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This page contains information on magic cubes discovered in 2008 and 2009.
| More on Compact and Complete | For hypercubes of all dimensions |
| Smallest Order-4 Multiply | Smallest product to date for order-4 |
| Frost Order-9 Cube Model | The Whipple model de-mystified? |
| Pantriagonal Associated order-4 | First example for this order? |
| Anti-magic cubes? | Anti-magic squares. Why not cubes? |
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.
Following is a condensation of the results of these 50+ emails to January 8,
2010.
Compact
Compact implies that the corners of all sub-hypercubes of dimension n sum
to the same value, where n = the dimension of the main hypercube.
The order of the sub-hypercube (hereafter shown as sub-cube) is shown as
compact_m
If all sub-cubes of multiple orders sum correctly, they may be shown as compact_m1,
m2, m3, … (i.e. compact_1,2,3,…)
For an order-16 cube:
Compact_2 which results in sub-cubes 4,6,8,10,12,14,and 16 also being correct
Compact_3 which results in sub-cubes 7,11,and 15 also being correct.
Compact_5 which results in sub-cube 13 also being correct. 21 would be the next one (on a larger cube).
Compact_9. (25 would be the next in this series.)
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.
The more general statement for cubes would be: Only three types of compact may
be present in a cube of order 2k for k>2. If there are
three, then one of the three must be compact_(2(k-1) + 1).
A general statement for tesseracts would be: Only four types of compact may be
present in a tesseract of order 2k for k>3. If there
are four, then one of the four must be compact_2(k-1) + 1). etc.
The above implies the sub-cubes are the same dimension as the hosting hypercube. In the event you wish to indicate a sub-hypercube of lower dimension (for instance the sub-squares in the planes of a cube), this may be indicated thus; 2compact_2 or 2compact2 (or simply 2compact_2).
Compactplus
If all possible orders of sub-cubes from 2 to m are compact, the Hypercube is compactplus.
The smallest possible order of each dimension of Nasik magic hypercube is always compactplus. i.e order-4 square, order-8 cube, order-16 tesseract, etc.
Higher orders of Nasik hypercubes are NOT compactplus! (This statement is not yet proven conclusively as of January, 2010.)
Addendum March 1, 2010:
In an email dated Feb. 27/10, Dwane Campbell proved that order-16 is the only
nasik tesseract that is compactplus.
i.e. This tesseract is compact_2,3,5,9 so the 16 corners of all sub-tesseracts
from 2 to 16 sum to the same constant.
As this is now proven for the three lowest dimension of magic hypercube, I think
we may assume it is general for all dimensions!
Examples of this feature using nasik squares for simplicity.
Order-4 is the smallest possible nasik square. |
Order-8 is compact_2, 5 but NOT 3 and 7, so is not compactplus. (Corners of all sub-squares of orders 2, 4, 5, 6, and 8 sum correctly in this order-8 square because compact_2 obviously includes all even orders up to m.) |
Complete
The terminology for the complete feature is similar to that for the compact
feature.
A magic square that is 2complete_2 or 2complete_2 would be complete in the
traditional sense. The first or superscripted 2 indicates that the feature is
present in two dimensions i.e. the diagonal of a square.
The second 2 indicates that the complete feature is summing two numbers.
A 3complete_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 1complete_4 indicates a figure in which all groups of 4 numbers
evenly spaced in any monagonal add to the same constant.
The order-4 and order-8 squares above are both 2complete_2.
However, tradition does not require the qualifiers because in both cases we are
referring to the diagonals, and in both cases the interval between the two
numbers of each pair is m/2. So it is sufficient to simply say these
squares are complete! Since the order-8 square is 2complete_2, it is
also 2complete_4 because two sets of complementary pairs evenly
spaced in the same diagonal always add to S/4.
| If an order 8 square is 2complete_2,
then r,c 1,1 and 5,5 are a complementary pair and sum to s/4. r,c 3,3 and 7,7 are also a complementary pair, making the square also 2complete_4. However r,c 1,1; 3,3; 5,5; and 7,7 can add to S/2 even when the individual pairs do not add to S/4 (see the example). Point-of-interest: This square is compact_2. All 2x2 squares sum to S/2. |
A
2complete_4 simple magic square.
|
The relationship between Complete and Compact is illustrated by the following quote from a Dwane Campbell email:
Order-4 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. Order-8 is
compact_2, 5 but NOT 3 and 7, so is not compactplus. (Corners of all sub-squares
of orders 2, 4, 5, 6, and 8 sum correctly in this order-8 square because
compact_2 obviously includes all even orders up to m.)
When a cube of order m is complete then all pairs of numbers spaced an (m/2,m/2,m/2)
vector apart will add to the same constant, C. As a result of the complete
function the cube must then be compact_(m/2+1). Each set of opposite
corners of an order (m/2+1) sub cube will add to C, therefore the eight
corners of the sub cubes will always add to 4C. For instance an order 12 cube
that is complete must also be compact_7. It may or may not be compact_3. This
latter fact points out that the converse of statements 1, 2, and 3 may not be
true, i.e. compact_3 always leads to compact_7 but compact_7 does not always
require that the hypercube also be compact_3. The converse is true, however, for
order 2n hypercubes.
[1] Dwane Campbell’s web site is at
http://home.earthlink.net/~dwanecampbel/index.html
[2] Mitsutoshi Nakamura’s web site is at
http://homepage2.nifty.com/googol/magcube/en/
[3] Aale de winkel's Encyclopedia web site is at
http://www.magichypercubes.com/Encyclopedia/index.html
Smallest Order-4 Multiply
In mid January, Christian Boyer posted his latest Update
to
www.multimagie.com/English/MultiplicCubes.htm
Included were two interesting order-4 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 2compact_2.
Of course, in a multiply magic hypercube we are concerned with products, just as
we are dealing with sums in an additive magic hypercube!
Both cubes have the same highest number, 364, but the
Alekseyev cube has a product that is exactly one-half of Boyer’s cube.
Compare 364 with Sayles’ and Trenkler’s 7560.[1]
The magic product for these two cubes is the same. 57,153,600.
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. wrap-around 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
The horizontal planes are not 2compact_2 or _3. (I did not check the other two orientations.)
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 Order-9 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 order-9 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.
For example, the central number in the model cube is 365. The corresponding number on the opposite face of this plane (the non-normal cube) is 445. In a handwritten note accompanying the cube model, Frost explained the relationship between these two central numbers thus:
365 on one side is 445 on the other side because 365 = 4 x 9 x 9 + 4 x 9 + 5
These (the second set) numbers form the well known Frost Order-9 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 order-9 cube constructed. [4]
Tam believes that Frost first constructed the non-normal nasik cube, then converted it to the normal cube and constructed the model. He had already done this with his order-7 cube. [3]
Actually, the date the model was constructed is still unknown!
Each plane appears on a vertical glass panel parallel to the front of the model.
Because my cube test program works by entering the horizontal planes, I have listed the planes as if the model had been rotated (rolled) back 90 degrees.
Frost’s Nasik Order-9 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 595Frost's conversion algorithm
An algorithm for converting a normal (i.e consecutive numbers) magic hypercube to a non-normal hypercube, has been explained above. Here I show an example using magic squares for simplicity.
Frost (1878) explained the method of construction of his order-7 pantriagonal magic cube which is probably the same as the construction of the order-9 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 non-normal 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 non-normal nasik magic cube on my Frost page. [4]
Point of interest - Both of Frost's cubes are associated. Neither of them are compact.
The two squares shown here are nasik magic and not associated. Because they are the smallest possible order of nasik square, they are compact and complete. This also illustrates that compact and complete work for non-normal squares as well (as we would expect).
[1] Nicholas Tam is a Canadian who has a wide range of pursuits. He has a website (nothing on magic cubes) here.
[2] A photograph of this cube may be seen at http://www.multimagie.com/English/Frost.htm
[3] A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.
He mentions that he presented a similar model of his order-7 cube to the South Kensington Museum. This is now lost.
[4] Frost's published order-9 cube is at here.
[5] Whipple Museum of the History of Science at http://www.hps.cam.ac.uk/whipple/. The accession number for Frost's cube is Wh.1251.
Pantriagonal Associated order-4
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?
It is the first pantriagonal associated order-4 magic cube that I have seen. It is not compact or complete.
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 order-8 (2 types) (December 2009).
He also added a magic tesseract of the class Diag+Pan3 in November 2009. It is order-16, associated and non-compact. Mitsutoshi now has an example of each of the 18 classes of magic tesseract!
Actually a visit to his update page will show that he has been very actively investigating magic hpercubes over the last several years. [1]
[1] Mitsutoshi Nakamura’s web site is at http://homepage2.nifty.com/googol/magcube/en/history.htm
Anti-magic 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 anti-magic square: [1] Heterosquare: similar to a magic square except all rows, columns and main diagonals have different sums. Anti-magic 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 order-4 anti-magic squares (pandiagonal only) by Dwane Campbell's son, Neil. [3] Here is a pan-antimagic square that is also 2compact_2.
 | This is an order-4 pan-anti-magic square with 32 sums in consecutive order, skipping the expected magic constant for order-4 magic squares. The sums range from 14 to 46 skipping the magic constant 30. This is the only possible square of this type (except, of course, for rotations or reflections). Can a hetero or anti-magic cube exist?
|
[1] My anti-magic square page is at http://www.magic-squares.net/anti_ms.htm [2] http://ion.uwinnipeg.ca/~vlinek/jcormie/ [3] Neil Campbell's anti-magic order-4 squares are at http://home.earthlink.net/~dwanecampbel/antimagic.html
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Harvey Heinz harveyheinz@shaw.ca
This page originated Febuary 2010
This page last updated March 10, 2010
Copyright © 2004 by Harvey D. Heinz