Office Applications and Entertainment, Magic Squares

29.0 Magic Squares, Higher Order, Composed, Single Even Order

29.1 Magic Squares (10 x 10)

The 4 x 4 Sub Squares as constructed in Section 22.1 and used for the construction of double even Composed Magic Squares can also be used for the construction of single even Magic Squares.

Both the normal Magic Square (MC = 260) and non normal Magic Square (MC = 404) shown below are suitable for the construction of 10 x 10 Magic Squares.

The right square can be completed with a 10 x 10 center cross based on the consecutive integers 33 ... 68, as discussed in detail in Section 10.3.5.

After adding 18 to all elements of the left square, it can be completed with a 10 x 10 center cross based on the integers 1 ... 18 and 83 ... 100, as shown below.

Each center cross corresponds with (8!) * (8!) = 1,6 10⁹ center crosses, which can be obtained by permutation of the horizontal and vertical pairs.

Both squares can be transformed into a Bordered Magic Square as shown below:

Each border corresponds with (8!) * (8!) = 1,6 10⁹ borders, which can be obtained by permutation of the horizontal and vertical pairs.

Each 10 x 10 Magic Square shown above corresponds with 4! * 384⁴ = 0,5 10¹², for each border or center cross.

29.2 Magic Squares (14 x 14)

Both the normal Magic Square (MC = 870) and non normal Magic Square (MC = 1182) shown below are suitable for the construction of 14 x 14 Magic Squares.

The right square can be completed with a 14 x 14 center cross based on the consecutive integers 73 ... 124:

MC = 1379

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

After adding 26 to all elements of the left square, it can be completed with a 14 x 14 center cross based on the integers 1 ... 26 and 171 to 196:

MC = 1379

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

Each center cross corresponds with (12!) * (12!) = 2,3 10¹⁷ center crosses, which can be obtained by permutation of the horizontal and vertical pairs.

Both squares can be transformed into a Bordered Magic Square as shown below:

MC = 1379

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

MC = 1379

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

Each border corresponds with (12!) * (12!) = 2,3 10¹⁷ borders, which can be obtained by permutation of the horizontal and vertical pairs.

Each 14 x 14 Magic Square shown above corresponds with 9! * 384⁹ = 6,6 10²⁸, for each border or center cross.

Comparable 14 x 14 Composed Magic Squares can be constructed based on:

• the center cross based on the integers 1 ... 26 and 171 to 196 and
• the order 6 Sub Squares of the Composed Magic Squares as discussed in Section 22.5

of which a few examples are shown in Attachment 27.2.

29.3 Magic Squares (18 x 18)

After adding 34 to all elements of the normal Composed Magic Square (16 x 16, MC = 2056) shown in Section 22.3, it can be completed with an 18 x 18 center cross based on the integers 1 ... 34 and 291 to 324:

MC = 2925

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

For n = 18 a non normal Magic Square (16 x 16, MC = 2600) can be constructed, which can be completed with an 18 x 18 center cross based on the consecutive integers 129 ... 196:

MC = 2925

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

Each center cross corresponds with (16!) * (16!) = 4,4 10²⁶ center crosses, which can be obtained by permutation of the horizontal and vertical pairs.

Both squares can be transformed into a Bordered Magic Square as shown below:

MC = 2925

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

MC = 2925

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

Each border corresponds with (16!) * (16!) = 4,4 10²⁶ borders, which can be obtained by permutation of the horizontal and vertical pairs.

Each 18 x 18 Magic Square shown above corresponds with 16! * 384¹⁶ = 4,7 10⁵⁴, for each border or center cross.

29.4 Magic Squares (22 x 22)

After adding 42 to all elements of the normal Composed Magic Square (20 x 20, MC = 4010) shown in Section 22.4, it can be completed with a 18 x 18 center cross based on the integers 1 ... 42 and 443 to 484:

MC = 5335

46 47 437 440 54 55 429 432 62 63 452 435 442 44 49 427 434 52 57 419 426 34 48 45 439 438 56 53 431 430 64 61 450 441 436 50 43 433 428 58 51 425 420 36 86 87 397 400 94 95 389 392 102 103 448 395 402 84 89 387 394 92 97 379 386 38 88 85 399 398 96 93 391 390 104 101 446 401 396 90 83 393 388 98 91 385 380 40 126 127 357 360 134 135 349 352 142 143 444 355 362 124 129 347 354 132 137 339 346 42 31 455 29 457 27 459 25 461 23 474 463

