Cube_Update-3

Introduction

Material about magic cubes continues to appear. This update contains material I have received during the last nine months of 2004.

An Order 16 Bordered Magic Cube	New Definitions
A New Magic Cube Class	Other News and Acknowledgements

An Order 16 Bordered Magic Cube

On July 16, 2004, I received the following email from Mitsutoshi Nakamura:

I have made an order-16 'bordered' diagonal magic cube

(Bordered16.xls). It consists of numbers 1 to 4096. S = 32776.

This cube is a diagonal magic cube and contains within it

the following subcubes that consist of consecutive integers:

an order-4  simple   magic cube (from 2017 to 2080), S = 8194

an order-6  diagonal magic cube (from 1941 to 2156), S = 12291

an order-8  diagonal magic cube (from 1793 to 2304), S = 16388

an order-10 diagonal magic cube (from 1549 to 2548), S = 20485

an order-12 diagonal magic cube (from 1185 to 2912), S = 24582

an order-14 diagonal magic cube (from  677 to 3420). S = 28679

The cube also contains a lot of order-4 magic squares.

I have confirmed we can construct such a cube for any even order greater than 4.

With best regards,

Mitsutoshi Nakamura

I have analyzed these cubes using my standard test spreadsheets.
The order 4 cube is indeed simple. It contains 4 planar and 2 oblique simple magic squares.
The orders (m) 6 to 16 cubes each contain 3m order m planar and 6 oblique simple magic squares, so each of these is a proper diagonal magic cube. Because in each case, the cube contains the minimum requirement for that class, the cube is said to be a proper diagonal magic cube. (The order 4 cube is not proper because if contains some magic squares, and so exceeds the minimum requirement for a simple magic cube.)
The magic sums of the 7 cubes are correct as per the equation (mⁿ⁺¹⁾ + m)/2 + m(a-1), where m = order, n = dimension, and a = starting number.

This is a bordered (or concentric) magic cube because the middle numbers of the series are all in the center cube. The lowest and highest numbers are in the outside shell!
If the numbers were mixed throughout the various cubes, this would be an inlaid cube. [1]

Good work Mitsutoshi!
An archive file of these 7 cubes, each in a test spreadsheet is available for downloading (1218 Kb)

[1] John R. Hendricks, Inlaid Magic Squares and Cubes, 0-9684700-1-7, 1999, pp 36-37.

New Definitions

Proper – Refers to a cube that contains exactly the minimum requirements for that class of cube. i.e. a proper simple or pantriagonal magic cube would contain no magic squares, a proper diagonal magic cube would contain exactly 3m plus 6 simple magic squares, etc.
This term was coined by Mitsutoshi Nakamura in an email of April 15, 2004.

s-magic cube – A magic cube in which the six surfaces are magic squares. All diagonal, pandiagonal and perfect cubes are s-magic, but cubes exist in which the interior planes are not magic squares. See the Worthington cube of 1910.
This term was coined by Walter Trump in 2004.

Pan-2,3-agonal – used by Nakamura on his site to designate what I refer to as Perfect. i.e. a combination Pantriagonal and Pandiagonal magic cube. That way he completely avoids the confusion over the term Perfect. (Historically, the term Perfect was used by different authors to define cubes with a variety of characteristics.)
I now avoid confusion with the term perfect by referring to them as nasik magic cubes. See my Theory of Paths Nasik article.

A New Magic Cube Class

PantriagDiag – A magic cube that is a combination Pantriagonal and Diagonal cube. All main and broken triagonals must sum correctly.
In addition, it will contain 3m order m simple magic squares in the orthogonal planes, and 6 order m pandiagonal magic squares in the oblique planes.

This is number 4 in what is now 6 classes of magic cubes. So far, very little is known of this class of cube. The only one constructed so far (that I am aware of) is by Mitsutoshi Nakamura see below) and is as order 8 and not associated.

An enhanced feature of this particular cube is that all 2x2 and corners of 5x5 cubes sum to S. this particular cube is also complete (every pantriagonal contains m/2 complement pairs, spaced m/2 apart).
It is also proper, because it contains the bare minimum magic squares required for this class of magic cube.

My Summary Page table of First Cube of Each Order for Each Class. is now out-of-date. It should have another column for the PantriagDiag cubes. However, at present there is only one such cube known (for order 8) so I will leave the table as is.

An Order 8 Pantriagonal Diagonal Magic Cube

I                                           II
  1  352  300  117  422  251  143  466      492  181  193  416   79  274  358   59
254  419  471  138  345    8  116  301      279   74   62  355  180  493  409  200
349    4  120  297  250  423  467  142      184  489  413  196  275   78   58  359
418  255  139  470    5  348  304  113       75  278  354   63  496  177  197  412
303  114    6  347  140  469  417  256      198  411  495  178  353   64   76  277
468  141  249  424  119  298  350    3       57  360  276   77  414  195  183  490
115  302  346    7  472  137  253  420      410  199  179  494   61  356  280   73
144  465  421  252  299  118    2  351      357   60   80  273  194  415  491  182
III                                         IV
230  443  463  146  321   32  108  309      271   82   38  379  172  501  385  224
 25  328  308  109  446  227  151  458      500  173  217  392   87  266  382   35
442  231  147  462   29  324  312  105       83  270  378   39  504  169  221  388
325   28  112  305  226  447  459  150      176  497  389  220  267   86   34  383
460  149  225  448  111  306  326   27       33  384  268   85  390  219  175  498
311  106   30  323  148  461  441  232      222  387  503  170  377   40   84  269
152  457  445  228  307  110   26  327      381   36   88  265  218  391  499  174
107  310  322   31  464  145  229  444      386  223  171  502   37  380  272  81
V                                           VI
373   44   96  257  210  399  507  166      160  449  437  236  315  102   18  335
394  215  163  510   45  372  264   89       99  318  330   23  456  153  237  436
 41  376  260   93  398  211  167  506      452  157  233  440  103  314  334   19
214  395  511  162  369   48   92  261      319   98   22  331  156  453  433  240
 91  262  370   47  512  161  213  396      434  239  155  454   21  332  320   97
168  505  397  212  259   94   42  375      333   20  104  313  234  439  451  158
263   90   46  371  164  509  393  216      238  435  455  154  329   24  100  317
508  165  209  400   95  258  374   43       17  336  316  101  438  235  159  450
VII                                          VIII
402  207  187  486   53  364  288   65      123  294  338   15  480  129  245  428
365   52   72  281  202  407  483  190      136  473  429  244  291  126   10  343
206  403  487  186  361   56   68  285      295  122   14  339  132  477  425  248
 49  368  284   69  406  203  191  482      476  133  241  432  127  290  342   11
192  481  405  204  283   70   50  367      341   12  128  289  242  431  475  134
 67  286  362   55  488  185  205  404      426  247  131  478   13  340  296  121
484  189  201  408   71  282  366   51        9  344  292  125  430  243  135  474
287   66   54  363  188  485  401  208      246  427  479  130  337   16  124  293

Other News and Acknowledgements

Other news

Christian Boyer posted a page of recent magic cube news on November 1, 2004. Find it at www.multimagie.com/

Acknowledgements

I wish to acknowledge with thanks, the contributions to filling my Summary Page table of First Cube of Each Order for Each Class.

Mitsutoshi Nakamura has been mentioned above. He supplied cubes for the last 7 vacant cells of the table (the last one in conjunction with Abhinav Soni) His site includes cubes definitions, algorithms, theorems, etc., and is at http://homepage2.nifty.com/googol/magcube/en/

Abhinav Soni supplied an amazing 16 magic cubes for the above table. He no longer has a web page (Nov.09)