Magic Cubes - Order 7

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With order 7 we have a new class of magic cube, the pandiagonal cube.
Order 3: Simple, associated only
Order 4: Simple not associated, simple associated, pantriagonal not associated
Order 5: Simple not associated, simple associated, pantriagonal not associated, pantriagonal associated, diagonal
Order 6: Simple not associated, simple associated, pantriagonal not associated, (NO pantriagonal cubes associated?)
Order 7: Simple not associated, simple associated, pantriagonal not associated, pantriagonal associated, diagonal associated, pandiagonal associated , pandiagonal not associated (NO diagonal cubes not associated?)

Also, more numbers and more planes, make possible a larger variety of minor features within the main classifications.
On this page I present several of each of the four main classifications. All have different minor features.

Weidemann Simple

1922 Not associated. Simple magic with 19 magic squares.

Soni Simple

2001 Associated. Simple magic with 19 magic squares.

Hendricks Pantriagonal

1973 Associated. 3 simple magic squares.

Hendricks Pantriagonal-2

1991 Associated. 3 simple and 3 pandiagonal magic squares.

Poyo Pantriagonal

? Not associated. 1 simple magic square.

Trump Diagonal

2003 Associated.  21 orthogonal and 6 oblique simple magic squares.

Wynne Pandiagonal

1975 Associated. 21 pandiagonal and 6 simple  magic squares.

Iriyama Pandiagonal

2001 Associated. 22 pandiagonal and 5 simple magic squares.

de Winkel Pandiagonal

2003 Not Associated. 21 pandiagonal and 6 simple  magic squares.

Weidemann Simple

This is a simple magic cube. It is not associated. The 7 horizontal planes and the 7 vertical planes parallel to the sides of the cube are pandiagonal magic squares. In addition, 1 oblique plane is pandiagonal magic and 4 are simple magic. All pantriagonals in 2 of the 4 directions are correct.

Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922

Horizontal plane I -                  II
326   89  195  252    8  114  220     122  228  334   97  154  253   16
 43  100  206  312   75  181  287     189  288    2  108  214  320   83
 61  167  273   29  135  241  298     200  306   69  175  274   37  143
121  227  333   96  153  259   15     260   23  129  235  341   55  161
188  294    1  107  213  319   82     327   90  196  246    9  115  221
199  305   68  174  280   36  142      44  101  207  313   76  182  281
266   22  128  234  340   54  160      62  168  267   30  136  242  299
III                                   IV
261   24  130  236  342   56  155      57  163  269   32  138  244  301
328   91  190  247   10  116  222     124  230  336   92  149  255   18
 45  102  208  314   77  176  282     184  290    4  110  216  322   78
 63  162  268   31  137  243  300     202  308   64  170  276   39  145
123  229  335   98  148  254   17     262   25  131  237  343   50  156
183  289    3  109  215  321   84     329   85  191  248   11  117  223
201  307   70  169  275   38  144      46  103  209  315   71  177  283
V                                     VI
203  302   65  171  277   40  146      48  105  204  310   73  179  285
263   26  132  238  337   51  157      59  165  271   34  140  239  296
323   86  192  249   12  118  224     126  225  331   94  151  257   20
 47  104  210  309   72  178  284     186  292    6  112  211  317   80
 58  164  270   33  139  245  295     197  303   66  172  278   41  147
125  231  330   93  150  256   19     264   27  133  232  338   52  158
185  291    5  111  217  316   79     324   87  193  250   13  119  218
VII - Bottom 
187  293    7  106  212  318   81
198  304   67  173  279   42  141
265   28  127  233  339   53  159
325   88  194  251   14  113  219
 49   99  205  311   74  180  286
 60  166  272   35  134  240  297
120  226  332   95  152  258   21

Soni Simple

This simple magic cube is associated. The 7 horizontal planes and the 7 vertical planes parallel to the front are pandiagonal magic squares.. The central vertical plane parallel to the sides and 4 of the 6 oblique planes are simple magic squares. The vertical planes parallel to the sides of the cube have no correct pandiagonals with the exception of the central plane which has the 2 main diagonal correct. Only 1 of the 4 sets of pantriagonals are correct.

