Definition:Perfect Magic Cube

Definition

A perfect magic cube is an arrangement of the first $n^3$ (strictly) positive integers into an $n \times n \times n$ cubic array such that:

the sum of the entries in each row in each of the $3$ dimensions
the sum of the entries along the main diagonal of each plane
the sum of the entries along the space diagonals

are the same.

Order

An $n \times n \times n$ magic cube is called an order $n$ magic cube.

Examples

Order $1$

The Order $1$ perfect magic cube is trivial:

$\begin{array}{|c|} \hline 1 \\ \hline \end{array}$

Order $7$

$\begin{array}{|c|c|c|c|c|c|c|} \hline 327 & 41 & 98 & 99 & 156 & 213 & 270 \\ \hline 52 & 109 & 166 & 223 & 280 & 330 & 44 \\ \hline 169 & 226 & 283 & 340 & 5 & 62 & 119 \\ \hline 293 & 301 & 8 & 65 & 122 & 179 & 236 \\ \hline 18 & 75 & 132 & 189 & 239 & 247 & 304 \\ \hline 135 & 192 & 200 & 257 & 314 & 28 & 78 \\ \hline 210 & 260 & 317 & 31 & 88 & 145 & 153 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|c|c|c|c|} \hline 113 & 170 & 227 & 284 & 341 & 6 & 63 \\ \hline 237 & 294 & 295 & 9 & 66 & 123 & 180 \\ \hline 305 & 19 & 76 & 133 & 183 & 240 & 248 \\ \hline 79 & 136 & 193 & 201 & 258 & 315 & 22 \\ \hline 154 & 204 & 261 & 318 & 32 & 89 & 146 \\ \hline 271 & 328 & 42 & 92 & 100 & 157 & 214 \\ \hline 45 & 53 & 110 & 167 & 224 & 274 & 331 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|c|c|} \hline 249 & 306 & 20 & 77 & 127 & 184 & 241 \\ \hline 23 & 80 & 137 & 194 & 202 & 259 & 309 \\ \hline 147 & 148 & 205 & 262 & 319 & 33 & 90 \\ \hline 215 & 272 & 329 & 36 & 93 & 101 & 158 \\ \hline 332 & 46 & 54 & 111 & 168 & 218 & 275 \\ \hline 57 & 114 & 171 & 228 & 285 & 342 & 7 \\ \hline 181 & 238 & 288 & 296 & 10 & 67 & 124 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|c|c|c|c|} \hline 91 & 141 & 149 & 206 & 263 & 320 & 34 \\ \hline 159 & 216 & 273 & 323 & 37 & 94 & 102 \\ \hline 276 & 333 & 47 & 55 & 112 & 162 & 219 \\ \hline 1 & 58 & 115 & 172 & 229 & 286 & 343 \\ \hline 125 & 182 & 232 & 289 & 297 & 11 & 68 \\ \hline 242 & 250 & 307 & 21 & 71 & 128 & 185 \\ \hline 310 & 24 & 81 & 138 & 195 & 203 & 253 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|c|c|} \hline 220 & 277 & 334 & 48 & 56 & 106 & 163 \\ \hline 337 & 2 & 59 & 116 & 173 & 230 & 287 \\ \hline 69 & 126 & 176 & 233 & 290 & 298 & 12 \\ \hline 186 & 243 & 251 & 308 & 15 & 72 & 129 \\ \hline 254 & 311 & 25 & 82 & 139 & 196 & 197 \\ \hline 35 & 85 & 142 & 150 & 207 & 264 & 321 \\ \hline 103 & 160 & 217 & 267 & 324 & 38 & 95 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|c|c|c|c|} \hline 13 & 70 & 120 & 177 & 234 & 291 & 299 \\ \hline 130 & 187 & 244 & 252 & 302 & 16 & 73 \\ \hline 198 & 255 & 312 & 26 & 83 & 140 & 190 \\ \hline 322 & 29 & 86 & 143 & 151 & 208 & 265 \\ \hline 96 & 104 & 161 & 211 & 268 & 325 & 39 \\ \hline 164 & 221 & 278 & 335 & 49 & 50 & 107 \\ \hline 281 & 338 & 3 & 60 & 117 & 174 & 231 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|c|c|} \hline 191 & 199 & 256 & 313 & 27 & 84 & 134 \\ \hline 266 & 316 & 30 & 87 & 144 & 152 & 209 \\ \hline 40 & 97 & 105 & 155 & 212 & 269 & 326 \\ \hline 108 & 165 & 222 & 279 & 336 & 43 & 51 \\ \hline 225 & 282 & 339 & 4 & 61 & 118 & 175 \\ \hline 300 & 14 & 64 & 121 & 178 & 235 & 292 \\ \hline 74 & 131 & 188 & 245 & 246 & 303 & 17 \\ \hline \end{array}$

Order $8$

$\begin{array}{|c|c|c|c|c|c|c|c|} \hline 512 & 2 & 510 & 4 & 5 & 507 & 7 & 505 \\ \hline 504 & 10 & 502 & 12 & 13 & 499 & 15 & 497 \\ \hline 17 & 495 & 19 & 493 & 492 & 22 & 490 & 24 \\ \hline 25 & 487 & 27 & 485 & 484 & 30 & 482 & 32 \\ \hline 33 & 479 & 35 & 477 & 476 & 38 & 474 & 40 \\ \hline 41 & 471 & 43 & 469 & 468 & 46 & 466 & 48 \\ \hline 464 & 50 & 462 & 52 & 53 & 459 & 55 & 457 \\ \hline 456 & 58 & 454 & 60 & 61 & 451 & 63 & 449 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|c|c|c|c|c|} \hline 65 & 447 & 67& 445 & 444 & 70 & 442 & 72 \\ \hline 73 & 439 & 75& 437 & 436 & 78 & 434 & 80 \\ \hline 432 & 82 & 430& 84 & 85 & 427 & 87 & 425 \\ \hline 424 & 90 & 422& 92 & 93 & 419 & 95 & 417 \\ \hline 416 & 98 & 414& 100 & 101 & 411 & 103 & 409 \\ \hline 408 & 106 & 406& 108 & 109 & 403 & 111 & 401 \\ \hline 113 & 399 & 115& 397 & 396 & 118 & 394 & 120 \\ \hline 121 & 391 & 123& 389 & 388 & 126 & 386 & 128 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|c|c|c|} \hline 129 & 383 & 131 & 381 & 380 & 134 & 378 & 136 \\ \hline 137 & 375 & 139 & 373 & 372 & 142 & 370 & 144 \\ \hline 368 & 146 & 366 & 148 & 149 & 363 & 151 & 361 \\ \hline 360 & 154 & 358 & 156 & 157 & 355 & 159 & 353 \\ \hline 352 & 162 & 350 & 164 & 165 & 347 & 167 & 345 \\ \hline 344 & 170 & 342 & 172 & 173 & 339 & 175 & 337 \\ \hline 177 & 335 & 179 & 333 & 332 & 182 & 330 & 184 \\ \hline 185 & 327 & 187 & 325 & 324 & 190 & 322 & 192 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|c|c|c|c|c|} \hline 320 & 194 & 318 & 196 & 197 & 315 & 199 & 313 \\ \hline 312 & 202 & 310 & 204 & 205 & 307 & 207 & 305 \\ \hline 209 & 303 & 211 & 301 & 300 & 214 & 298 & 216 \\ \hline 217 & 295 & 219 & 293 & 292 & 222 & 290 & 224 \\ \hline 225 & 287 & 227 & 285 & 284 & 230 & 282 & 232 \\ \hline 233 & 279 & 235 & 277 & 276 & 238 & 274 & 240 \\ \hline 272 & 242 & 270 & 244 & 245 & 267 & 247 & 265 \\ \hline 264 & 250 & 262 & 252 & 253 & 259 & 255 & 257 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|c|c|c|} \hline 256 & 258 & 254 & 260 & 261 & 251 & 263 & 249 \\ \hline 248 & 266 & 246 & 268 & 269 & 243 & 271 & 241 \\ \hline 273 & 239 & 275 & 237 & 236 & 278 & 234 & 280 \\ \hline 281 & 231 & 283 & 229 & 228 & 286 & 226 & 288 \\ \hline 289 & 223 & 291 & 221 & 220 & 294 & 218 & 296 \\ \hline 297 & 215 & 299 & 213 & 212 & 302 & 210 & 304 \\ \hline 208 & 306 & 206 & 308 & 309 & 203 & 311 & 201 \\ \hline 200 & 314 & 198 & 316 & 317 & 195 & 319 & 193 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|c|c|c|c|c|} \hline 321 & 191 & 323 & 189 & 188 & 326 & 186 & 328 \\ \hline 329 & 183 & 331 & 181 & 180 & 334 & 178 & 336 \\ \hline 176 & 338 & 174 & 340 & 341 & 171 & 343 & 169 \\ \hline 168 & 346 & 166 & 348 & 349 & 163 & 351 & 161 \\ \hline 160 & 354 & 158 & 356 & 357 & 155 & 359 & 153 \\ \hline 152 & 362 & 150 & 364 & 365 & 147 & 367 & 145 \\ \hline 369 & 143 & 371 & 141 & 140 & 374 & 138 & 376 \\ \hline 377 & 135 & 379 & 133 & 132 & 382 & 130 & 384 \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|c|c|c|} \hline 385 & 127 & 387 & 125 & 124 & 390 & 122 & 392 \\ \hline 393 & 119 & 395 & 117 & 116 & 398 & 114 & 400 \\ \hline 112 & 402 & 110 & 404 & 405 & 107 & 407 & 105 \\ \hline 104 & 410 & 102 & 412 & 413 & 99 & 415 & 97 \\ \hline 96 & 418 & 94 & 420 & 421 & 91 & 423 & 89 \\ \hline 88 & 426 & 86 & 428 & 429 & 83 & 431 & 81 \\ \hline 433 & 79 & 435 & 77 & 76 & 438 & 74 & 440 \\ \hline 441 & 71 & 443 & 69 & 68 & 446 & 66 & 448 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|c|c|c|c|c|} \hline 64 & 450 & 62 & 452 & 453 & 59 & 455 & 57 \\ \hline 56 & 458 & 54 & 460 & 461 & 51 & 463 & 49 \\ \hline 465 & 47 & 467 & 45 & 44 & 470 & 42 & 472 \\ \hline 473 & 39 & 475 & 37 & 36 & 478 & 34 & 480 \\ \hline 481 & 31 & 483 & 29 & 28 & 486 & 26 & 488 \\ \hline 489 & 23 & 491 & 21 & 20 & 494 & 18 & 496 \\ \hline 16 & 498 & 14 & 500 & 501 & 11 & 503 & 9 \\ \hline 8 & 506 & 6 & 508 & 509 & 3 & 511 & 1 \\ \hline \end{array}$

Also known as

A perfect magic cube is often referred to just as a magic cube, but the subject is a large one, and there are several varieties of magic cube which are almost perfect.

Historical Note

The first perfect magic cube to be found appears to be the order $7$ one as reported by Andrew Hollingworth Frost in The Quarterly Journal of Pure and Applied Mathematics in $1866$.

The next one was the order $8$ one discovered by Gustavus Frankenstein, as reported in The Cincinnati Commercial in $1875$.

David Wells reports in his Curious and Interesting Numbers, 2nd ed. of $1997$ that the first to be published was in $1905$, but it is clear that more recent research supersedes his information.

This appears to be a misprint for the one of order $9$ discovered by Charles Planck, and published by him in his The Theory of Path Nasiks of $1905$.

He also reports, accurately for the time, that it was not known whether or not there exist perfect magic cubes whose order is $5$ or $6$.

However, since $1997$ an example of each has been found.