Magic Cubes - Order 8

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With order 8, we are introduced to two new classifications, The diagonal cube, and the perfect cube.
Both of the order 8 diagonal cubes I have appear already on other pages so I will not reproduce them here.
I have no order 8 pantriagonal magic cubes that are associated, or that contain magic squares. Are there any of either?
I have not seen any order 8 pandiagonal magic cubes or, for that matter, such cubes in higher orders.
The perfect cube I show is not associated, because an associated perfect magic cube can appear first only in order 9. Much more information on perfect cubes and listings for such cube may be seen on my perfect and perfect2 pages.
A feature called compact that I did test for in order 8 is for corners of smaller order cube arrays within the summing to S.

Andrews 1908 Simple magic cube, no magic squares.
Hendricks Inlaid 1993 Simple magic cube, no magic squares.
Hetherington 1997 Diagonal magic cube, 30 magic squares.
Hendricks Inlaid 2 1999 Pantriagonal magic cube, no magic squares.
Soni 2001 Pantriagonal magic cube, no magic squares.
Hendricks Perfect (nasik) 1998 Perfect (nasik) magic cube, 30 pandiagonal magic squares.

Andrews

This cube has the minimum requirements required to be magic.

This simple magic cube, first published in 1908 is associated. It contains no magic squares, but the 8 corners of all orders 3 and 7 sub cubes sum to S, so this cube is compact_3.  See my definitions page.

W. S. Andrews, Magic Squares & Cubes, Open Court, 1908, 193+ pages.
W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, 419+ pages .
     

Horizontal plane I - Top                   II
  1  511  510    4    5  507  506    8     448   66   67  445  444   70   71  441
504   10   11  501  500   14   15  497      73  439  438   76   77  435  434   80
496   18   19  493  492   22   23  489      81  431  430   84   85  427  426   88
 25  487  486   28   29  483  482   32     424   90   91  421  420   94   95  417
 33  479  478   36   37  475  474   40     416   98   99  413  412  102  103  409
472   42   43  469  468   46   47  465     105  407  406  108  109  403  402  112
464   50   51  461  460   54   55  457     113  399  398  116  117  395  394  120
 57  455  454   60   61  451  450   64     392  122  123  389  388  126  127  385
III                                        IV
384  130  131  381  380  134  135  377     193  319  318  196  197  315  314  200
137  375  374  140  141  371  370  144     312  202  203  309  308  206  207  305
145  367  366  148  149  363  362  152     304  210  211  301  300  214  215  297
360  154  155  357  356  158  159  353     217  295  294  220  221  291  290  224
352  162  163  349  348  166  167  345     225  287  286  228  229  283  282  232
169  343  342  172  173  339  338  176     280  234  235  277  276  238  239  273
177  335  334  180  181  331  330  184     272  242  243  269  268  246  247  265
328  186  187  325  324  190  191  321     249  263  262  252  253  259  258  256
V                                          VI
257  255  254  260  261  251  250  264     192  322  323  189  188  326  327  185
248  266  267  245  244  270  271  241     329  183  182  332  333  179  178  336
240  274  275  237  236  278  279  233     337  175  174  340  341  171  170  344
281  231  230  284  285  227  226  288     168  346  347  165  164  350  351  161
289  223  222  292  293  219  218  296     160  354  355  157  156  358  359  153
216  298  299  213  212  302  303  209     361  151  150  364  365  147  146  368
208  306  307  205  204  310  311  201     369  143  142  372  373  139  138  376
313  199  198  316  317  195  194  320     136  378  379  133  132  382  383  129
VII                                        VIII - Bottom
128  386  387  125  124  390  391  121     449   63   62  452  453   59   58  456
393  119  118  396  397  115  114  400      56  458  459   53   52  462  463   49
401  111  110  404  405  107  106  408      48  466  467   45   44  470  471   41
104  410  411  101  100  414  415   97     473   39   38  476  477   35   34  480
 96  418  419   93   92  422  423   89     481   31   30  484  485   27   26  488
425   87   86  428  429   83   82  432      24  490  491   21   20  494  495   17
433   79   78  436  437   75   74  440      16  498  499   13   12  502  503    9
 72  442  443   69   68  446  447   65     505    7    6  508  509    3    2  512
            rule-w.gif (2726 bytes)

Hendricks Simple Inlaid Cube

This is a simple magic cube and is not associated. S = 1026

It contains no magic squares. All order 5 sub-cubes have corners summing to S, so this cube is compact_5.

However, this cube is inlaid. Each of the 8 octants of this cube is itself an order 4 pantriagonal magic cube with S = 1026. One of the octants is shown below the listing for this order 8 cube.

John R. Hendricks, An Inlaid Magic Cube, JRM 25:4, 1993, pp 286-288.     

Horizontal plane I - Top                   II
375  200  314  137   73  506    8  439     249  330  184  263  455  120  394   57
257  178  336  255   63  400  114  449     143  320  194  369  433    2  512   79
192  271  241  322  386   49  463  128     306  129  383  208   16  447   65  498
202  377  135  312  504   71  441   10     328  247  265  186  122  457   55  392
210  353  159  304  496   95  417   18     352  239  273  162   98  465   47  416
168  279  233  346  410   41  471  104     298  153  359  216   24  423   89  490
281  170  344  231   39  408  106  473     151  296  218  361  425   26  488   87
367  224  290  145   81  482   32  431     225  338  176  287  479  112  402   33
III                                        IV
272  191  321  242   50  385  127  464     130  305  207  384  448   15  497   66
378  201  311  136   72  503    9  442     248  327  185  266  458  121  391   56
199  376  138  313  505   74  440    7     329  250  264  183  119  456   58  393
177  258  256  335  399   64  450  113     319  144  370  193    1  434   80  511
169  282  232  343  407   40  474  105     295  152  362  217   25  426   88  487
223  368  146  289  481   82  432   31     337  226  288  175  111  480   34  401
354  209  303  160   96  495   17  418     240  351  161  274  466   97  415   48
280  167  345  234   42  409  103  472     154  297  215  360  424   23  489   90
V                                          VI
131  308  206  381  445   14  500   67     269  190  324  243   51  388  126  461
245  326  188  267  459  124  390   53     379  204  310  133   69  502   12  443
332  251  261  182  118  453   59  396     198  373  139  316  508   75  437    6
318  141  371  196    4  435   77  510     180  259  253  334  398   61  451  116
294  149  363  220   28  427   85  486     172  283  229  342  406   37  475  108
340  227  285  174  110  477   35  404     222  365  147  292  484   83  429   30
237  350  164  275  467  100  414   45     355  212  302  157   93  494   20  419
155  300  214  357  421   22  492   91     277  166  348  235   43  412  102  469
VII                                        VIII - Bottom
252  331  181  262  454  117  395   60     374  197  315  140   76  507    5  438
142  317  195  372  436    3  509   78     260  179  333  254   62  397  115  452
307  132  382  205   13  446   68  499     189  270  244  323  387   52  462  125
325  246  268  187  123  460   54  389     203  380  134  309  501   70  444   11
349  238  276  163   99  468   46  413     211  356  158  301  493   94  420   19
299  156  358  213   21  422   92  491     165  278  236  347  411   44  470  101
150  293  219  364  428   27  485   86     284  171  341  230   38  405  107  476
228  339  173  286  478  109  403   36     366  221  291  148   84  483   29  430
The top back left octant of the above cube is an order 4 pantriagonal magic cube. Because it is an inlay, the numbers used are not consecutive, and so it is not a normal cube. S = 1026.

Any of the eight order 4 cubes may be rotated or reflected to any of it's 48 aspects, it may be transformed to a different pandiagonal magic cube by translocation of planes, or the 8 cubes may be rearranged within the order 8 cube in any manner, without destroying the magic of the order 8 cube.

 
I – Top                        II                                III                                IV - Bottom
375  200  314  137     249  330  184  263     272  191  321  242     130  305  207  384
257  178  336  255     143  320  194  369     378  201  311  136     248  327  185  266
192  271  241  322     306  129  383  208     199  376  138  313     329  250  264  183
202  377  135  312     328  247  265  166     177  258  256  335     319  144  370  193

                  

Hetherington Diagonal Cube

This cube is classed as a diagonal magic cube, with extra features. This cube was never published, but I constructed it from instructions by Charles Hetherington that were received by Mutsumi Suzuki in Aug. 1997.

The 8 horizontal planes and 8 vertical planes parallel with the front of the cube, are simple magic squares.
The 8 vertical planes parallel with the sides of the cube are pandiagonal magic squares.
The 6 oblique planes are simple magic squares.
Corners of all orders 3, 5 and 7 sub-cubes sum to S so the cube is compact_3, 5. The 7 is implied because all compact_3 are compact_7 (The reverse is not always true). When I mention the compact feature, wrap-around is always assumed to apply. That means that there are m3  sub-cubes of the order specified that are correct.

Matsumi Suzuki’s excellent site is now available at http://mathforum.com/te/exchange/hosted/suzuki/MagicSquare.html

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

                 

Hendricks pantriagonal Inlaid

All 256 main and broken triagonals are correct in this cube, so this is a pantriagonal magic cube.

Planar squares are not magic because of incorrect diagonals. Because all pantriagonals are correct, all pandiagonals are correct in the 6 oblique squares. However, 4 squares have incorrect rows and 2 have incorrect columns. i.e. there are no magic squares in this cube.

Corner sums of all orders-3 and 7 cubes equal S so it is compact_3. Also, this cube is termed complete because every pantriagonal contains m/2 complement pairs with the two members of each pair spaced m/2 apart.

This cube is similar to the first Hendricks cube shown inasmuch as the 8 octants are inlaid order 4 magic cubes.
However, as stated above this cube is pantriagonal. This permits it to be transformed to a different magic cube by moving planes from one side of the cube to the opposite side. This is similar to moving rows or columns in a pandiagonal magic square from one side to the opposite side, to obtain a different pandiagonal magic square. By doing this, however, the inlaid feature is lost and the octants will no longer be magic cubes.

Of course, the order 8 cube may also be transformed by translocation of planes in any of the order 4 cubes also, because they are all pantriagonal as well. Furthermore, the cube may also be changed by rotating or reflecting any (or all) of the 8 order 4 cubes to any of their 48 aspects and/or exchanging positions of the individual cubes. These transformations will, in almost all cases, destroy the pantriagonal feature of the order 8 cube.

John R. Hendricks, Inlaid Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7, 188+ pages.

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

Following is the top right back quadrant of the above order 8 pantriagonal magic cube. Each of these octants are also pandiagonal magic and also compact_3 and complete. These order 4 cubes are not normal because the numbers used in each cube are not consecutive.

I                     II                     III                   IV
 92  306  205  423    271  101  410  244    226  396  119  285    437  223  292   74
247  413   98  268    420  202  309   95     77  295  220  434    282  116  399  229
394  228  287  117    221  439   76  290    308   90  421  207    103  269  242  412
293   79  436  218    114  284  231  397    415  245  266  100    204  418   93  311

                    

Soni pantriagonal cube

This pantriagonal magic cube is not associated and contains no magic squares. In fact, I have not seen any order 8 pantriagonal magic cubes that are associated or that contain any magic squares.
Corners of all cubes of orders 2, 4, 5, 6, and 8 (including wrap-around) all sum correctly, so this cube is compact_2,5. This cube is also complete.

In additions All planar diagonal pairs sum to two times the magic constant, S.
There are horizontal bent diagonals (V shaped)
on all horizontal planes, and vertical planes parallel to the front of the cube starting all cells of columns 1 and 5,
and on all vertical planes parallel to the sides, starting on all cells of columns 3 and 7.
There are no planes that have vertical bent diagonals starting on all cells of any particular row or column

Constructed with his HyperMagicCube.exe program. Obtainable from his magic cubes site. (No longer available)

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

Addendum November 1, 2006
I received a similar cube to Soni's, hand constructed by 84 year old Arsène Durupt. His cube differs in features only in that it is not complete, and order-5 sub-cube corners sum incorrectly. Also, bent diagonals of planes parallel to the sides of the cube start on a different column then in the other two orientations.

                  

Hendricks Perfect (nasik)

Order 8 is the lowest possible for a perfect magic cube and order 9 is the lowest possible for an associated perfect magic cube. See my perfect and perfect2 pages for more information on this class of cube.

All 24 planar squares are pandiagonal magic as are also the 6 oblique squares and the 42 broken (2 segment) oblique squares. The 264 pantriagonals also all sum correctly.

Corners of all orders 2, 3, 4, 5, 6, 7 and 8 also sum correctly to 2052. This is always the case with the lowest order of each nasik perfect hypercube.  By newly adopted terminology, this cube is compactplus i.e. compact_2, 3, 5, which is all possible for this order of cube. See definitions in my glossary.

So 30 order-8 pandiagonal magic squares each have 8 rows, 8 columns and 16 pandiagonals sum correctly as do 256 pantriagonals. Total combinations so far = 960 lines. Corners of 7 orders of cubes = 7 times 256 (counting wrap-around) = 1792 corners. Total sums = 2752 (plus possible other combinations not yet counted, or even discovered).

John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9 

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

This page was originally posted February 2003
It was last updated October 19, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz