276 31 300 41 274 33 302 39 272 35 304 216
126 28 299 44 279 30 301 42 277 32 303 40 275 34 305 38 273 199
127 296 27 280 47 294 25 282 49 292 23 284 51 290 21 286 53 198
145 281 46 297 26 283 48 295 24 285 50 293 22 287 52 291 20 180
309 10 11 12 13 14 15 323 308 307 306 289 181 73 55 54 321 324

Each pair of order 18 Orthogonal Non-Latin Borders corresponds with 8 * (16!)2 = 3,502 1027 Borders.

18.2.6 Composed Magic Squares

Magic Center Cross (2 x 18)

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 8 Orthogonal Semi-Latin Squares (A8, B8).

Following numerical example is based on Composed Magic Sub Squares and the balanced Sub Series {1 ... 16}:

A
8 7 10 9 6 5 12 11 15 2 4 3 14 13 2 1 16 15
9 10 7 8 11 12 5 6 13 4 13 14 3 4 15 16 1 2
7 8 9 10 5 6 11 12 12 5 3 4 13 14 1 2 15 16
10 9 8 7 12 11 6 5 11 6 14 13 4 3 16 15 2 1
8 7 10 9 6 5 12 11 17 0 4 3 14 13 2 1 16 15
9 10 7 8 11 12 5 6 0 17 13 14 3 4 15 16 1 2
7 8 9 10 5 6 11 12 7 10 3 4 13 14 1 2 15 16
10 9 8 7 12 11 6 5 8 9 14 13 4 3 16 15 2 1
8 7 6 5 4 3 1 16 0 15 17 0 17 17 17 17 0 3
9 10 11 12 13 14 16 1 2 17 0 17 0 0 0 0 17 14
8 7 10 9 6 5 12 11 17 0 4 3 14 13 2 1 16 15
9 10 7 8 11 12 5 6 17 0 13 14 3 4 15 16 1 2
7 8 9 10 5 6 11 12 0 17 3 4 13 14 1 2 15 16
10 9 8 7 12 11 6 5 17 0 14 13 4 3 16 15 2 1
8 7 10 9 6 5 12 11 0 17 4 3 14 13 2 1 16 15
9 10 7 8 11 12 5 6 17 0 13 14 3 4 15 16 1 2
7 8 9 10 5 6 11 12 0 17 3 4 13 14 1 2 15 16
10 9 8 7 12 11 6 5 0 17 14 13 4 3 16 15 2 1
B
8 9 7 10 8 9 7 10 17 0 8 9 7 10 8 9 7 10
7 10 8 9 7 10 8 9 17 0 7 10 8 9 7 10 8 9
10 7 9 8 10 7 9 8 17 0 10 7 9 8 10 7 9 8
9 8 10 7 9 8 10 7 17 0 9 8 10 7 9 8 10 7
6 11 5 12 6 11 5 12 15 2 6 11 5 12 6 11 5 12
5 12 6 11 5 12 6 11 9 8 5 12 6 11 5 12 6 11
12 5 11 6 12 5 11 6 0 17 12 5 11 6 12 5 11 6
11 6 12 5 11 6 12 5 0 17 11 6 12 5 11 6 12 5
17 17 17 17 17 17 0 0 0 0 0 1 1 7 13 14 15 0
0 0 0 0 0 0 17 17 17 17 17 16 16 10 4 3 2 17
4 13 3 14 4 13 3 14 3 14 4 13 3 14 4 13 3 14
3 14 4 13 3 14 4 13 4 13 3 14 4 13 3 14 4 13
14 3 13 4 14 3 13 4 5 12 14 3 13 4 14 3 13 4
13 4 14 3 13 4 14 3 5 12 13 4 14 3 13 4 14 3
2 15 1 16 2 15 1 16 6 11 2 15 1 16 2 15 1 16
1 16 2 15 1 16 2 15 6 11 1 16 2 15 1 16 2 15
16 1 15 2 16 1 15 2 7 10 16 1 15 2 16 1 15 2
15 2 16 1 15 2 16 1 8 9 15 2 16 1 15 2 16 1
M = A + 18 * B + 1
153 170 137 190 151 168 139 192 322 3 149 166 141 194 147 164 143 196
136 191 152 171 138 193 150 169 320 5 140 195 148 167 142 197 146 165
188 135 172 155 186 133 174 157 319 6 184 131 176 159 182 129 178 161
173 154 189 134 175 156 187 132 318 7 177 158 185 130 179 160 183 128
117 206 101 226 115 204 103 228 288 37 113 202 105 230 111 200 107 232
100 227 116 207 102 229 114 205 163 162 104 231 112 203 106 233 110 201
224 99 208 119 222 97 210 121 8 317 220 95 212 123 218 93 214 125
209 118 225 98 211 120 223 96 9 316 213 122 221 94 215 124 219 92
315 314 313 312 311 310 2 17 1 16 18 19 36 144 252 270 271 4
10 11 12 13 14 15 323 308 309 324 307 306 289 181 73 55 54 321
81 242 65 262 79 240 67 264 72 253 77 238 69 266 75 236 71 268
64 263 80 243 66 265 78 241 90 235 68 267 76 239 70 269 74 237
260 63 244 83 258 61 246 85 91 234 256 59 248 87 254 57 250 89
245 82 261 62 247 84 259 60 108 217 249 86 257 58 251 88 255 56
45 278 29 298 43 276 31 300 109 216 41 274 33 302 39 272 35 304
28 299 44 279 30 301 42 277 126 199 32 303 40 275 34 305 38 273
296 27 280 47 294 25 282 49 127 198 292 23 284 51 290 21 286 53
281 46 297 26 283 48 295 24 145 180 285 50 293 22 287 52 291 20

Each pair of order 18 Orthogonal Non-Latin Center Crosses corresponds with 8 * (16!)2 = 3,502 1027 Center Crosses.

18.2.7 Composed Magic Squares

Magic Sub Squares Order 6

Order 18 Magic Squares M composed of order 6 Magic Sub Squares can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B).

A numerical example, based on order 6 Semi-Latin Magic Squares with Symmetrical Diagonals - as discussed in Section 6.2.3 - for the untegers {0, 1, ... 17} is shown below:

A
17 1 15 2 16 0 14 4 12 5 13 3 11 7 9 8 10 6
0 16 2 15 1 17 3 13 5 12 4 14 6 10 8 9 7 11
0 1 2 15 16 17 3 4 5 12 13 14 6 7 8 9 10 11
17 1 2 15 16 0 14 4 5 12 13 3 11 7 8 9 10 6
0 16 15 2 1 17 3 13 12 5 4 14 6 10 9 8 7 11
17 16 15 2 1 0 14 13 12 5 4 3 11 10 9 8 7 6
17 1 15 2 16 0 14 4 12 5 13 3 11 7 9 8 10 6
0 16 2 15 1 17 3 13 5 12 4 14 6 10 8 9 7 11
0 1 2 15 16 17 3 4 5 12 13 14 6 7 8 9 10 11
17 1 2 15 16 0 14 4 5 12 13 3 11 7 8 9 10 6
0 16 15 2 1 17 3 13 12 5 4 14 6 10 9 8 7 11
17 16 15 2 1 0 14 13 12 5 4 3 11 10 9 8 7 6
17 1 15 2 16 0 14 4 12 5 13 3 11 7 9 8 10 6
0 16 2 15 1 17 3 13 5 12 4 14 6 10 8 9 7 11
0 1 2 15 16 17 3 4 5 12 13 14 6 7 8 9 10 11
17 1 2 15 16 0 14 4 5 12 13 3 11 7 8 9 10 6
0 16 15 2 1 17 3 13 12 5 4 14 6 10 9 8 7 11
17 16 15 2 1 0 14 13 12 5 4 3 11 10 9 8 7 6
B = T(A)
17 0 0 17 0 17 17 0 0 17 0 17 17 0 0 17 0 17
1 16 1 1 16 16 1 16 1 1 16 16 1 16 1 1 16 16
15 2 2 2 15 15 15 2 2 2 15 15 15 2 2 2 15 15
2 15 15 15 2 2 2 15 15 15 2 2 2 15 15 15 2 2
16 1 16 16 1 1 16 1 16 16 1 1 16 1 16 16 1 1
0 17 17 0 17 0 0 17 17 0 17 0 0 17 17 0 17 0
14 3 3 14 3 14 14 3 3 14 3 14 14 3 3 14 3 14
4 13 4 4 13 13 4 13 4 4 13 13 4 13 4 4 13 13
12 5 5 5 12 12 12 5 5 5 12 12 12 5 5 5 12 12
5 12 12 12 5 5 5 12 12 12 5 5 5 12 12 12 5 5
13 4 13 13 4 4 13 4 13 13 4 4 13 4 13 13 4 4
3 14 14 3 14 3 3 14 14 3 14 3 3 14 14 3 14 3
11 6 6 11 6 11 11 6 6 11 6 11 11 6 6 11 6 11
7 10 7 7 10 10 7 10 7 7 10 10 7 10 7 7 10 10
9 8 8 8 9 9 9 8 8 8 9 9 9 8 8 8 9 9
8 9 9 9 8 8 8 9 9 9 8 8 8 9 9 9 8 8
10 7 10 10 7 7 10 7 10 10 7 7 10 7 10 10 7 7
6 11 11 6 11 6 6 11 11 6 11 6 6 11 11 6 11 6
M = A + 18 * B + 1
324 2 16 309 17 307 321 5 13 312 14 310 318 8 10 315 11 313
19 305 21 34 290 306 22 302 24 31 293 303 25 299 27 28 296 300
271 38 39 52 287 288 274 41 42 49 284 285 277 44 45 46 281 282
54 272 273 286 53 37 51 275 276 283 50 40 48 278 279 280 47 43
289 35 304 291 20 36 292 32 301 294 23 33 295 29 298 297 26 30
18 323 322 3 308 1 15 320 319 6 311 4 12 317 316 9 314 7
270 56 70 255 71 253 267 59 67 258 68 256 264 62 64 261 65 259
73 251 75 88 236 252 76 248 78 85 239 249 79 245 81 82 242 246
217 92 93 106 233 234 220 95 96 103 230 231 223 98 99 100 227 228
108 218 219 232 107 91 105 221 222 229 104 94 102 224 225 226 101 97
235 89 250 237 74 90 238 86 247 240 77 87 241 83 244 243 80 84
72 269 268 57 254 55 69 266 265 60 257 58 66 263 262 63 260 61
216 110 124 201 125 199 213 113 121 204 122 202 210 116 118 207 119 205
127 197 129 142 182 198 130 194 132 139 185 195 133 191 135 136 188 192
163 146 147 160 179 180 166 149 150 157 176 177 169 152 153 154 173 174
162 164 165 178 161 145 159 167 168 175 158 148 156 170 171 172 155 151
181 143 196 183 128 144 184 140 193 186 131 141 187 137 190 189 134 138
126 215 214 111 200 109 123 212 211 114 203 112 120 209 208 117 206 115

The balanced series {0, 1, 2, 3, 4 ... 17} has been split into three balanced sub series, each summing to s6 = 51:

     {0, 1, 2, 15, 16, 17}, {3, 4, 5, 12, 13, 14} and {6, 7, 8, 9, 10, 11}

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

The 560 unique sets of balanced series, which can be used for the construction method illustrated above, are shown in Attachment 22.9.1.

Attachment 22.9.2 shows a few more order 18 Magic Squares composed of nine Magic Sub Squares, rather based on Mutual Orthogonal - than on Self Orthogonal Semi-Latin Squares.

Each constructed square corresponds with numerous Composed Magic Squares, which can be obtained by permutation of the sub squares and (or) selecting other aspects (of the sub squares).

18.2.8 Composed Magic Squares

Magic Sub Squares Order 4 and 14

Order 18 Composed Magic Squares, can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B) for miscellaneous types of Sub Squares.

The example shown below is based on:

  • Five Order  4 Latin Diagonal Pan Magic Sub Square and
  • One  Order 14 Semi-Latin Diagonal Composed Magic Sub Square

as discussed in resp. Section 4.2.2 and Section 14.2.7,

A
2 1 18 17 4 3 16 15 6 5 14 13 8 9 10 11 12 7
17 18 1 2 15 16 3 4 13 14 5 6 11 12 7 9 8 10
1 2 17 18 3 4 15 16 5 6 13 14 8 7 9 10 11 12
18 17 2 1 16 15 4 3 14 13 6 5 11 10 12 8 7 9
2 1 18 17 4 3 16 15 6 5 14 13 8 9 10 11 12 7
17 18 1 2 15 16 3 4 13 14 5 6 11 12 8 10 9 7
1 2 17 18 3 4 15 16 5 6 13 14 11 8 9 10 7 12
18 17 2 1 16 15 4 3 14 13 6 5 8 9 11 7 10 12
2 1 18 17 4 3 16 15 6 5 14 13 7 8 12 10 11 9
17 18 1 2 15 16 3 4 13 14 5 6 12 9 7 8 10 11
1 2 17 18 3 4 15 16 5 6 13 14 12 10 7 11 9 8
18 17 2 1 16 15 4 3 14 13 6 5 7 11 12 9 8 10
2 18 1 17 3 15 4 16 5 13 6 14 12 8 9 10 11 7
1 2 17 18 3 16 15 4 5 6 14 13 7 11 9 10 8 12
17 2 18 1 15 4 16 3 14 6 13 5 7 8 10 9 11 12
1 17 2 18 16 3 4 15 5 14 6 13 12 8 10 9 11 7
18 17 2 1 4 15 3 16 14 13 5 6 7 11 10 9 8 12
18 1 17 2 16 4 15 3 14 5 13 6 12 11 9 10 8 7
B = T(A)
2 17 1 18 2 17 1 18 2 17 1 18 2 1 17 1 18 18
1 18 2 17 1 18 2 17 1 18 2 17 18 2 2 17 17 1
18 1 17 2 18 1 17 2 18 1 17 2 1 17 18 2 2 17
17 2 18 1 17 2 18 1 17 2 18 1 17 18 1 18 1 2
4 15 3 16 4 15 3 16 4 15 3 16 3 3 15 16 4 16
3 16 4 15 3 16 4 15 3 16 4 15 15 16 4 3 15 4
16 3 15 4 16 3 15 4 16 3 15 4 4 15 16 4 3 15
15 4 16 3 15 4 16 3 15 4 16 3 16 4 3 15 16 3
6 13 5 14 6 13 5 14 6 13 5 14 5 5 14 5 14 14
5 14 6 13 5 14 6 13 5 14 6 13 13 6 6 14 13 5
14 5 13 6 14 5 13 6 14 5 13 6 6 14 13 6 5 13
13 6 14 5 13 6 14 5 13 6 14 5 14 13 5 13 6 6
8 11 8 11 8 11 11 8 7 12 12 7 12 7 7 12 7 12
9 12 7 10 9 12 8 9 8 9 10 11 8 11 8 8 11 11
10 7 9 12 10 8 9 11 12 7 7 12 9 9 10 10 10 9
11 9 10 8 11 10 10 7 10 8 11 9 10 10 9 9 9 10
12 8 11 7 12 9 7 10 11 10 9 8 11 8 11 11 8 8
7 10 12 9 7 7 12 12 9 11 8 10 7 12 12 7 12 7
M = A + (B - 1) * 18
20 289 18 323 22 291 16 321 24 293 14 319 26 9 298 11 318 313
17 324 19 290 15 322 21 292 13 320 23 294 317 30 25 297 296 10
307 2 305 36 309 4 303 34 311 6 301 32 8 295 315 28 29 300
306 35 308 1 304 33 310 3 302 31 312 5 299 316 12 314 7 27
56 253 54 287 58 255 52 285 60 257 50 283 44 45 262 281 66 277
53 288 55 254 51 286 57 256 49 284 59 258 263 282 62 46 261 61
271 38 269 72 273 40 267 70 275 42 265 68 65 260 279 64 43 264
270 71 272 37 268 69 274 39 266 67 276 41 278 63 47 259 280 48
92 217 90 251 94 219 88 249 96 221 86 247 79 80 246 82 245 243
89 252 91 218 87 250 93 220 85 248 95 222 228 99 97 242 226 83
235 74 233 108 237 76 231 106 239 78 229 104 102 244 223 101 81 224
234 107 236 73 232 105 238 75 230 103 240 77 241 227 84 225 98 100
128 198 127 197 129 195 184 142 113 211 204 122 210 116 117 208 119 205
145 200 125 180 147 214 141 148 131 150 176 193 133 191 135 136 188 192
179 110 162 199 177 130 160 183 212 114 121 203 151 152 172 171 173 156
181 161 164 144 196 165 166 123 167 140 186 157 174 170 154 153 155 169
216 143 182 109 202 159 111 178 194 175 149 132 187 137 190 189 134 138
126 163 215 146 124 112 213 201 158 185 139 168 120 209 207 118 206 115

Attachment 18.27,1 shows a few more order 18 Self Orthogonal Composed Semi-Latin Squares, with order 4 and 14 Sub Squares, which can be found with procedure CompLat18.

The enclosed squares are the first occurring for each of the Balanced Series defining the Corner Squares (ref. Attachment 18.27.2).

Each square shown corresponds with numerous squares, which can be found by selecting other aspects of the corner squares in combination with procedure CompLat18.

18.2.9 Simple Magic Squares

Latin Patterns

A nice numerical example of the construction of an order 18 Simple Magic Square M, based on a pair of Orthogonal Squares (A, B) with a Non-Latin Square A and a Latin Diagonal Square B, is shown below:

A
6 16 5 9 15 6 9 4 17 2 15 10 13 4 10 14 3 13
18 2 12 18 6 6 2 16 1 18 3 17 13 13 1 7 17 1
6 5 2 10 13 6 3 4 15 4 15 16 13 6 9 17 14 13
1 11 9 4 18 15 3 9 4 15 10 16 4 1 15 10 8 18
9 2 11 13 7 11 10 15 6 13 4 9 8 12 6 8 17 10
8 2 17 14 10 18 10 3 18 1 16 9 1 9 5 2 17 11
7 11 8 2 6 8 16 14 5 14 5 3 11 13 17 11 8 12
14 9 15 14 3 5 9 16 3 16 3 10 14 16 5 4 10 5
11 16 2 2 15 4 18 1 7 12 18 1 15 4 17 17 3 8
18 17 12 8 14 15 15 4 12 7 15 4 4 5 11 7 2 1
2 7 14 8 2 7 16 1 16 3 18 3 12 17 11 5 12 17
14 15 11 3 5 6 12 12 14 5 7 7 13 14 16 8 4 5
14 9 1 14 12 7 1 8 18 1 11 18 12 7 5 18 10 5
8 2 12 18 5 15 6 11 6 13 8 13 4 14 1 7 17 11
6 15 10 11 9 2 6 18 9 10 1 13 17 10 8 9 4 13
17 16 12 7 18 14 13 11 3 16 8 6 5 1 12 7 3 2
1 7 3 14 11 10 6 12 7 12 7 13 9 8 5 16 12 18
11 9 15 2 2 16 16 13 10 9 6 3 3 17 17 4 10 8
B
18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
17 3 15 5 6 7 8 9 1 18 10 11 12 13 14 4 16 2
16 4 5 6 7 8 9 1 2 17 18 10 11 12 13 14 15 3
15 5 6 7 8 9 1 2 16 3 17 18 10 11 12 13 14 4
5 6 7 8 9 1 2 16 15 4 3 17 18 10 11 12 13 14
6 7 8 9 1 2 16 15 14 5 4 3 17 18 10 11 12 13
7 8 9 1 2 16 15 14 13 6 5 4 3 17 18 10 11 12
8 9 1 2 16 15 14 13 12 7 6 5 4 3 17 18 10 11
9 1 2 16 15 14 13 12 11 8 7 6 5 4 3 17 18 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
10 18 17 3 4 5 6 7 8 11 12 13 14 15 16 2 1 9
11 10 18 17 3 4 5 6 7 12 13 14 15 16 2 1 9 8
12 11 10 18 17 3 4 5 6 13 14 15 16 2 1 9 8 7
13 12 11 10 18 17 3 4 5 14 15 16 2 1 9 8 7 6
14 13 12 11 10 18 17 3 4 15 16 2 1 9 8 7 6 5
4 14 13 12 11 10 18 17 3 16 2 1 9 8 7 6 5 15
3 15 14 13 12 11 10 18 17 2 1 9 8 7 6 5 4 16
2 16 4 14 13 12 11 10 18 1 9 8 7 6 5 15 3 17
M = A + (B - 1) * 18
312 304 275 261 249 222 207 184 179 146 141 118 103 76 64 50 21 13
306 38 264 90 96 114 128 160 1 324 165 197 211 229 235 61 287 19
276 59 74 100 121 132 147 4 33 292 321 178 193 204 225 251 266 49
253 83 99 112 144 159 3 27 274 51 298 322 166 181 213 226 242 72
81 92 119 139 151 11 28 285 258 67 40 297 314 174 186 206 233 244
98 110 143 158 10 36 280 255 252 73 70 45 289 315 167 182 215 227
115 137 152 2 24 278 268 248 221 104 77 57 47 301 323 173 188 210
140 153 15 32 273 257 243 232 201 124 93 82 68 52 293 310 172 185
155 16 20 272 267 238 234 199 187 138 126 91 87 58 53 305 309 170
18 35 48 62 86 105 123 130 156 169 195 202 220 239 263 277 290 307
164 313 302 44 56 79 106 109 142 183 216 219 246 269 281 23 12 161
194 177 317 291 41 60 84 102 122 203 223 241 265 284 34 8 148 131
212 189 163 320 300 43 55 80 108 217 245 270 282 25 5 162 136 113
224 200 192 180 311 303 42 65 78 247 260 283 22 14 145 133 125 101
240 231 208 191 171 308 294 54 63 262 271 31 17 154 134 117 94 85
71 250 228 205 198 176 319 299 39 286 26 6 149 127 120 97 75 254
37 259 237 230 209 190 168 318 295 30 7 157 135 116 95 88 66 288
29 279 69 236 218 214 196 175 316 9 150 129 111 107 89 256 46 296

Square B contains eightteen patterns - 1 (grey) thru 18 (blue) - each populated with eightteen times the same number.

To enable the construction of a (correct) Simple Magic Square M, the comparable patterns in square A should each be populated with the numbers 1 thru 18 (Latin Patterns).

Square B is both column - and row symmetric (rows 1/10, 2/18, 3/17, 4/16 ... 9/11 from top to bottom).

As in the example shown above Square A is Column Symmetric, the resulting Simple Magic Square M is column symmetric as well.

18.4   Summary

The obtained results regarding the order 18 (Semi) Latin - and related Magic Squares, as deducted and discussed in previous sections, can be summarized as folows:

  • The construction of order 18 Simple Magic Squares based on Self Orthogonal Latin Diagonal Squares (Historical).
  • The construction of order 18 Almost Associated Magic Squares based on Orthogonal Semi Latin Diagonal Squares.
  • The construction of order 18 Concentric Borders based on Non Latin but Orthogonal Borders.
  • The construction of order 18 Concentric Center Crosses based on Non Latin but Orthogonal Center Crosses.
  • The construction of order 18 Composed Magic Squares based on Orthogonal Semi Latin Diagonal Squares.
  • The construction of order 18 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 higher order (Semi) Latin - and related (Pan) Magic Squares.

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