# Recreational Math I: Magic Squares: the “really good” kind – Part 7

Welcome to part 7, where the magic squares are 7×7. I don’t know if there is any numerological significance to that, but it wasn’t intended. Although, if someone wanted to make something of it, 7 was the number of known planets in medieval times, as well as the number of known elements, and the number of days in the week. The rest is all alchemy.

The algorithm for a 7×7 matrix is the same as for 5×5. In fact, it should work for any odd-ordered matrix. Do the first based on a random sequence of the numbers between 1 and 7:

 1 6 2 3 4 7 5 7 5 1 6 2 3 4 3 4 7 5 1 6 2 6 2 3 4 7 5 1 5 1 6 2 3 4 7 4 7 5 1 6 2 3 2 3 4 7 5 1 6

Now, do the second matrix using 0 and every seventh number after it up to 42:

 7 42 0 21 14 35 28 0 21 14 35 28 7 42 14 35 28 7 42 0 21 28 7 42 0 21 14 35 42 0 21 14 35 28 7 21 14 35 28 7 42 0 35 28 7 42 0 21 14

The resulting matrix is magic, but not “hyper-magic” like the 5x5s were:

 8 48 2 24 18 42 33 7 26 15 41 30 10 46 17 39 35 12 43 6 23 34 9 45 4 28 19 36 47 1 27 16 38 32 14 25 21 40 29 13 44 3 37 31 11 49 5 22 20

The matrix above was done using Excel, and checked using VBA, where I checked it for “magic”. The loss of magic is unfortunate, but I suppose inevitable when you scale up. There are (7!)2 = 25,401,660 magic squares possible by this algorithm. The VBA proved handy later when I used dynamic programming to produce any odd-ordered magic square of any size, and investigated the results.

What if I made a mistake and got the shifting wrong, such as shifting from the second number from the left instead of the right? The result is variable; full magic can only be restored if the rightward diagonal is all 4/s:

 7 3 6 2 5 1 4 3 6 2 5 1 4 7 6 2 5 1 4 7 3 2 5 1 4 7 3 6 5 1 4 7 3 6 2 1 4 7 3 6 2 5 4 7 3 6 2 5 1

Why does this work? This is because the sum of the numbers 1 to 7 in any order is 28. Notice that whatever number is in the opposite diagonal is repeated 7 times. There is only one number, when multiplied by 7, equals 28, and that is 4. All other numbers will result in a loss of magic. This appears to be a way to maintain magic in all odd-ordered magic squares past order 7.

When added to the second preliminary square, both diagonals became magic, as well as the columns and rows. Notice that the opposite diagonal consists of all 4’s. We lose variety with this restriction, but not by much: 7!6! = 3,628,800. Still enough to occupy you on a rainy day.

9×9 magic squares won’t work so well, because the second square is shifted by 3, which is a number that divides 9. This will result in certain rows repeating in one of the pre-squares; and the result generally is that one of the diagonals won’t total the magic number (369 in this case).

 13 66 78 59 27 8 34 38 46 74 55 22 3 33 41 54 17 70 26 7 29 37 49 12 69 77 63 32 45 53 16 65 73 58 21 6 48 15 68 81 62 25 2 28 40 64 76 57 24 5 36 44 52 11 61 20 1 31 39 51 14 72 80 9 35 43 47 10 67 75 60 23 42 50 18 71 79 56 19 4 30

The magic number can be known in advance of any magic square because an order-n magic square obeys the formula (n × (n2 + 1))/2. The best bet is to look for odd-ordered n × n squares where 3 does not divide n. In my VBA program, I made double sure by sticking to magic squares where the order is prime. Here is an 11 × 11 square, where the magic number is 671:

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

Finally, to stretch things out to the boldly absurd, what about an order-23 square whose magic number is 6095?

 92 369 279 275 108 466 226 403 125 327 360 316 55 457 485 14 192 182 234 425 36 145 524 283 271 115 461 210 413 131 328 364 311 56 442 498 17 193 181 232 428 31 159 510 80 381 103 473 214 409 138 323 348 321 62 443 502 12 194 166 245 431 32 158 508 83 376 297 257 228 395 126 335 352 317 69 438 486 22 200 167 249 426 33 143 521 86 377 296 255 106 468 129 330 366 303 57 450 490 18 207 162 233 436 39 144 525 81 378 281 268 109 469 227 393 365 301 60 445 504 4 195 174 237 432 46 139 509 91 384 282 272 104 470 212 406 132 331 63 446 503 2 198 169 251 418 34 151 513 87 391 277 256 114 476 213 410 127 332 350 314 488 15 201 170 250 416 37 146 527 73 379 289 260 110 483 208 394 137 338 351 318 58 447 196 171 235 429 40 147 526 71 382 284 274 96 471 220 398 133 345 346 302 68 453 489 19 236 433 35 148 511 84 385 285 273 94 474 215 412 119 333 358 306 64 460 484 3 206 177 45 154 512 88 380 286 258 107 477 216 411 117 336 353 320 50 448 496 7 202 184 231 417 507 72 390 292 259 111 472 217 396 130 339 354 319 48 451 491 21 188 172 243 421 41 161 386 299 254 95 482 223 397 134 334 355 304 61 454 492 20 186 175 238 435 27 149 519 76 266 99 478 230 392 118 344 361 305 65 449 493 5 199 178 239 434 25 152 514 90 372 287 464 218 404 122 340 368 300 49 459 499 6 203 173 240 419 38 155 515 89 370 290 261 113 399 136 326 356 312 53 455 506 1 187 183 246 420 42 150 516 74 383 293 262 112 462 221 324 359 307 67 441 494 13 191 179 253 415 26 160 522 75 387 288 263 97 475 224 400 135 308 66 439 497 8 205 165 241 427 30 156 529 70 371 298 269 98 479 219 401 120 337 362 452 500 9 204 163 244 422 44 142 517 82 375 294 276 93 463 229 407 121 341 357 309 51 10 189 176 247 423 43 140 520 77 389 280 264 105 467 225 414 116 325 367 315 52 456 495 180 242 424 28 153 523 78 388 278 267 100 481 211 402 128 329 363 322 47 440 505 16 190 430 29 157 518 79 373 291 270 101 480 209 405 123 343 349 310 59 444 501 23 185 164 252 141 528 85 374 295 265 102 465 222 408 124 342 347 313 54 458 487 11 197 168 248 437 24

That probably ran into the neighbouring columns, but it’s just for illustration. The code, which I have used to generate magic squares of as high an order as 113 (magic number 721,505) is fewer than 170 lines in VBA for Excel 2007. And yes, even the above square, and the order 113 square are magic.

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