33 421 424 70 71 413 416 78 79 405 408 451 60 65 411 418 68 73 403 410 76 81 35 423 422 72 69 415 414 80 77 407 406 449 66 59 417 412 74 67 409 404 82 75 37 381 384 110 111 373 376 118 119 365 368 447 100 105 371 378 108 113 363 370 116 121 39 383 382 112 109 375 374 120 117 367 366 445 106 99 377 372 114 107 369 364 122 115 41 341 344 150 151 333 336 158 159 325 328 443 140 145 331 338 148 153 323 330 156 161 21 1 465 19 467 17 469 15 471 13 473

454 30 456 28 458 26 460 24 462 11 464 128 125 359 358 136 133 351 350 144 141 32 361 356 130 123 353 348 138 131 345 340 10 166 167 317 320 174 175 309 312 182 183 476 315 322 164 169 307 314 172 177 299 306 8 168 165 319 318 176 173 311 310 184 181 478 321 316 170 163 313 308 178 171 305 300 6 206 207 277 280 214 215 269 272 222 223 480 275 282 204 209 267 274 212 217 259 266 4 208 205 279 278 216 213 271 270 224 221 482 281 276 210 203 273 268 218 211 265 260 2

22 484 20 466 18 468 16 470 14 472 12 453 343 342 152 149 335 334 160 157 327 326 475 146 139 337 332 154 147 329 324 162 155 9 301 304 190 191 293 296 198 199 285 288 477 180 185 291 298 188 193 283 290 196 201 7 303 302 192 189 295 294 200 197 287 286 479 186 179 297 292 194 187 289 284 202 195 5 261 264 230 231 253 256 238 239 245 248 481 220 225 251 258 228 233 243 250 236 241 3 263 262 232 229 255 254 240 237 247 246 483 226 219 257 252 234 227 249 244 242 235

For n = 22 a non normal Magic Square (20 x 20, MC = 4850) can be constructed, which can be completed with a 22 x 22 center cross based on the consecutive integers 201 ... 284:

MC = 5335

4 5 479 482 12 13 471 474 20 21 252 477 484 2 7 469 476 10 15 461 468 234 6 3 481 480 14 11 473 472 22 19 250 483 478 8 1 475 470 16 9 467 462 236 44 45 439 442 52 53 431 434 60 61 248 437 444 42 47 429 436 50 55 421 428 238 46 43 441 440 54 51 433 432 62 59 246 443 438 48 41 435 430 56 49 427 422 240 84 85 399 402 92 93 391 394 100 101 244 397 404 82 87 389 396 90 95 381 388 242 231 255 229 257 227 259 225 261 223 274 263

233 463 466 28 29 455 458 36 37 447 450 251 18 23 453 460 26 31 445 452 34 39 235 465 464 30 27 457 456 38 35 449 448 249 24 17 459 454 32 25 451 446 40 33 237 423 426 68 69 415 418 76 77 407 410 247 58 63 413 420 66 71 405 412 74 79 239 425 424 70 67 417 416 78 75 409 408 245 64 57 419 414 72 65 411 406 80 73 241 383 386 108 109 375 378 116 117 367 370 243 98 103 373 380 106 111 365 372 114 119 221 201 265 219 267 217 269 215 271 213 273

254 230 256 228 258 226 260 224 262 211 264 86 83 401 400 94 91 393 392 102 99 232 403 398 88 81 395 390 96 89 387 382 210 124 125 359 362 132 133 351 354 140 141 276 357 364 122 127 349 356 130 135 341 348 208 126 123 361 360 134 131 353 352 142 139 278 363 358 128 121 355 350 136 129 347 342 206 164 165 319 322 172 173 311 314 180 181 280 317 324 162 167 309 316 170 175 301 308 204 166 163 321 320 174 171 313 312 182 179 282 323 318 168 161 315 310 176 169 307 302 202

222 284 220 266 218 268 216 270 214 272 212 253 385 384 110 107 377 376 118 115 369 368 275 104 97 379 374 112 105 371 366 120 113 209 343 346 148 149 335 338 156 157 327 330 277 138 143 333 340 146 151 325 332 154 159 207 345 344 150 147 337 336 158 155 329 328 279 144 137 339 334 152 145 331 326 160 153 205 303 306 188 189 295 298 196 197 287 290 281 178 183 293 300 186 191 285 292 194 199 203 305 304 190 187 297 296 198 195 289 288 283 184 177 299 294 192 185 291 286 200 193