Hendricks' Inlaid Magic Cubes

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John Hendricks is well known among magic square hobbyists as a prolific designer and writer on the subject.

Over many years he has come up with many innovative ideas. These include:

  • A completely new way to illustrate the tesseract on paper.
  • A coordinated definition of the main types of magic cubes, that is consistent and extends to all dimensions of magic hypercubes.
  • An order 25 bimagic cube and two perfect magic tesseracts.
  • Inlaid magic squares, cubes, and recently, tesseracts.

On this page I will illustrate some of John’s inlaid magic cube creations. All are taken from a book [1] [2] he has published on this subject. As is normal for my web pages, my purpose is mainly to illustrate. For a detailed discussion of construction methods, and more inlaid magic cubes, please refer to his book.

John Hendricks has a web site at http://members.shaw.ca/johnhendricksmath/. However, his books are now all out of print.

I will also include 1 order 8 cube constructed by myself.  Actually this is better described as a composition magic cube.

[1] John R. Hendricks, Inlaid Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7, 188+ pages.
[2] John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated by Holger Danielsson.

The 28-in-1 bent triagonal cube

27 order 4 cubes inlaid in an order 8 cube.

A versatile magic cube

Contains 1 pantriagonal cube and 12 pandiagonal magic squares.

A more versatile magic cube

Contains 8 pantriagonal cubes and 48 pandiagonal magic squares.

8 Order 4 cubes = 1 order 8

The 8 octants are each an order 4 pantriagonal, compact and complete magic cube.

An inlaid magic tesseract

An order 6 with inlaid order 3.

John Hendricks - original page

This page (1998) on my Geocites site has many examples of squares, cubes, tesseracts. Also history, etc.

 The 28-in-1 bent triagonal cube

 Earlier in his book, John described an order 8 magic cube wherein each of the 8 octants consisted of an order 4 magic cube. The cube described here does a little better then that.

This is a semi-pantriagonal magic cube. It’s claim to fame is that it contains bent triagonals. These are similar to the bent diagonals of Franklin's famous magic squares. However, the four straight triagonals are also correct so this cube is truly magic. Examples of bent triagonals, starting at the top left corner are 336,177, 250, 263, then 3 sets of four numbers that return to top corners; 218, 295, 368, 145; 506, 7, 80, 433; and 39, 474, 401, 112.
There are also 3 sets of four numbers that end up at bottom corners. They are 254, 259, 332, 181; 3, 510, 437, 76; and 291, 222, 149, 364. Finally we have a set of four numbers (478, 35, 108, 405) which is the completion of the straight triagonal.

The above feature insures that the main triagonals are correct for order 4 cubes located as the 8 octants of the order 8 cube. However, because all broken triagonals that start at all the odd numbered rows, columns and planes also sum correctly, we actually have many more inlaid order 4 cubes that are magic. In fact, there are 9 cubes that start with the top left corner on the top plane at rows and columns 1, 3 and 5. Similarly, 9 that start with that corner on the 3rd plane from the top, and 9 with that corner on the 5th plane from the top (but see the note below regarding wrap-around).
John Hendricks refers to this cube as his 28-in-1!

An additional feature is that the corners of all order 3 and 7 cubes (within the main cube) total to the magic constant of 2052. Corners of most of the cubes of orders 2, 4, 6, and 8 also sum correctly. Interestingly, corners of none of the cubes of order 5 sum to 2052!

From Inlaid Magic Squares and Cubes, 1999 pp 130-136
From Inlaid Magic Squares and Cubes, 2nd edition, 2000 pp 147-152

Listing for the order 8 magic cube

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

If we include wrap-around, there are many more order 4 magic cubes within the order 8 cube. I guess, though, that we could not really call it inlaid if a cube is in two or four parts! Here I show an example that is wrapped around from left to right. It starts in the top plane, row 3, column 7.

Top                   II                    III                   IV
392   72  185  377    385   65  142  384    122  442  327  135    127  447  322  130
448  128  129  321    441  121  136  328     66  386  383  191     71  391  378  186
 97  417  352  160    104  424  345  153    415   95  162  354    410   90  167  359
 89  409  360  168     96  416  353  161    423  103  154  346    418   98  159  351

All order 4 cubes shown here are also semi-pantriagonal and bent triagonal magic cubes.

