Office Applications and Entertainment, Magic Squares

28.0 Magic Squares, Higher Order, Composed Border (2)

28.1 Introduction

The Composed Borders as constructed in previous section can be combined with transformations of Composed Inlaid Magic Squares with Associated Borders (ref. Section 26).

Following sections will show some examples of the resulting Inlaid Magic Squares with Composed Borders and inlays with different magic Sums.

28.2 Magic Squares (12 x 12)

Inlaid Magic Squares of order 8, with four 3^th order Embedded Magic Squares with different Magic Sums and an Associated Border, as described in Section 8.8.5, can be transformed into Inlaid Magic Squares with a Center Cross:

The resulting square can be combined with a composed border as constructed in Section 27.4:

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

s3 210 261 174 225

Attachment 28.2 shows a few 12^th order Inlaid Magic Squares for miscellaneous Inlays with different Magic Sums.

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

28.3 Magic Squares (16 x 16)

Inlaid Magic Squares of order 12, with four 5^th order Embedded Magic Squares with different Magic Sums and an Associated Border, as described in Section 26.5, can be transformed into Inlaid Magic Squares with a Center Cross:

Inlaid 12 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

s5 291 423 302 434

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

The resulting square can be combined with a composed border as constructed in Section 27.5:

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

s5 571 703 582 714

Attachment 28.3 shows a few 16^th order Inlaid Magic Squares for miscellaneous Inlays with different Magic Sums.

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

28.4 Magic Squares (20 x 20)

Inlaid Magic Squares of order 16, with four 7^th order Embedded Magic Squares with different Magic Sums and an Associated Border, as described in Section 26.6, can be transformed into Inlaid Magic Squares with a Center Cross:

Inlaid 16 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

s7 952 840 959 847

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

The resulting square can be combined with a composed border as constructed in Section Section 27.6:

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

s7 1456 1344 1463 1351

Attachment 28.4 shows a few 20^th order Inlaid Magic Squares for miscellaneous Inlays with different Magic Sums.

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