Definition:Perfect Magic Cube

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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 $7$

Order $7$ perfect magic cube:

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


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


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


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


Order $8$

Order $8$ perfect magic cube:

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


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


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


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


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.


Also see


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.


Sources