Magic Cube Update6

More on Compact and Complete Before reading this section, you may wish to review my original discussion on compact and complete here. A lively email discussion was started in late
October, 2009 when Dwane Campbell sent an email to Aale de Winkel with the
subject: complete_p in compound order hypercubes. This discussion
involved about 8 people, but the main participants (and contributors) were
Dwane Campbell [1], Mitsutoshi
Nakamura [2], and Aale de Winkel
[3]. The subject contained the word
complete but much of the discussion involved the related subject of
compact.
Compact Compact implies that the corners of all
subhypercubes of dimension n sum to the same value, where n = the
dimension of the main hypercube. For an order16 cube:
More generally, depending on the order of the cube, other types of compact are possible. The next one is Compact_17. However, Mitsutoshi's proof confirms that the maximum number of compact types that can be present in a cube is three regardless of how many types are possible. For a magic square, only two types of compact are
possible, for a tesseract only 4 types are possible, etc. A general statement for tesseracts would be: Only four types of compact may be present in a tesseract of order 2^{k} for k>3. If there are four, then one of the four must be compact_2^{(k1)} + 1). etc. The above implies the subcubes are the same dimension as the hosting hypercube. In the event you wish to indicate a subhypercube of lower dimension (for instance the subsquares in the planes of a cube), this may be indicated thus; ^{2}compact_2 or ^{2}compact_{2} (or simply 2compact_2).
Compactplus If the corners of all possible orders of subcubes from 2 to m sum correctly, the Hypercube is compactplus.
Addendum March 1, 2010: Examples of this feature using nasik squares for simplicity.
Complete The terminology for the
complete feature is similar to that for the compact feature. A ^{3}complete_3
would indicate that the sum of three numbers evenly spaced in all of the
triagonals of a cube will always add to the same constant. This definition of
complete can also be seen in the monagonals. The order4 and order8
squares above are both ^{2}complete_2.
The relationship between Complete and Compact is illustrated by the following quote from a Dwane Campbell email:
Order4 is the smallest possible nasik square. It is compact_2, 3 which is all
the types that are possible for this order, so is compactplus. Order8 is
compact_2, 5 but NOT 3 and 7, so is not compactplus. (Corners of all subsquares
of orders 2, 4, 5, 6, and 8 sum correctly in this order8 square because
compact_2 obviously includes all even orders up to m.) [1]
Dwane Campbell’s web site is at
http://magictesseract.com Smallest Order4 Multiply
In mid January, Christian Boyer posted his latest
Update to
www.multimagie.com/English/MultiplicCubes.htm
Included were two interesting order4 multiplication magic cubes. The first one was constructed by Christian in 2007. The second by Max Alekseyev. Both cubes are interesting in light of the previous
discussion because they are compact_3. Furthermore, the four horizontal
planes of the Boyer cube are magic squares which are ^{2}compact_2. Both cubes have the same highest number, 364, but
the Alekseyev cube has a product that is exactly onehalf of Boyer’s cube. Boyer cube 52 168 15 132 42 11 234 160 165 72 16 91 48 130 308 9 36 55 273 32 260 144 3 154 56 26 264 45 33 84 80 78 231 12 96 65 8 182 220 54 312 120 21 22 30 66 39 224 40 156 44 63 198 60 112 13 6 77 195 192 364 24 18 110
Remember that we consider a corner of a square or
cube starting on any cell in the hypercube. i.e. wraparound is in effect!
Alekseyev cube
1 110 224 351 231 12 78 40 144 91 15 44 260 72 33 14 130 8 297 28 98 273 5 66 77 18 52 120 9 220 112 39 308 27 13 80 6 55 168 156 195 32 198 7 24 182 20 99 216 364 10 11 65 48 132 21 4 165 56 234 154 3 117 160
It is interesting to note that The Alekseyev cube's low P is not the lowest to date. Boyer constructed a cube in January, 2006 with a Max nb of 546 but with a P of only 6,486,480! [1] Sayles’ and Trenkler’s cubes may be seen on my Multiply cubes page. Frost Order9 Cube Model On November 3, 2009, I received an email from Nicholas Tam [1] advising me that he was researching A. H. Frost`s glass model of an order9 Magic cube. [2] Tam has transcribed the numbers on each side of each of
the 9 vertical plates. The numbers on one side range from 1 to 729 and form a
normal nasik magic cube. The numbers on the reverse of each plate are the
equivalent to the corresponding number on the other side, when it is reduced by
one, considered a base 10 number, converted to base 9, then 1 added. And with
nothing to indicate otherwise, we naturally assume all the numbers in both cubes
to be base 10. These (the second set) numbers form the well known Frost
Order9 nasik cube published in 1878. [3]
In this cube, the central number is 445. Because the numbers range from 1 to
889, the cube is not normal, but is thought to be the first nasik order9 cube
constructed. [4] Actually, the date the model was constructed is still
unknown! Frost’s Nasik Order9 from the Whipple museum. [5] I – Top (actually front plate of model) II 135 606 519 652 430 334 308 59 242 343 254 68 224 162 615 528 679 412 518 656 432 336 312 58 241 127 605 67 223 154 614 527 683 414 345 258 424 335 311 62 243 129 609 517 655 156 618 526 682 406 344 257 71 225 310 61 235 128 608 521 657 426 339 530 684 408 348 256 70 217 155 617 237 132 607 520 649 425 338 314 63 407 347 260 72 219 159 616 529 676 611 522 651 429 337 313 55 236 131 259 64 218 158 620 531 678 411 346 650 428 341 315 57 240 130 610 514 222 157 619 523 677 410 350 261 66 340 307 56 239 134 612 516 654 427 621 525 681 409 349 253 65 221 161 60 238 133 604 515 653 431 342 309 680 413 351 255 69 220 160 613 524 III IV 642 537 688 439 325 263 14 233 144 245 23 179 153 624 564 697 448 352 692 441 327 267 13 232 136 641 536 178 145 623 563 701 450 354 249 22 326 266 17 234 138 645 535 691 433 627 562 700 442 353 248 26 180 147 16 226 137 644 539 693 435 330 265 702 444 357 247 25 172 146 626 566 141 643 538 685 434 329 269 18 228 356 251 27 174 150 625 565 694 443 540 687 438 328 268 10 227 140 647 19 173 149 629 567 696 447 355 250 437 332 270 12 231 139 646 532 686 148 628 559 695 446 359 252 21 177 262 11 230 143 648 534 690 436 331 561 699 445 358 244 20 176 152 630 229 142 640 533 689 440 333 264 15 449 360 246 24 175 151 622 560 698 V VI 546 724 457 361 272 5 188 99 633 32 170 108 579 555 706 484 370 281 459 363 276 4 187 91 632 545 728 100 578 554 710 486 372 285 31 169 275 8 189 93 636 544 727 451 362 553 709 478 371 284 35 171 102 582 181 92 635 548 729 453 366 274 7 480 375 283 34 163 101 581 557 711 634 547 721 452 365 278 9 183 96 287 36 165 105 580 556 703 479 374 723 456 364 277 1 182 95 638 549 164 104 584 558 705 483 373 286 28 368 279 3 186 94 637 541 722 455 583 550 704 482 377 288 30 168 103 2 185 98 639 543 726 454 367 271 708 481 376 280 29 167 107 585 552 97 631 542 725 458 369 273 6 184 378 282 33 166 106 577 551 707 485 VII VIII 715 466 397 290 41 197 90 588 501 206 117 570 510 661 475 379 317 50 399 294 40 196 82 587 500 719 468 569 509 665 477 381 321 49 205 109 44 198 84 591 499 718 460 398 293 664 469 380 320 53 207 111 573 508 83 590 503 720 462 402 292 43 190 384 319 52 199 110 572 512 666 471 502 712 461 401 296 45 192 87 589 54 201 114 571 511 658 470 383 323 465 400 295 37 191 86 593 504 714 113 575 513 660 474 382 322 46 200 297 39 195 85 592 496 713 464 404 505 659 473 386 324 48 204 112 574 194 89 594 498 717 463 403 289 38 472 385 316 47 203 116 576 507 663 586 497 716 467 405 291 42 193 88 318 51 202 115 568 506 662 476 387 IX – Bottom (actually back vertical plate of model) 421 388 299 77 215 126 597 492 670 303 76 214 118 596 491 674 423 390 216 120 600 490 673 415 389 302 80 599 494 675 417 393 301 79 208 119 667 416 392 305 81 210 123 598 493 391 304 73 209 122 602 495 669 420 75 213 121 601 487 668 419 395 306 125 603 489 672 418 394 298 74 212 488 671 422 396 300 78 211 124 595 Frost's conversion algorithm An algorithm for converting a normal (i.e consecutive numbers) magic hypercube to a nonnormal hypercube, has been explained above. Here I show an example using magic squares for simplicity. Frost (1878) explained the method of construction of his order7 pantriagonal magic cube which is probably the same as the construction of the order9 cube. However, this narrative is quite difficult to understand, so I leave it to someone else to explain. [3] The following illustration demonstrates a method to construct and convert between nonnormal and normal versions of a magic hypercube (using a square for simplicity). This is not the same as the method Frost used, but works for any number base. The auxiliary squares used are normally in the form of Latin squares i.e. 1 of each value on each line. Converting a plane of the above cube in a like manner will show a match with a plane of Frost’s nonnormal nasik magic cube on my Frost page. [4] Point of interest  Both of Frost's cubes are associated.
Neither of them are compact. [1] Nicholas Tam is a Canadian who has a wide range of
pursuits. He has a website (nothing on magic cubes)
here.
Pantriagonal Associated order4 In reviewing my notes recently, I came across these two unusual pantriagonal magic cubes constructed by Mitsutoshi Nakamura. This first cube, constructed in 2004, has 8 orthogonal planes horizontally symmetrical. It is compact_2 but not complete. 1 32 33 64 62 35 30 3 4 29 36 61 63 34 31 2 56 41 24 9 11 22 43 54 53 44 21 12 10 23 42 55 13 20 45 52 50 47 18 15 16 17 48 49 51 46 19 14 60 37 28 5 7 26 39 58 57 40 25 8 6 27 38 59 This cube constructed in 2007, is associated (center
symmetric). Is this the first such cube? 1 55 14 60 31 38 20 41 46 28 33 23 52 9 63 6 40 29 43 18 53 3 58 16 11 50 8 61 26 48 21 35 30 44 17 39 4 57 15 54 49 7 62 12 47 22 36 25 59 2 56 13 42 32 37 19 24 45 27 34 5 51 10 64 A recent visit to Mitsutoshi’s site shows new constructions for associated magic cubes of order8 (2 types) (December 2009). He also added a magic tesseract of the class
Diag+Pan3 in November 2009. It is order16, associated and noncompact.
Mitsutoshi now has an example of each of the 18 classes of magic tesseract! [1] Mitsutoshi Nakamura’s web site is at http://homepage2.nifty.com/googol/magcube/en/history.htm
Antimagic cubes? Much work has been done with these types of number squares. But so far, I have seen no example of a cube with this these properties. Definition of an antimagic square: [1] Heterosquare: similar to a magic square except all rows, columns and main diagonals have different sums. Antimagic Square: similar to a heterosquare except all rows, columns and main diagonals have consecutive sums. As mentioned, much work has been done on this subject. In particular, by John Cormie and Václav Linak of the University of Winnipeg in 1999. [2] More recently, an investigation of order4 antimagic squares (pandiagonal only) by Dwane Campbell's son, Neil. [3] Here is a panantimagic square that is also^{ 2}compact_2.
[1] My antimagic square page is at here. [2] http://ion.uwinnipeg.ca/~vlinek/jcormie/ [3] Neil Campbell's antimagic order4 squares are at http://magictesseract.com
