# The Heinz X6 Magic Cube

A 4x4x4 magic cube consists of an array of cells that each hold a number. These numbers are such that each row, column, pillar, and each of the four triagonals sum to the constant.
Imagine a structure such that each cell was actually itself a small cube. If we place a number on each of the six surfaces of each cubelet, it is possible to have 6 magic cubes, one of which is represented by each face of the cubelets. Herein is described a physical model of such a cube.

 This model is constructed using sixty-four 3/4" wooden blocks and connected by 1/8" hardwood dowels showing rows, columns, pillars and triagonals. Numbers used are 1 to 384, which is 43 times 6. This model was completed on July 28, 2002 The six faces of each cubelet are each painted a different color. The 64 cubelet faces of each color form a pantriagonal magic cube with two special features. It is compact because all 2x2 square arrays within the cube, in all 3 orientations, sum to the constant. This includes wraparound. It is complete because every pantriagonal contains m/2 complement pairs spaced m/2 apart. Each of the six cubes is also pantriagonal, because all two and three segment lines parallel to each of the 4 triagonals sums to the correct value! Unfortunately, the magic constants for these 6 squares are not the same, but vary from 760 to 780. They are:  White = 760, Blue = 764, Red = 768, Pink = 772, Green = 776, Yellow = 780. Some examples of the 2x2 square arrays are: white, 271 + 133 + 115 + 241 = 760; green, 305 + 95 + 77 + 299 = 776; green (wrap-around), 101 + 281 + 95 + 299 = 776. Compare these with the large copy of picture 1. Combinations in each cube that equal the magic constant are: rows, columns, pillars = 3 x 42, pantriagonals =  4m2, so total lines = 112. 2x2 squares that sum correctly: 43 x 3 orientations = 192. total combinations for each cube = 304. The six faces of each of the 64 cubelets sum to 1155. This is because the numbers appearing on opposite faces of each cubelet  are members of a complement pair (3 times 385 = 1155). (Click on a picture for an enlarged view.)

The Six cubes listed

 White  S = 760 Top                   Top – 1                 1  355  217  187    367   37  151  205     373   31  157  199     19  337  235  169      55  301  271  133    313   91   97  259     331   73  115  241     61  295  277  127     Bottom + 1            Bottom 109  247  325   79    283  121   67  289 265  139   49  307    103  253  319   85 163  193  379   25    229  175   13  343 223  181    7  349    145  211  361   43

Blue    S = 764
Top                   Top - 1               Bottom + 1            Bottom
2  356  218  188    368   38  152  206    110  248  326   80    284  122   68  290
374
32  158  200     20  338  236  170    266  140   50  308    104  254  320   86
56
302  272  134    314   92   98  260    164  194  380   26    230  176   14  344
332
74  116  242     62  296  278  128    224  182    8  350    146  212  362   44

Red     S = 768
Top                   Top - 1               Bottom + 1            Bottom
3  357  219  189    369   39  153  207    111  249  327   81    285  123   69  291
375
33  159  201     21  339  237  171    267  141   51  309    105  255  321   87
57
303  273  135    315   93   99  261    165  195  381   27    231  177   15  345
333   75  117  243     63  297  279  129    225  183    9  351    147  213  363   45

Pink     S = 772
Top                   Top - 1               Bottom + 1            Bottom
94  316  262  100    304   58  136  274    178  232  346   16    196  166   28  382
298
64  130  280     76  334  244  118    214  148   46  364    184  226  352   10
40  370  208  154
358    4  190  220    124  286  292   70    250  112   82  328
340   22  172  238
34  376  202  160    256  106   88  322    142  268  310   52

Green  S = 776
Top                   Top - 1               Bottom + 1            Bottom
95  317  263  101    305   59  137  275    179  233  347   17    197  167   29  383
299
65  131  281     77  335  245  119    215  149   47  365    185  227  353   11
41
371  209  155    359    5  191  221    125  287  293   71    251  113   83  329
341   23  173  239     35  377  203  161    257  107   89  323    143  269  311   53

Yellow S = 780
Top                   Top - 1               Bottom + 1            Bottom
240  174   24  342    162  204  378   36    324   90  108  258     54  312  270  144
156
210  372   42    222  192    6  360     72  294  288  126    330   84  114  252
282
132   66  300    120  246  336   78    366   48  150  216     12  354  228  186
102  264  318   96    276  138   60  306     18  348  234  180    384   30  168  198

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 14, 2009
Copyright © 2003 by Harvey D. Heinz