Abhinav Soni  HyperMagicCube.exe program.   Obtainable from his geocities magic cubes page. (Sorry. No longer available)

Horizontal plane I - Top                                 II
 71  243   16  188  304  133  249     332  112  277   50  222   44  167
185  308  130  246   75  240   20      54  219   48  164  336  109  274
250   72  244   17  189  305  127     168  333  106  278   51  223   45
 21  186  302  131  247   76  241     275   55  220   49  165  330  110
128  251   73  245   18  183  306      46  162  334  107  279   52  224
242   15  187  303  132  248   77     111  276   56  221   43  166  331
307  129  252   74  239   19  184     218   47  163  335  108  280  53
III                                   IV
201   23  195  311  140  256   78     119  284   57  229    2  174  339
315  137  253   82  198   27  192     226    6  171  343  116  281   61
 79  202   24  196  312  134  257     340  113  285   58  230    3  175
193  309  138  254   83  199   28      62  227    7  172  337  117  282
258   80  203   25  190  313  135     169  341  114  286   59  231    4
 22  194  310  139  255   84  200     283   63  228    1  173  338  118
136  259   81  197   26  191  314       5  170  342  115  287   60  225
V                                     VI
 30  153  318  147  263   85  208     291   64  236    9  181  297  126
144  260   89  205   34  150  322      13  178  301  123  288   68  233
209   31  154  319  141  264   86     120  292   65  237   10  182  298
316  145  261   90  206   35  151     234   14  179  295  124  289   69
 87  210   32  148  320  142  265     299  121  293   66  238   11  176
152  317  146  262   91  207   29      70  235    8  180  296  125  290
266   88  204   33  149  321  143     177  300  122  294   67  232   12
VII 
160  325  105  270   92  215   37
267   96  212   41  157  329  102
 38  161  326   99  271   93  216
103  268   97  213   42  158  323
217   39  155  327  100  272   94
324  104  269   98  214   36  159
 95  211   40  156  328  101  273

Hendricks Pantriagonal

All 4 sets of pantriagonals (and therefore all oblique pandiagonals) are correct, which makes this a pantriagonal magic cube.
There are no other features except that it is associated, so the 3 central orthogonal planes are simple magic squares. (Compare with features of the next pantriagonal cube.)

J. R. Hendricks, Magic Cubes of Odd Order, JRM 6:4, 1973, pp 268-272 and Magic Square Course, 1991,  p. 366

Plane I - Top                         II
 98  134  177  220  263  306    6     104  196  232  275  318   18   61
139  182  218  261  304    4   96     194  237  280  316   16   59  102
180  223  266  302    2   94  137     235  278  321   21   57  100  192
221  264  307    7   92  135  178     276  319   19   62  105  190  233
262  305    5   97  140  176  219     317   17   60  103  195  238  274
303    3   95  138  181  224  260      15   58  101  193  236  279  322
  1   93  136  179  222  265  308      63   99  191  234  277  320   20
III                                   IV
159  202  294  330   30   73  116     214  257  300   49   85  128  171
200  292  335   35   71  114  157     255  298   47   90  133  169  212
290  333   33   76  119  155  198     296   45   88  131  174  217  253
331   31   74  117  160  203  288      43   86  129  172  215  258  301
 29   72  115  158  201  293  336      91  127  170  213  256  299   48
 77  113  156  199  291  334   34     132  175  211  254  297   46   89
118  161  197  289  332   32   75     173  216  259  295   44   87  130
V                                     VI
269  312   12   55  147  183  226     324   24   67  110  153  245  281
310   10   53  145  188  231  267      22   65  108  151  243  286  329
  8   51  143  186  229  272  315      70  106  149  241  284  327   27
 56  141  184  227  270  313   13     111  154  239  282  325   25   68
146  189  225  268  311   11   54     152  244  287  323   23   66  109
187  230  273  309    9   52  144     242  285  328   28   64  107  150
228  271  314   14   50  142  185     283  326   26   69  112  148  240
VII
 36   79  122  165  208  251  343 
 84  120  163  206  249  341   41 
125  168  204  247  339   39   82 
166  209  252  337   37   80  123 
207  250  342   42   78  121  164 
248  340   40   83  126  162  205 
338   38   81  124  167  210  246 

Hendricks Pantriagonal-2

This pantriagonal magic cube is associated, so the 3 central orthogonal planes are simple magic squares. In addition, 3 of the 6 oblique arrays are pandiagonal magic squares because all rows and columns sum correctly.
The pandiagonals of all 6 oblique arrays sum correctly because they are the pantriagonals of the cube.
All the pandiagonals in one direction are correct in each of the 21 planar arrays.

J. R. Hendricks,  Magic Square Course, 1991,  p. 367-369

Plane I - Top                         II
236   14  177  298  125  288   66     302  129  250   77  240   18  188
 72  242   20  183  304  131  252     194  315  135  256   83  197   24
258   78  199   26  196  310  137      30  151  321  141  262   89  210
143  264   91  205   32  153  316     216   36  157  327  105  268   95
329  100  270   97  211   38  159      52  222   49  163  333  111  274
165  335  106  276   54  224   44     287   58  228    6  169  339  117
  1  171  341  119  282   60  230     123  293   64  234   12  182  296
III                                   IV
 81  202   22  192  313  140  254     154  317  144  265   85  206   33
260   87  208   35  149  319  146      39  160  323  101  271   98  212
103  273   93  214   41  155  325     218   45  166  336  107  277   55
331  109  279   50  220   47  168      61  231    2  172  342  113  283
174  337  115  285   63  226    4     289   67  237    8  178  299  126
 10  180  301  121  291   69  232     132  246   73  243   21  184  305
245   16  186  307  127  248   75     311  138  259   79  200   27  190
V                                     VI
269   96  217   37  158  328   99      48  162  332  110  280   51  221
112  275   53  223   43  164  334     227    5  175  338  116  286   57
340  118  281   59  229    7  170      70  233   11  181  295  122  292
176  297  124  294   65  235   13     249   76  239   17  187  308  128
 19  189  303  130  251   71  241     134  255   82  203   23  193  314
198   25  195  309  136  257   84     320  147  261   88  209   29  150
 90  204   31  152  322  142  263     156  326  104  267   94  215   42
VII
114  284   62  225    3  173  343
300  120  290   68  238    9  179
185  306  133  247   74  244   15
 28  191  312  139  253   80  201
207   34  148  318  145  266   86
 92  213   40  161  324  102  272
278   56  219   46  167  330  108

Poyo Pantriagonal

This pantriagonal magic cube is not associated 1 planar square (the right face) is simple magic.
Only 1 of the 40 other planar main diagonals are correct. Only 16 of the 252 broken diagonals are correct.
Because it is a pantriagonal magic cube, the 49 triagonals in each of four directions all sum correctly to 1204.

F. Poyo  Pantriagonal   (from his Web site which is no longer available)

Plane I - Top                         II
278  335   49   50  107  164  221     333   47   55  112  162  219  276
321   35   85  142  150  207  264      33   90  147  148  205  262  319
 21   71  128  185  242  250  307      76  133  183  240  248  305   19
 57  114  171  228  285  342    7     119  169  226  283  340    5   62
100  157  214  271  328   42   92     155  212  269  326   40   97  105
192  200  257  314   28   78  135     198  255  312   26   83  140  190
235  292  300   14   64  121  178     290  298   12   69  126  176  233
III                                   IV
 45   53  110  167  224  274  331      51  108  165  222  279  336   43
 88  145  153  210  260  317   31     143  151  208  265  322   29   86
131  188  245  246  303   17   74     186  243  251  308   15   72  129
174  231  281  338    3   60  117     229  286  343    1   58  115  172
217  267  324   38   95  103  160     272  329   36   93  101  158  215
253  310   24   81  138  195  203     315   22   79  136  193  201  258
296   10   67  124  181  238  288       8   65  122  179  236  293  301
V                                     VI
106  163  220  277  334   48   56     168  218  275  332   46   54  111
149  206  263  320   34   91  141     204  261  318   32   89  146  154
241  249  306   20   77  127  184     247  304   18   75  132  189  239
284  341    6   63  113  170  227     339    4   61  118  175  225  282
327   41   98   99  156  213  270      39   96  104  161  211  268  325
 27   84  134  191  199  256  313      82  139  196  197  254  311   25
 70  120  177  234  291  299   13     125  182  232  289  297   11   68
VII
223  280  330   44   52  109  166
266  316   30   87  144  152  209
302   16   73  130  187  244  252
  2   59  116  173  230  287  337
 94  102  159  216  273  323   37
137  194  202  259  309   23   80
180  237  294  295    9   66  123

Trump Diagonal

Walter Trump's order 7 diagonal magic cube is called that because all planar diagonals sum correctly to S. It is not associated.
It contains 21 + 6 simple magic squares and uses the consecutive numbers 0 - 342.

From an email on Sept. 3, 2003 

I - Top                               II 
289  125   11  232   68  297  175     182   18  247  132  304  239   75
206   84  320  149   34  263  141      99  327  213   91  270  156   41
 55  276  162   47  219  105  333     340  169   62  283  112    5  226
 14  242  128  307  185   71  250     257  135   21  200   78  314  192
323  159   94  266  102   37  216     223   52  330  166   44  273  109
172    1  286  115  336  229   58      65  293  179    8  236  122  294
138  310  196   81  253  195   24      31  203  145  317  153   88  260
III                                   IV
211   96  325  154   39  268  104     111  332  218   54  275  161   46
177   13  291  120  299  234   63      70  249  184   20  241  127  306
 26  198  133  312  190   83  255     262  140   33  205   90  319  148
335  164   50  278  107   42  221     228   57  342  171    0  285  114
245  130   16  244   73  302  187     194   23  252  137  309  202   80
143  315  208   86  265  151   29      36  215  101  322  158   93  272
 60  281  174    3  224  117  338     296  181   67  288  124   10  231
V                                     VI
  4  225  118  339  168   61  282      82  254  189   25  197  139  311
313  191   77  256  134   27  199      48  220  106  334  163   49  277
155   40  269   98  326  212   97     233   69  298  176   12  290  119
121  300  235   64  292  178    7     150   28  264  142  321  207   85
 87  259  152   30  209  144  316     116  337  230   59  280  173    2
279  108   43  222   51  329  165     301  186   72  251  129   15  243
238   74  303  188   17  246  131     267  103   38  210   95  324  160
VII - Bottom
318  147   89  261  146   32  204
284  113    6  227   56  341  170
126  305  240   76  248  183   19
 92  271  157   35  214  100  328
  9  237  123  295  180   66  287
201   79  308  193   22  258  136
167   45  274  110  331  217   53

Wynne Pandiagonal

This is a pandiagonal magic cube because all 21 planar squares are pandiagonal magic. The 6 oblique squares are simple magic. All oblique squares are associated but only the3 center planar squares are. This cube is associated.

This cube has the same features as the Frost cube of 1878 and Langman cube of 1962. The Frost pandiagonal cube of 1866 and the Leeflang cube of 1978 were slightly better with all pantriagonals in two orientations correct (instead of just one). Also, one of the oblique planes of the Frost cube was pandiagonal magic (instead of simple magic).

Bayard E. Wynne, Perfect Magic Cubes of Order Seven, JRM 8:4, 1975-76, pp 285-293
NOTE that this title is another ambiguous use of the term perfect.
A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-102
A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.

Top                                   II
326   42   94  104  156  215  267     114  173  225  284  343    3   62
 52  111  163  222  274  333   49     232  291  301   10   69  121  180
170  229  281  340    7   59  118     308   17   76  128  187  239  249
288  298   14   66  125  177  236      83  135  194  197  256  315   24
 21   73  132  184  243  246  305     152  204  263  322   31   90  142
139  191  201  253  312   28   80     270  329   38   97  100  159  211
208  260  319   35   87  146  149      45   55  107  166  218  277  336
III                                   IV
252  304   20   72  131  183  242      89  141  151  210  262  321   30
 27   79  138  190  200  259  311     158  217  269  328   37   96   99
145  148  207  266  318   34   86     276  335   44   54  106  165  224
214  273  325   41   93  103  155       2   61  113  172  231  283  342
332   48   51  110  162  221  280     120  179  238  290  300    9   68
 58  117  169  228  287  339    6     245  248  307   16   75  127  186
176  235  294  297   13   65  124     314   23   82  134  193  203  255
V                                     VI
220  279  331   47   50  109  168       8   67  126  178  237  289  299
338    5   57  116  175  227  286     133  185  244  247  306   15   74
 64  123  182  234  293  296   12     202  254  313   22   81  140  192
189  241  251  303   19   71  130     320   29   88  147  150  209  261
258  310   26   78  137  196  199      95  105  157  216  268  327   36
 33   85  144  154  206  265  317     164  223  275  334   43   53  112
102  161  213  272  324   40   92     282  341    1   60  119  171  230
VII
195  198  257  309   25   84  136
264  316   32   91  143  153  205
 39   98  101  160  212  271  323
108  167  219  278  330   46   56
226  285  337    4   63  115  174
295   11   70  122  181  233  292
 77  129  188  240  250  302   18

Iriyama Pandiagonal

This associated pandiagonal magic cube was obtained from Mutsumi Suzukiís Web site in 2002.

All planar squares are pandiagonal, also 1 oblique square. The other 5 oblique squares are simple magic, with 4 of these having all the pandiagonals in one direction correct. All pantriagonals in 1 of the 4 directions are correct.
Point of interest. A. H. Frost published a pandiagonal magic cube with these features in 1866.

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

Top                                   II
229  248  316   48   67  135  161     113  188  207  275  301   26   94
 41   60  128  154  222  290  309     268  343   19   87  106  181  200
196  215  283  302   34   53  121      80   99  174  242  261  336   12
295   27   95  114  189  208  276     235  254  329    5   73  141  167
107  182  201  269  337   20   88      47   66  134  160  228  247  322
262  330   13   81  100  175  243     153  221  289  315   40   59  127
 74  142  168  236  255  323    6     308   33   52  120  195  214  282
III                                   IV
  4   72  147  166  234  253  328     287  306   31   50  125  193  212
159  227  246  321   46   65  140      92  118  186  205  280  299   24
314   39   58  133  152  220  288     198  273  341   17   85  111  179
126  194  213  281  307   32   51      10   78  104  172  240  266  334
274  300   25   93  119  187  206     165  233  259  327    3   71  146
 86  112  180  199  267  342   18     320   45   64  139  158  226  252
241  260  335   11   79  105  173     132  151  219  294  313   38   57
V                                     VI
171  239  265  333    9   84  103      62  130  149  224  292  311   36
326    2   77  145  164  232  258     217  285  304   29   55  123  191
138  157  225  251  319   44   70      22   97  116  184  210  278  297
293  312   37   63  131  150  218     177  203  271  339   15   90  109
 56  124  192  211  286  305   30     332    8   83  102  170  245  264
204  279  298   23   98  117  185     144  163  238  257  325    1   76
 16   91  110  178  197  272  340     250  318   43   69  137  156  231
VII
338   21   89  108  176  202  270
101  169  244  263  331   14   82
256  324    7   75  143  162  237 
 68  136  155  230  249  317   49
223  291  310   42   61  129  148
 35   54  122  190  216  284  303
183  209  277  296   28   96  115

de Winkel Pandiagonal

Aale de Winkel's order 7 pandiagonal magic cube is also pandiagonal because all 3m orthogonal planes are pandiagonal magic squares. It is NOT associated.

It contains 21 pandiagonal and 6 simple magic squares and uses the consecutive numbers 0 - 342.

From an email on Sept. 3, 2003 

I - Top                               II
 21   57  114  202  179  285  339     230  186  292  318    7   64  100
117  199  175  281  338   27   60     288  317   13   67  103  227  182
181  284  341   24   56  113  198      10   63   99  226  188  291  320
337   23   62  116  201  178  280     102  229  185  287  316    9   69
 59  112  197  177  286  340   26     184  293  319   12   66   98  225
200  180  283  336   22   58  118     315    8   65  104  228  187  290
282  342   25   61  115  196  176      68  101  224  183  289  321   11
III                                   IV 
304   14   50  128  237  193  271     135  244  172  257  311    0   78
 53  131  234  189  267  303   20     168  253  310    6   81  138  241
233  195  270  306   17   49  127     313    3   77  134  240  174  256
266  302   16   55  130  236  192      83  137  243  171  252  309    2
 19   52  126  232  191  272  305     239  170  258  312    5   80  133
132  235  194  269  301   15   51     255  308    1   79  139  242  173
190  268  307   18   54  129  231       4   82  136  238  169  254  314
V                                     VI
264  297   28   85  142  223  158      92  121  209  165  250  325   35
 34   88  145  220  154  260  296     206  161  246  324   41   95  124
141  219  160  263  299   31   84     249  327   38   91  120  205  167
157  259  295   30   90  144  222      37   97  123  208  164  245  323
298   33   87  140  218  156  265     119  204  163  251  326   40   94
 86  146  221  159  262  294   29     166  248  322   36   93  125  207
217  155  261  300   32   89  143     328   39   96  122  203  162  247
VII - Bottom 
151  278  332   42   71  107  216
331   48   74  110  213  147  274
 70  106  212  153  277  334   45
215  150  273  330   44   76  109
279  333   47   73  105  211  149
 43   72  111  214  152  276  329
108  210  148  275  335   46   75

This page was originally posted February 2003
It was last updated October 16, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz