Magic Cube Update-6

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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:

  1. Compact_2 which results in sub-cubes 4,6,8,10,12,14,and 16 also being correct
  2. Compact_3 which results in sub-cubes 7,11,and 15 also being correct.
  3. Compact_5 which results in sub-cube 13 also being correct. 21 would be the next one (on a larger cube).
  4. 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 the corners of all possible orders of sub-cubes from 2 to m sum correctly, 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.

1 

8

10

15

12

13

3

6

7

2

16

9

14

11

5

4

 

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.

1

16

17

32

53

60

37

44

63

50

47

34

11

6

27

22

3

14

19

30

55

58

39

42

61

52

45

36

9

8

25

24

12

5

28

21

64

49

48

33

54

59

38

43

2

15

18

31

10

7

26

23

62

51

46

35

56

57

40

41

4

13

20

29

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.
0 31 34 61 36 59 6 25
47 48 13 18 11 20 41 54
24 7 58 37 60 35 30 1
55 40 21 10 19 12 49 46
17 14 51 44 53 42 23 8
62 33 28 3 26 5 56 39
9 22 43 52 45 50 15 16
38 57 4 27 2 29 32 63

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://magictesseract.com
[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
  • Constructed in 2007
  • Max nb = 364
  • P = 17,297,280
  • The 4 horizontal planes are simple magic squares that are 2compact_3 (because the product of the 4 corners of all 3x3 squares = P.
  • The cube is Compact_3. The product of 8 corners of all 3x3x3 sub-cubes = P2.

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
  • Constructed in January 2010
  • Max nb = 364
  • P = 8,648,640.
  • The cube is Compact_3. The product of 8 corners of all 3x3x3 sub-cubes = P2.
  • 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  595

Frost'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).

Model of the 9x9x9 magic cube made by A. H. Frost c. 1877 (Wh.1251) Whipple Museum of the History of Science, University of Cambridge (click to enlarge)

a. Normal (consecutive) side

b. Non-consecutive side

Frost's cube as seen from two opposite sides: one with numbers running from 1 to 729 (left), the othe with numbers from 1 to 889.

[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?

Yes! Peter Bartsch constructed several heterocubes in 2003-2004. One is almost anti-magic! And two consist of prime numbers.
See Update-1.

 

[1] My anti-magic square page is at here.
[2] http://ion.uwinnipeg.ca/~vlinek/jcormie/
[3] Neil Campbell's anti-magic order-4 squares are at http://magictesseract.com

This page was originally posted Feb. 2010
It was last updated February 11, 2013
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz