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22.0  Magic Squares, Higher Order, Inlaid

22.11 Introduction

In previous sections Composed Simple Inlaid Magic Squares have been discussed for miscellaneous orders.

Following sections will show some examples of Composed Inlaid Magic Squares with Associated Borders.

22.12 Magic Squares (8 x 8)

Inlaid Magic Squares of order 8, with four 3th order Embedded Magic Squares with different Magic Sums and an Associated Border, have been described in Section 8.8.5.

Examples of such Inlaid Magic Squares of order 8 are shown in Attachment 8.6.14.

22.13 Magic Squares (9 x 9)

Inlaid Magic Squares of order 9, with four Embedded Magic Squares with different Magic Sums and an Associated Border, have been described in Section 9.7.6.

Examples of such Inlaid Magic Squares of order 9 are shown in Attachment 9.7.7.

Alternatively, the Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on resp. order 3 and 4 Magic Lines for the integers 0 ... 8 as shown in Attachment 18.6.1.

Attachment 18.6.2 shows a few 9th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with two each 3th order Semi Magic Center Squares (6 magic lines).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border

22.14 Magic Squares (11 x 11)

Inlaid Magic Squares of order 11, with Embedded Magic Squares of order 4 and 5 (overlapping) with different Magic Sums and an Associated Border, have been described in Section 11.3.2.

Examples of such Inlaid Magic Squares of order 11 are shown in Attachment 14.9.9b.

22.15 Magic Squares (12 x 12)

The 12th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 5th order Pan Magic Center Squares with Magic Sums s(1) = 291, s(2) = 423, s(3) = 302, and s(4) = 434.

 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
 291 423 302 434

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

```s(1) = 5 * s12 / 6 - s(4)
s(2) = 5 * s12 / 6 - s(3)
```

With s12 = 870 the Magic Sum of the 12th order Inlaid Magic Square.

The 5th order Pan Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 5th order Magic Lines for the integers 0 ... 11 as shown in Attachment 18.3.1.

Attachment 18.3.2 shows a few 12th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border

22.16 Magic Squares (16 x 16)

The 16th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 7th order Ultra Magic Center Squares with Magic Sums s(1) = 952, s(2) = 840, s(3) = 959 and s(4) = 847.

 69 80 68 71 73 75 78 66 252 17 175 99 246 216 138 61 60 250 221 129 63 19 166 104 108 54 24 170 109 241 223 131 140 111 243 214 136 58 29 161 172 141 49 31 163 102 248 218 220 168 106 253 209 143 51 22 28 211 134 56 26 173 97 255
 178 190 187 185 183 180 192 181 225 95 147 6 40 122 205 12 10 45 113 207 227 86 152 204 198 232 90 157 1 47 115 156 159 3 38 120 202 237 81 124 125 193 239 83 150 8 42 92 88 154 13 33 127 195 230 44 35 118 200 234 93 145 15 236
 21 32 162 110 251 217 135 52 213 247 212 144 50 30 171 105 165 59 25 167 100 256 210 142 133 98 254 219 137 55 20 176 101 132 64 18 174 107 249 215 53 169 103 244 224 130 62 27 245 222 139 57 23 164 112 242 76 65 77 74 72 70 67 79
 240 82 158 11 41 119 196 229 7 36 128 194 238 91 153 37 203 233 87 148 16 34 126 85 146 14 43 121 199 228 96 117 116 208 226 94 155 9 39 149 89 151 4 48 114 206 235 197 46 123 201 231 84 160 2 5 191 179 182 184 186 189 177 188
 952 840 959 847

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

```s(1) = 7 * s16 / 8 - s(4)
s(2) = 7 * s16 / 8 - s(3)
```

With s16 = 2056 the Magic Sum of the 16th order Inlaid Magic Square.

Attachment 18.4.2 shows a few 16th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with Pan Magic Center Squares (Non Symmetric).

The 7th order Pan Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 7th order Magic Lines for the integers 0 ... 15 as shown in Attachment 18.4.1.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

22.17 Magic Squares (20 x 20)

The 20th order Inlaid Magic Square shown below, originally published by John Hendricks, is composed out of an Associated Border and four each 9th order Pan Magic Center Squares with Magic Sums s(1), s(2), s(3) and s(4).

 295 283 292 288 291 294 282 296 284 300 46 270 37 313 65 389 59 238 147 201 226 393 45 229 159 218 267 21 310 77 146 209 279 38 307 61 390 57 233 145 206 78 387 41 230 157 213 265 29 319 266 141 210 277 33 305 69 399 58 227 26 317 73 385 49 239 158 207 261 30 306 225 149 219 278 27 301 70 397 53 66 39 318 67 381 50 237 153 205 269 386 47 221 150 217 273 25 309 79 398
 120 104 116 102 114 111 108 112 103 115 130 377 93 325 9 359 178 247 181 346 13 345 169 259 198 127 361 90 337 166 189 139 378 87 321 10 357 173 245 246 338 7 341 170 257 193 125 369 99 186 241 190 137 373 85 329 19 358 167 126 97 333 5 349 179 258 187 121 370 366 165 249 199 138 367 81 330 17 353 86 379 98 327 1 350 177 253 185 129 326 347 161 250 197 133 365 89 339 18 6
 395 271 24 308 76 392 42 223 154 220 75 388 56 232 142 203 274 40 311 64 315 212 262 23 314 80 391 44 228 156 35 63 394 60 231 144 208 276 32 302 275 160 211 264 28 316 72 382 43 234 215 304 68 396 52 222 143 214 280 31 155 236 152 202 263 34 320 71 384 48 235 22 303 74 400 51 224 148 216 272 55 54 240 151 204 268 36 312 62 383 286 298 289 293 290 287 299 285 297 281
 131 364 88 336 12 342 163 254 200 15 8 356 172 242 183 134 380 91 324 335 192 122 363 94 340 11 344 168 256 95 323 14 360 171 244 188 136 372 82 375 260 191 124 368 96 332 2 343 174 135 84 328 16 352 162 243 194 140 371 195 176 252 182 123 374 100 331 4 348 255 362 83 334 20 351 164 248 196 132 175 354 180 251 184 128 376 92 322 3 355 101 117 105 119 107 110 113 109 118 106
 1719 1899 1710 1890
 The relation between the Magic Sums s(1), s(2), s(3) and s(4) is: ```s(1) = 9 * s20 / 10 - s(4) s(2) = 9 * s20 / 10 - s(3) ``` With s20 = 4010 the Magic Sum of the 20th order Inlaid Magic Square. Attachment 18.5.2 shows a few 20th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4). Attachment 18.5.3 shows a few more 20th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with Simple Magic Center Squares for which one third of the diagonals sum to the Magic Sum. The 9th order (Pan) Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 9th order Magic Lines for the integers 0 ... 19 as shown in Attachment 18.5.1. Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.