 A versatile magic cube

 This is an order 8 simple magic cube with an order 4 pantriagonal magic cube in the center and 12 4x4 pandiagonal magic squares in the planes parallel to each face. The inner cube sums to 1026 in rows, columns, pillars and all pantriagonals. All 12 pandiagonal magic squares sum to 1026 in rows, columns and all pandiagonals. The cube as a whole sums to 2052 in rows, columns pillars and the 4 triagonals.
Following is a list of the actual numbers shown as horizontal planes. The inlaid cube and the horizontal magic squares are shown in blue. As with all the examples in John’s book, he shows how to obtain a variety of magic cubes of this type by using different solution sets.

I call this a versatile magic cube because it may be easily changed to a different cube by changes to the components. The order 4 inner cube may be placed in any of its 48 aspects by rotations and/or reflections. In addition, because it is pantriagonal, any of the 64 numbers may be brought to the upper left hand corner. This gives 48 x 64 = 3072 different inner cubes.

Each of the 3 sets of 4 magic squares may have their layers interchanged (4!=24 ways). In addition, there are 8 aspects due to rotations and/or reflections and each of 16 numbers may be brought to the top left corner. Note however that the 4 squares in the stack must be treated as a unit when making these changes.

Finally, the stacks themselves may be interchanged in 3!=6 ways. So there are 18,432 variations involving operations to the squares. 3072 x 18432 makes 56,623,104 variations to this one order 8 magic cube. Then this complete cube also has 48 aspects. However, these are not normally considered when counting magic object variations.

John R. Hendricks, Inlaid Magic Squares and Cubes, 1999, pp 137-147
John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd edition, 2000, pp 153-165
 

Top  Top pantriagonal magic square        II  2nd pantriagonal magic square
505   72  264  185  200  377  392   57      2  447  255  322  319  130  127  450
 16  433  241  336  305  144  113  464    503   74  266  183  202  375  394   55
489   88  217  359  170  280  425   24     18  431  224  354  175  273   87  490
 32  417  304  146  351  225   96  481    487   90  297  151  346  232  418   31
465  112  343  233  296  154  401   48     42  407  338  240  289  159  111  466
 40  409  162  288  209  367  104  473    479   98  167  281  216  362  410   39
 56  393  272  177  208  369   73  505    463  114  242  335  306  143  434   15
449  128  249  328  313  136  448    1     58  391  263  186  199  378   71  506
III  Top plane of 4x4x4 cube              IV  2nd plane of 4x4x4 cube
  5  444  309  132  379  206   69  508    507   70  203  382  133  308  443    6
500   77  269  188  323  246  396   53     14  435  243  326  189  268  118  459
 28  476  148  291  221  366  101  421    493   45  286  173  339  228  404   84
422  102  230  341  171  284  475   27     83  403  364  219  293  150   46  494
107  427  347  236  278  165   22  470    414   94  213  358  156  299  483   35
469   21  301  158  356  211  428  108     36  484  163  276  238  349   93  413
461  116  252  333  182  259  437   12     51  398  262  179  332  253   75  502
 60  389  196  373  142  315  124  453    454  123  318  139  372  197  390   59
V  3rd plane of 4x4x4 cube                VI  Bottom plane of 4x4x4 cube
  4  445  134  307  204  381   68  509    510   67  380  205  310  131  446    3
501   76  190  267  244  325  397   52     11  438  324  245  270  187  115  462
406   86  235  348  166  277  491   43     99  419  357  214  300  155   30  478
 44  492  157  302  212  355   85  405    477   29  275  164  350  237  420  100
485   37  292  147  365  222  412   92     20  468  174  285  227  340  109  429
 91  411  342  229  283  172   38  486    430  110  220  363  149  294  467   19
460  117  331  254  261  180  436   13     54  395  181  260  251  334   78  499
 61  388  371  198  317  140  125  452    451  126  141  316  195  374  387   62
VII  3rd horizontal magic square          VIII  Bottom pantriagonal magic square
  7  442  314  135  250  327  122  455    512   65  193  384  257  192  385   64
498   79  207  370  271  178  399   50      9  440  312  137  248  329  120  457
 47  402  290  160  337  239  106  471    472  105  295  153  344  234  408   41
474  103  215  361  168  282  415   34     33  416  210  368  161  287   97  480
 23  426  176  274  223  353   82  495    496   81  169  279  218  360  432   17
482   95  345  231  298  152   43   26     25  424  352  226  303  145   89  488
458  119  311  138  247  330  439   10     49  400  201  376  265  184   80  497
 63  386  194  383  258  191   66  511    456  121  320  129  256  321  441    8

To the left is the two layer ‘expansion shell’. Above are the 'parts', In the center is the order 4 pantriagonal magic cube. It is surrounded by 12 pandiagonal magic squares.
To get a feel for this cube, compare some of the numbers in the above illustrations with the text listing for the magic cube.

 A more versatile magic cube

In his book John shows an order 10 with an inlaid order 6 cube and 12 order 4 magic squares, so it is the same style as the order 8 I have just described.
He then describes a more elaborate cube which uses just a single layer expansion shell. It is an order 12 simple magic cube with eight order 4 inlaid pantriagonal magic cubes and 48 order 4 pandiagonal magic squares.

The constant sum for the order 12 is 10,374 which is the required sum for a normal order 12. The sum for the order 4 cubes and squares is 3,458.

Here I will just show the top horizontal layer of the cube (including the top four pandiagonal magic squares.

Variations of this cube just by operations on the 8 inlaid cubes are;

  • Each sub-cube may translocate planar faces in 64 ways.
  • Each sub-cube may rotate and/or reflect in 48 ways.
  • Sub-cubes may be placed in different locations in the main cube in 8! ways.

This gives a total of 123,863,040 possible variations involving the inlaid cubes only. Still available are the variations involving the 48 pantriagonal magic squares!

 942  1230   355   222  1651   643  1075    67  1518  1363   510   798
 966   474  1400   401  1183   619  1110   174  1700   101  1483   763
 751  1267   317  1340   534  1122   607  1543    41  1616   258   978
 882  1328   546  1255   329   703  1026  1628   246  1555    29   847
 859   389  1195   462  1412  1014   715   113  1471   186  1688   870
 775   487  1386  1495    90  1098   666  1674   199   378  1207   919
 811  1242   343   234  1639  1062   630    55  1530  1351   522   955
 727   306  1280   569  1303  1146   583     6  1580   269  1603  1002
 990  1435   437  1172   414   595  1134  1711   161  1448   138   739
 835  1160   426  1423   449  1038   691  1460   126  1723   149   894
 906   557  1315   294  1292   679  1050   281  1591    18  1568   823
 930   499  1374  1507    78   655  1087  1662   211   366  1219   786
This is the top horizontal plane. Blue numbers are the top pandiagonal magic squares. Black numbers are the top layer of the expansion shell. Following is the top left back inlaid order 4 pantriagonal magic cube. It's top left corner is row 2, column 2 of the 2nd plane from the top of the order 12 cube.

Top                       II                        III                       IV
 563  1308   422  1165    1429   446  1284   299     312  1295   433  1418    1154   409  1319   576
 289  1274   456  1439    1175   432  1298   553     566  1309   419  1164    1428   443  1285   302
1296   311  1417   434     410  1153   575  1320    1307   564  1166   421     445  1430   300  1283
1310   565  1163   420     444  1427   301  1286     312  1295   433  1418     431  1176   554  1297

This complete cube complete with illustrations is also shown on my cube_12.htm page.

John R. Hendricks, Inlaid Magic Squares and Cubes, 1999, pp 163-182
John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd edition, 2000, pp 185-207

 

 8 Order 4 cubes = 1 order 8

 Here 8 order 4 cubes are constructed from a pattern cube so all have identical features. Each is pantriagonal, compact and complete. They were then combined as octants to form an order 8 magic cube. [1]

The order 8 cube, however, is just simple magic. It has none of the special features of the small cubes, presumably because of discontinuances between the small cube interfaces.

This shows the placement of the sub-cubes in the octants of the order 8 cube. The cube to the left was inserted into the lower left front octant, as indicated by the 1 in the corner.

The eight order 4 cubes were constructed by putting the first 8 numbers in pattern position 1 of each of the 8 cubes in turn. Then the next 8 numbers were inserted into pattern position two, but in reverse order. The same procedure was followed, always reversing the order for each pass, until all 512 numbers were used and the 8 order 4 cubes were completed.

The eight cubes were rotated or reflected as necessary when being placed in the order 8 cube, so that the numbers from 1 to 8 appeared in the corners of the large cube.

Pattern Cube 
Plane 1 -Top       Plane 2            Plane 3            Plane 4-Bottom
 1  48  49  32     60  21  12  37     13  36  61  20     56  25   8  41
63  18  15  34      6  43  54  27     51  30   3  46     10  39  58  23
 4  45  52  29     57  24   9  40     16  33  64  17     53  28   5  44
62  19  14  35      7  42  55  26     50  31   2  47     11  38  59  22
Cube 1
Plane 1 -Top          Plane 2               Plane 3               Plane 4 - Bottom
  1  384  385  256    480  161   96  289     97  288  481  160    448  193   64  321
497  144  113  272     48  337  432  209    401  240   17  368     80  305  464  177
 32  353  416  225    449  192   65  320    128  257  512  129    417  224   33  352
496  145  112  273     49  336  433  208    400  241   16  369     81  304  465  176
Cube 2
Plane 1 -Top          Plane 2               Plane 3               Plane 4 - Bottom
  2  383  386  255    479  162   95  290     98  287  482  159    447  194   63  322
498  143  114  271     47  338  431  210    402  239   18  367     79  306  463  178
 31  354  415  226    450  191   66  319    127  258  511  130    418  223   34  351
495  146  111  274     50  335  434  207    399  242   15  370     82  303  466  175
Cube 3
Plane 1 -Top          Plane 2               Plane 3               Plane 4 - Bottom
  3  382  387  254    478  163   94  291     99  286  483  158    446  195   62  323
499  142  115  270     46  339  430  211    403  238   19  366     78  307  462  179
 30  355  414  227    451  190   67  318    126  259  510  131    419  222   35  350
494  147  110  275     51  334  435  206    398  243   14  371     83  302  467  174

Etc

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

There is a lot of number manipulation required to assign the required numbers to each order 4 cube, but this drudgery was largely eliminated by using a spreadsheet.
The individual order 4 cubes were entered manually into the order 8 test spreadsheet , while mentally reflecting or rotating them as required to put the numbers 1 to 8 in the corners of the large cube. The whole project, from conception to final testing, took only about 4 hours.

The concept of using sub-cubes with equal constants may be extended to other orders of cubes. Only even order sub-cubes may be used because each cube must consist of complete complement pairs.

 [1] H. Heinz, Jan. 6, 2003. See also my Composition cubes page.

 An inlaid magic tesseract

 In 1999 John Hendricks published an order 6 magic tesseract with an order 3 inlaid magic tesseract.
I have included it on this page to illustrate a consistency between dimensions (a tesseract is a 4 dimensional cubical object). I will show that the order 3 tesseract, cube and square are all associated.

Just as you can have an inlaid magic square in one quadrant of a larger magic square, so too can you have an inlaid magic cubes within the larger cube. And, now the world’s first example of an inlaid magic tesseract of order 3 situated within the world’s first magic tesseract of order six.

A square has quadrants, a cube has octants and a tesseract has hexadecimants. In this tesseract the inlaid order 3 tesseract is in hexidecimant 6.

The magic sum for the sixth-order tesseract is 3891 and the magic sum for the inner magic sub-tesseract revealed by the red numbers, is 1824.

Click on these thumbnails for a larger view. Then use your browser back button to return here.

The order 6 tesseract uses the numbers from 1 to 46 = 1 to 1296. It is not associated.

However, we know that the inlaid order 3 tesseract should be associated, because all order 3 magic hypercubes are. A quick look at diametrically opposite corners show that all sum to 1216, which is the sum of the first and last numbers (570 + 646) used in this (order 3) tesseract.

Similarly, we can confirm that the order 3 cube shown is one of the four center cubes of the order 3 tesseract, therefore it should also be associated. Again we can confirm by summing opposite corners. In this case they should also sum to 1216, the sum of the first and last numbers used in this construction.

Finally, we can confirm that the square shown, because it is one of 3 central squares of the order 3 cube, is magic and also associated.

The complete Inlaid Order 6 Magic Tesseract was supplied as an 8 page chart insert with these books;
John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, 142++ pages.
John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated by Holger Danielsson.
It may be downloaded as a PDF document here.

This page was originally posted December 2002
It was last updated October 19, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz