MostPerfect Magic Cubes

For example; multiply, border, inlaid, multimagic, prime number, composition, etc., magic squares all have the equivalent in magic cubes. This page will explore the 3 dimensional equivalent of the mostperfect magic square. For convenience in explaining the features of a mostperfect magic square, I will reproduce here an excerpt from the relevant page of my magic squares site. For more information go to my mostperfect squares page. Features of Mostperfect magic squares
Definition 1. Every 2 x 2 block of cells (including wraparound) sum to 2T (where T= n^{2} + 1) 2. Any pair of integers distant ½n along a diagonal sum to T 3. Doublyeven pandiagonal normal magic squares (i.e. orders 4, 8, 12, etc. using integers from 1 to n^{2} ) See
When we extend the above requirements for mostperfect to 3 dimensions we have
All 48 pandiagonal magic squares of order4 are
mostperfect (and belong to Group I)! Order4 This order 4 cube has the characteristics of a most perfect magic cube, except that it is not a perfect magic cube. Cube_4HendricksJRM1 Pantriagonal and a Group I cube 01 32 49 48 62 35 14 19 04 29 52 45 63 34 15 18 56 41 08 25 11 22 59 38 53 44 05 28 10 23 58 39 13 20 61 36 50 47 02 31 16 17 64 33 51 46 03 30 60 37 12 21 07 26 55 42 57 40 09 24 06 27 54 43 NOTE: Not all order 4 pantriagonal cubes belong to group I. The following cube is an example. Although it is pantriagonal, notice that not all 2 x 2 subsquares sum to S. Also, not all pairs of integers distant ½n along a pantriagonal sum to T I choose this particular cube as an example, because of another outstanding feature. This numbers in this cube by Guenter Stertenbrink are arranged in a chess knight closed tour! Cube_4Stertenbrink2.xls Pantriagonal but not a group I cube 20 41 14 55 15 54 17 44 42 19 56 13 53 16 43 18 39 30 57 04 60 01 38 31 29 40 03 58 02 59 32 37 10 51 24 45 21 48 11 50 52 09 46 23 47 22 49 12 61 08 35 26 34 27 64 05 07 62 25 36 28 33 06 63 I have included these order 4 cubes as examples. However, a condition for mostperfect cube is that they belong to the perfect cube class, so we cannot consider order 4 (or 12, 20 etc.). Order8 Order8 Perfect (nasik) magic cubes
Order8 Pantriagonal magic cubes
Therefore only order 8 perfect magic cubes may be
considered as candidates for the title “mostperfect’. ADDENDUM: Compactplus and complete are intimate features of mostperfect. See new material on these features here. An example cube. Plane I Top II 31 358 477 161 26 355 476 168 508 136 63 326 509 129 58 323 306 395 244 80 311 398 245 73 237 81 298 403 236 88 303 406 359 478 165 25 354 475 164 32 132 64 327 510 133 57 322 507 394 243 76 312 399 246 77 305 85 297 402 235 84 304 407 238 479 166 29 353 474 163 28 360 60 328 511 134 61 321 506 131 242 75 308 400 247 78 309 393 301 401 234 83 300 408 239 86 167 30 357 473 162 27 356 480 324 512 135 62 325 505 130 59 74 307 396 248 79 310 397 241 405 233 82 299 404 240 87 302 Horizontal Plane III IV 50 331 500 144 55 334 501 137 493 145 42 339 492 152 47 342 295 414 229 89 290 411 228 96 196 128 263 446 197 121 258 443 330 499 140 56 335 502 141 49 149 41 338 491 148 48 343 494 415 230 93 289 410 227 92 296 124 264 447 198 125 257 442 195 498 139 52 336 503 142 53 329 45 337 490 147 44 344 495 150 231 94 293 409 226 91 292 416 260 448 199 126 261 441 194 123 138 51 332 504 143 54 333 497 341 489 146 43 340 496 151 46 95 294 413 225 90 291 412 232 444 200 127 262 445 193 122 259 Horizontal Plane V VI 39 350 485 153 34 347 484 160 452 192 7 382 453 185 2 379 266 435 204 120 271 438 205 113 213 105 274 427 212 112 279 430 351 486 157 33 346 483 156 40 188 8 383 454 189 1 378 451 434 203 116 272 439 206 117 265 109 273 426 211 108 280 431 214 487 158 37 345 482 155 36 352 4 384 455 190 5 377 450 187 202 115 268 440 207 118 269 433 277 425 210 107 276 432 215 110 159 38 349 481 154 35 348 488 380 456 191 6 381 449 186 3 114 267 436 208 119 270 437 201 429 209 106 275 428 216 111 278 Horizontal Plane VII VIII 10 371 460 184 15 374 461 177 469 169 18 363 468 176 23 366 287 422 221 97 282 419 220 104 252 72 319 390 253 65 314 387 370 459 180 16 375 462 181 9 173 17 362 467 172 24 367 470 423 222 101 281 418 219 100 288 68 320 391 254 69 313 386 251 458 179 12 376 463 182 13 369 21 361 466 171 20 368 471 174 223 102 285 417 218 99 284 424 316 392 255 70 317 385 250 67 178 11 372 464 183 14 373 457 365 465 170 19 364 472 175 22 103 286 421 217 98 283 420 224 388 256 71 318 389 249 66 315 Order12 There are NO order 12 Perfect magic cubes, because it is
impossible to construct a cube of this order that is both Pantriagonal and
Pandiagonal. Aale de Winkel's Pantriagonal magic cube (on my Order12 Cubes page) comes close!
Order16 Order16 Perfect magic cubes
Order16 Pantriagonal magic cubes
Summary As is to be expected, higher dimension magic objects
increases the complexity. It seems to me that to be called mostperfect, the cube should be a subset of the perfect magic cube. This is the highest class of magic cube, just as the pandiagonal magic square is the highest class of square. Although, as shown previously, the required features for a mostperfect magic square are available in both the perfect and the pantriagonal magic cubes. Order4 All order 4 group I cubes fulfill the 3 requirements as called for in the most perfect square. Of course, these cubes do not belong to the ‘perfect’ class, so I suggest they not be called mostperfect. Order8 All order 8 perfect magic cubes meet all the requirements for mostperfect magic squares. This assumes that requirement: 1. Has been adapted for cubes from “every 2x2 block of
cells…” to corner cells of every 2x2x2 block of cells”. The fact that all order 8 nasik perfect magic cubes are mostperfect is consistent with magic squares where all the members of the lowest order possible (order 4) are mostperfect. Also for order 8, 4T =S, which is consistent with order 4 mostperfect magic squares where 2T = S. Order12 There are no order 12 mostperfect magic cubes because
there are no order 12 perfect magic cubes. Order16 In this order there are perfect magic cubes, and a subset
of these are mostperfect by the definitions listed under order 8. Conclusions Requirements for a mostperfect magic cube. 1. Corners of every 2 x 2 x 2 block of cells (including
wraparound) sum to 4T (where T = m3 + 1). Kathleen Ollerinshaw made much use of doublyeven reversible squares for enumerating the mostperfect squares. However, they are not required in defining mostperfect squares. I have made no effort to investigate reversible cubes, although I suspect they do exist. Perhaps someone else will feel motivated to do so. This is a preliminary study only, and I invite others to investigate mostperfect magic cubes more thoroughly. As explained above, I suggest the following definitions for most perfect magic cubes. Others may have different opinions or further investigation may require changes. I invite others to study the subject further, and am prepared to modify this page as required by new findings. A note regarding definitions:
