# The Early Cubes

Early magic square development took place in Asia, notably in China, India, and the Arab world. The idea of extending this concept to 3 dimensions, however, seems to have originated in the West.
The first occurrence I have found of magic cubes was in France, by Pierre de Fermat (1601 – 1665), followed by Joseph Sauveur (1653 – 1716) and B. Violle (cc1838).

Then the emphasis seems to shift to England, with writings by Rev. A. H. Frost, followed by W. Firth (? – 1889), Dr. C. Planck (?), etc. By 1900 the investigation of magic cubes had progressed to many other countries.

It goes without saying, that this collection of early magic cubes is probably not complete.
I would be very interested in hearing from anyone who can provide additional information on this subject.

 Format 1640 This order 4 cube has no correct triagonals but does contain 8 simple magic squares. Sauveur 1710 All diagonals and triagonals are correct but only 1 row and 1 column of each orthogonal array. 21 arrays sum 1575. Violle order-4 1838 An order 4 cube with 14 of the 18  3 x 4 + 6 oblique arrays summing to 520. Violle order-5 1838 This cube is not magic but all diagonals and triagonals sum correctly. 21 planes = 1575 Violle order-6 1838 This cube has similar characteristics to Violle's order 4. 24 planes = 3906.  (3x 6+6 oblique) Violle order-7 1838 This cube has the same characteristics as Violle's order 5. 27 planes = 8428. (3x7+6) Frankenstein - 8 1875 This magic cube contains 30 simple magic squares. Triagonals are correct. Hugel order-3 1876 This is a disguised index # 1, but earliest record I could find of a magic order 3 cube. Frost order-4 1878 The 4 horizontal planes of this cube are pandiagonal magic squares. Triagonals are incorrect. Barnard order-4 1888 This cube is not magic by present standards, but does possess some unique features. Firth order-6 1889 This cube has the minimum requirements to make it simple magic. First published in 1917? Schubert order-4 1898 A simple, associated magic cube. Schubert order-5 1898 A simple, associated magic cube. Fourrey order-4 1899 Not magic but contains 8 magic squares. Same features as the Fermat cube. Andrews order-4 1908 A simple, associated magic cube. Andrews order-5 1908 This associated magic cube contains 10 pandiagonal and 5 simple magic squares. Worthington order-6 1910 The 6 faces of this simple magic cube are simple magic squares.

### Format 1640

 ``` I II III IV 4 62 63 1 53 11 10 56 60 6 7 57 13 51 50 16 41 23 22 44 32 34 35 29 17 47 46 20 40 26 27 37 21 43 42 24 36 30 31 33 45 19 18 48 28 38 39 25 64 2 3 61 9 55 54 12 8 58 59 5 49 15 14 52 ```This cube is not magic by present day definition because NO triagonals are correct. The 4 horizontal planes and the 4 vertical planes parallel to the sides are simple magic squares. All rows and columns sum correctly for 4 of the 6 oblique squares. This cube was described in a letter to Marin Mersenne dated Apr. 1, 1640 [1]. I obtained this cube from [2] after a tip from Christian Boyer [1] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, originally published 1917 (footnote p. 365). [2] Edauard Lucas, L’Arithmetique amusante (Amusing Arithmetic), Gauthier-Villars, 1895. (French).

### Sauveur 1710

his cube, published by Joseph Sauveur in 1710, is not magic by present day standards. However, it does have some interesting features.

This cube and the method of construction was graciously supplied by Christian Boyer [1] of France, who was successful in locating the original text of Joseph Sauveur in the Mémoires de l'Académie Royale des Sciences of 1710. He was not permitted to photocopy the document because of its age, so the following is prepared from his notes.

Definition by Joseph Sauveur (original French text) of what is a magic cube:

"Un cube magique est un cube divisé en cellules cubiques qui
renfermentchacun un nombre, qui est tel que la somme de tous
les nombres qui sont dans chaque couche carrée parallèle aux
trois bases, ou qui sont dans chacun des six plans ou couches
qui coupent les diagonales des bases opposées, est toujours la même".

So Sauveur means that, for example in this 5th-order cube, the 25 numbers of each square (the 5+5+5 layers
parallel to each side, and the 6 diagonal squares, or 15+6=21 squares) should have the same sum.

This definition was used by most of the magic cube constructors of the 18th and 19th century.

More facts on the Sauveur cube:

The 15 orthogonal planes each have these features:
Only I row and 1 column sums correctly (S = 315).
The total of the 5 rows is 5S.
Both main diagonals sum correctly.
All pandiagonals in one direction sum correctly.

The 6 oblique planes have these features:
The total of the 5 rows in each plane is 5S.
Both diagonals are correct in each plane.
3 of these planes are magic squares, because all rows and columns and 2 main diagonals sum correctly.
2 planes have all 5 rows, and all 5 pandiagonals in one direction, with correct sums.
1 plane has all 5 columns, and all 5 pandiagonals in one direction, with correct sums.

```Sauveur's cube [2]         formed by adding the numbers in these three subsidiary squares
Plane 1 - Top
62    2   32   92  122      50    0   25   75  100    10  0  5  15  20    2  2  2  2  2
114   54    9   44   99     100   50    0   25   75    10  0  5  15  20    4  4  4  4  4
90  105   60   20   50      75  100   50    0   25    10  0  5  15  20    5  5  5  5  5
36   76  106   66   21      25   75  100   50    0    10  0  5  15  20    1  1  1  1  1
13   28   83  118   73       0   25   75  100   50    10  0  5  15  20    3  3  3  3  3
Plane 2
74   14   29   84  119      50    0   25   75  100    20  10  0  5  15    4  4  4  4  4
125   65    5   35   95     100   50    0   25   75    20  10  0  5  15    5  5  5  5  5
96  111   51    6   41      75  100   50    0   25    20  10  0  5  15    1  1  1  1  1
48   88  103   58   18      25   75  100   50    0    20  10  0  5  15    3  3  3  3  3
22   37   77  107   67       0   25   75  100   50    20  10  0  5  15    2  2  2  2  2
Plane 3
70   25   40   80  110      50    0   25   75  100    15  20  10  0  5    5  5  5  5  5
116   71   11   26   81     100   50    0   25   75    15  20  10  0  5    1  1  1  1  1
93  123   63    3   33      75  100   50    0   25    15  20  10  0  5    3  3  3  3  3
42   97  112   52    7      25   75  100   50    0    15  20  10  0  5    2  2  2  2  2
19   49   89  104   59       0   25   75  100   50    15  20  10  0  5    4  4  4  4  4
Plane 4
56   16   46   86  101      50    0   25   75  100    5  15  20  10  0    1  1  1  1  1
108   68   23   38   78     100   50    0   25   75    5  15  20  10  0    3  3  3  3  3
82  117   72   12   27      75  100   50    0   25    5  15  20  10  0    2  2  2  2  2
34   94  124   64    4      25   75  100   50    0    5  15  20  10  0    4  4  4  4  4
10   45  100  115   55       0   25   75  100   50    5  15  20  10  0    5  5  5  5  5
Plane 5 - Bottom
53    8   43   98  113      50    0   25   75  100    0  5  15  20  10    3  3  3  3  3
102   57   17   47   87     100   50    0   25   75    0  5  15  20  10    2  2  2  2  2
79  109   69   24   39      75  100   50    0   25    0  5  15  20  10    4  4  4  4  4
30   85  120   75   15      25   75  100   50    0    0  5  15  20  10    5  5  5  5  5
1   31   91  121   61       0   25   75  100   50    0  5  15  20  10    1  1  1  1  1```
[1] Christian Boyer has recently made some amazing discoveries regarding multimagic squares and cubes.
[2] This from Christian Boyer email of March 16, 2003

### Violle order-4 1838

B. Violle published a monumental book on Magic squares in 1838. From it I have extracted the following cubes of orders 4, 5, 6, and 7. The method for constructing orders 8 and 9 was also presented, but I do not read French, so leave to someone else the pleasure of reconstructing these. The entire book of over 1000 pages is available free on the Web [1].

This cube is not considered magic by present definition. Even though the 4 triagonals are correct, rows, columns and pillars are not.

 ```I II III IV 13 32 48 61 57 44 28 9 53 40 24 5 1 20 36 49 2 19 35 50 54 39 23 6 58 43 27 10 14 31 47 62 3 18 34 51 55 38 22 7 59 42 26 11 15 30 46 63 16 29 45 64 60 41 25 12 56 37 21 8 4 17 33 52``` The total of the 16 cells in each of the 18 square arrays (the 3 x 4 orthogonal and the 6 oblique) sum to 4 x 130 = 520. Both diagonals of these 18 arrays also sum correctly to 130. All 4 triagonals sum correctly to 130. None of the 12 planar squares have any rows or columns that sum to 130! Two of the oblique squares have all rows summing correctly and four have all columns summing correctly. Violle cube 6 has exactly the same features except this one is associated and order 6 is not.

[1] Par B. Violle, Traité complet des Carrés Magiques, 1837, (French). This book is available on the Internet at at http://gallica.bnf.fr.as scanned pages. See my Cube_biblio page for download instructions.

### Violle order-5 1838

This cube is pantriagonal because ALL triagonals sum correctly to 315. However, it is not magic (by today's standards) because rows and columns are incorrect. It is not associated. Also, all diagonals including broken diagonal pairs, of the planar arrays sum correctly.

The 25 numbers in each of the 15 planar squares and 6 oblique squares sum to 1575 (5 times the correct constant of 315). This is a required feature of J. Sauveur's 1710 definition of a magic cube. However, a close inspection of the four Violle cubes, the Sauveur order 5 and the Fourrey order 3 reveal differences in other characteristics.

Only 1 row and 1 column of each planar square is correct. In fact the rows sum to 5 different values starting with 65 and increasing by 125. The columns also sum to 5 different values this time starting at 305 and increasing by 5 (the order of the cube). The 5 planes in each direction have different arrangements of these sums.

Four of the oblique squares have all rows summing correctly and two have all columns summing correctly. One pair of the oblique squares has the columns summing the same as the planar squares.

```I                          II                         III
15   24    8   17    1     88   97   81   95   79     36   50   34   43   27
45   29   38   47   31    118  102  111  125  109     66   55   64   73   57
75   59   68   52   61     23    7   16    5   14     96   85   94   78   87
80   89   98   82   91     28   37   46   35   44    101  115  124  108  117
110  119  103  112  121     58   67   51   65   74      6   20    4   13   22
IV                         V
114  123  107  116  105     62   71   60   69   53
19    3   12   21   10     92   76   90   99   83
49   33   42   26   40    122  106  120  104  113
54   63   72   56   70      2   11   25    9   18
84   93   77   86  100     32   41   30   39   48```
Par B. Violle, Traité complet des Carrés Magiques, 1837 (French).

### Violle order-6 1838

This is not considered magic by present definition. Even though the 4 triagonals are correct, rows, columns and pillars are not.

The total of the 36 cells in each of the 18 square arrays (the 3 x 6 orthogonal and the 6 oblique) sum to 6 x 651 = 3906. Both diagonals of these 24 arrays also sum correctly to 651.

All 4 triagonals sum correctly to 651. None of the 12 planar squares have any rows or columns that sum to 651! Two of the oblique squares have all rows summing correctly and four have all columns summing correctly. This cube has the exact same features as Violle order 4 except this one is not associated.

```I                               II                              III
1   32   33    4   35    6    187  188  207  208  191  210    193  200  201  202  197  198
42   68   70   39   71   37     48   44   64   63   47   61    162  164  166  165  161  157
78  107  106   75  104   73    120  119  136  135  116  133    126  131  130  129  122  121
109  143  142  111  140  114     79   83  100   99   80  102     85   95   94   93   86   90
150  176  177  148  179  145    156  152  171  172  155  169     54   56   57   58   53   49
181  215  213  184  212  186      7   11   27   28    8   30     13   23   21   22   14   18
IV                              V                               VI
19   14   15   16   23   24    205  206  189  190  209  192     31    2    3   34    5   36
168  158  160  159  167  163     66   62   46   45   65   43    180  146  148  177  149  175
132  125  124  123  128  127    102  101   82   81   98   79    108   77   76  105   74  103
91   89   88   87   92   96    133  137  118  117  134  120    139  113  112  141  110  144
60   50   51   52   59   55    174  170  153  154  173  151     72   38   39   70   41   67
199  197  195  196  200  204     25   29    9   10   26   12    211  185  183  214  182  216```
Par B. Violle, Traité complet des Carrés Magiques, 1837 (French).

### Violle order-7 1838

This cube is not considered magic by present definition. Even though the 4 triagonals are correct, rows columns and pillars are not. The total of the 49 cells in each of 27 square arrays sum to 7 x 1204 = 8428.

All pantriagonals sum correctly to 1204 as do all planar diagonals. None of the 21 planar squares have any rows or columns that sum to 1204! In fact the rows sum to 7 different values starting with 1057 and increasing by 49. The columns also sum to 7 different values this time starting at 1183 and increasing by 7 (the order of the cube). The 7 planes in each direction have different arrangements of these sums.

Four of the oblique squares have all rows summing correctly and two have all columns summing correctly. One pair of the oblique squares has the columns summing the same as the planar squares.
All pandiagonals of all planar and oblique squares are correct.
This cube has exactly the same characteristics as the Violle order 5.

Par B. Violle, Traité complet des Carrés Magiques, 1837 (French).

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

### Frankenstein - 8 1875

Gustavus Frankenstein published this cube in The Commercial, a Cincinnati city daily newspaper, March 11, 1875.
The instructions for building this cube later appeared as a lengthy footnote in Barnard's 1888 [1] paper. Because rows, columns, pillars, and triagonals all sum correctly to 2052, this cube is magic by the current definition.

This is the Frankenstein cube in the orientation that he published it. However, he numbered the planes from the bottom up. I reversed the numbering to be consistent with my other cube listings.
Benson and Jacoby [2] reconstructed this cube, presumably from instructions in a footnote of Barnard’s paper. The version they show is a different aspect of this cube i.e. rotated and reflected.

I am grateful to Christian Boyer for locating the original publication that this cube appeared in.
He has posted the complete text of the article here.

Note that Christian refers to it as the first perfect cube. However, by the new (Hendricks) definition it is classified as a diagonal cube because it contains 3m simple magic squares. (A perfect (nasik) cube contains 9m pandiagonal magic squares).

All 24 planar squares and the 6 oblique squares are simple magic. Also, corners of all 256 (including wrap-around) 5x5x5 cubes sum correctly to S.
Benson & Jacoby also refer to this as a perfect cube. They refer to their cube, which is perfect by the new definition, as pandiagonal perfect. (In that cube, all 30 planar squares are pandiagonal magic and all pantriagonals sum correctly [along with other features].)

Cubes with all planar squares simple magic, like this Frankenstein cube, had no name as yet under Hendricks new definitions. For a short time, I called them myers cubes in honor of a similar cube constructed by Richard Myers in 1970 and popularized by M. Gardner [3]. However, Aale de Winkel suggested the name diagonal magic cubes.
I have decided to use this name instead of myers because it is more descriptive (main diagonals of all planar squares sum correctly to S).
If all planar squares were pandiagonal magic, instead of simple, they would be called pandiagonal magic cubes.
And if the cube is pandiagonal magic AND all pantriagonals are also magic the cube is called perfect! A perfect cube has 9m pandiagonal magic squares [4][5]. See this article to understand why nasik is now the preferred term for this type of cube!

[1] F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4, 1888, pp 209-270 (footnote on pp. 244-248).
[2] W. H. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, 0-486-24140-8
[3] Martin Gardner, From his mathematical Games column in Scientific American, Jan. 1976.
[4] B. Rosser and R. J. Walker, A Continuation of The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered 729 – 753.
[5] F. Liao, T. Katayama, K. Takaba, Technical Report 99021, School of Informatics, Kyoto University, 1999.

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

### Hugel order-3 1876

 26  15   1     6  19  17    10   8  24 12   7  23    25  14   3     5  21  16  4  20  18    11   9  22    27  13   2 This is a disguised version of index # 1. It is the earliest record I could find of the order 3 magic cube. All rows, columns, pillars and 4 main diagonals sum to 42. T. Hugel, Das Problem der magishen Systeme, 1876, Verlag von A. H. Gottschick.

### Frost order-4 1878

Horizontal planes of this cube [1] are pandiagonal magic squares. Rows and columns of the other planes are correct, but not the diagonals. All 4 triagonals are incorrect so this cube is not magic by present standards.

It is interesting to note that Frost published two advanced truly magic cubes 12 years earlier [2]. In that paper [1], he published 6 other magic cubes, all with correct triagonals, and one of them a perfect magic cube! See my Frost page which includes all of these cubes.

```I                 II                III               IV
1  56  13  60    44  17  40  29    61  12  49   8    24  45  28  33
30  43  18  39     7  62  11  50    34  23  46  27    59   2  55  14
52   5  64   9    25  36  21  48    16  57   4  53    37  32  41  20
47  26  35  22    54  15  58   3    19  38  31  42    10  51   6  63```
[1] A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.
[2] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103

### Barnard order-4 1888

Dr. F. A. P. Barnard published a lengthy and important paper on magic squares and cube in 1888 [1]. He was 79 years old at the time.

Like Dr. A. H. Frost, he was years ahead of his time on the subject of cubes, displaying an order 8 and two order 11 perfect cubes. He seemed unaware that anyone else had worked on magic cubes, except for G. Frankenstein. For more information go to my Barnard page.

 This order 4 cube by Barnard is not magic by present standards. However, it does have special magic features!Horizontal planes rows only sum correctly to 130 Vertical planes parallel to the front. All are pandiagonal magic squares. Vertical planes parallel to the sides. Pandiagonals and columns are O.K. All pantriagonals in two of the four directions are correct. No pantriagonals in two of the four directions are correct. All 2x2 arrays in the eight vertical planes sum correct. Barnard, like Frost, did not seem to appreciate simple magic cubes. Almost all their energies went into the design of advanced magic cubes. [1] F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4, 1888, pp. 207-270.

### Firth order-6 1889

Constructed before 1889 by W. Firth (he died in 1889 [1, p. 298 footnote].

A simple magic cube. 24 planar arrays have rows and columns and the 4 main triagonals are correct. These are the minimal requirements by the current definition [2]. Oblique arrays; 4 with columns correct, 2 with rows correct. Not associated.

Dr. Planck wrote [1, page 373 footnote], “It was by this method that Firth in the 80’s constructed what was, almost certainly, the first correct magic cube of order 6.”

```Top                              II                             III
2    8  134  129  186  192      5    3  132  135  189  187    117  114  146  152   62   60
6    4  130  133  190  188      1    7  136  131  185  191    118  113  150  148   64   58
182  178   21   24  121  125    180  184   18   19  127  123     54   50  109  106  168  164
177  181   22   23  126  122    183  179   17   20  124  128     52   56  110  105  162  166
144  138  174  169   16   10    139  141  172  175   11   13    154  160   70   68   97  102
140  142  170  173   12   14    143  137  176  171   15    9    156  158   66   72   98  101
IV                               V                               VI
120  115  149  147   63   57    206  204   42   45   78   76    201  207   48   43   73   79
119  116  145  151   61   59    202  208   46   41   74   80    205  203   44   47   77   75
51   55  112  107  161  165     89   93  198  199   38   34     95   91  193  196   36   40
53   49  111  108  167  163     94   90  197  200   33   37     92   96  194  195   39   35
155  157   65   71  100  103     28   30   82   85  212  214     31   25   88   83  215  209
153  159   69   67   99  104     32   26   86   81  216  210     27   29   84   87  211  213```

[1] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, original 1917.
Pages 298 & 305. See Chapter XII  The ‘Theory of Reversions  pp. 295-320 (written by Dr. C. Planck).
[2] W. S. Andrews, Magic Squares & Cubes, Open Court, 1908, page 64.

### Schubert order-4 1898

This cube is magic but has no special features except it is associated. It is one of the first magic cubes by the current definition to be published.
The 2 members of each of the 12 planar diagonal pairs sums to the same value.

```I                  II                 III                IV
1  48  32  49     63  18  34  15     62  19  35  14      4  45  29  52
60  21  37  12      6  43  27  54      7  42  26  55     57  24  40   9
56  25  41   8     10  39  23  58     11  38  22  59     53  28  44   5
13  36  20  61     51  30  46   3     50  31  47   2     16  33  17  64```
Hermann Schubert, Mathematical Recreations and Essays, Open Court 1899 page 62.

### Schubert order-5 1898

The center plane in each dimension is magic (a feature of associated magic cubes).
The horizontal planes and vertical planes parallel to the front, have all diagonals in one direction correct.

The oblique squares: 2 are simple magic 2 have rows correct and 2 have columns correct. 1 oblique square has all pandiagonals in both directions correct 4 have all pandiagonals in one direction correct.
All the pantriagonals in 2 of the 4 directions sum correctly.

```Top                         II                        III
121   27   83   14   70      2   58  114   45   96     33   89   20   71  102
10   61  117   48   79     36   92   23   54  110     67  123   29   85   11
44  100    1   57  113     75  101   32   88   19     76    7   63  119   50
53  109   40   91   22     84   15   66  122   28    115   41   97    3   59
87   18   74  105   31    118   49   80    6   62     24   55  106   37   93
IV                          V
64  120   46   77    8     95   21   52  108   39
98    4   60  111   42    104   35   86   17   73
107   38   94   25   51     13   69  125   26   82
16   72  103   34   90     47   78    9   65  116
30   81   12   68  124     56  112   43   99    5```
Hermann Schubert, Mathematical Recreations and Essays, Open Court 1899 page 62.

### Fourrey order-4 1899

This cube has 8 simple magic squares, the 4 horizontal and the 4 parallel to the sides. However it is not magic by present standards because, although all orthogonal lines sum correctly, none of the 4 triagonals are do. Features are exactly the same as the Fermat cube.

```I                  II                 III                IV
4  62  63   1     57   7   6  60     32  34  35  29     37  27  26  40
41  23  22  44     56  10  11  53     17  47  46  20     16  50  51  13
21  43  42  24     12  54  55   9     45  19  18  48     52  14  15  49
64   2   3  61      5  59  58   8     36  30  31  33     25  39  38  28```

E. Fourrey Recreations arithmetiques  (Arithmetical Recreations) 8th edition Vuibert 2001.

### Andrews order-4 1908

This is a simple magic cube, having the basic characteristics only.
The 2 members of each of the 12 planar diagonal pairs sums to the same value.

```I                  II                 III                IV
1  63  60   6     48  18  21  43     32  34  37  27     49  15  12  54
62   4   7  57     19  45  42  24     35  29  26  40     14  52  55   9
56  10  13  51     25  39  36  30     41  23  20  46      8  58  61   3
11  53  50  16     38  28  31  33     22  44  47  17     59   5   2  64```
W. S. Andrews, Magic Squares & Cubes, Open Court, 1908.
W. S. Andrews  Magic Squares and Cubes  2nd edition  Dover Publ. 1960 p. 80 (First published in 1917)

### Andrews order-5 1908

This is an associated magic cube. It contains 5 pandiagonal magic squares (the 5 horizontal). As well, the other 2 central orthogonal squares and 4 of the 6 oblique squares are simple magic. The 3 central orthogonal squares and the 4 oblique magic squares are associated.

The cube is not pantriagonal magic because all the triagonals in only 1 of the 4 directions is correct.

```Top                        II                         III
67   98  104   10   36    106   12   43   74   80     50   51   82  113   19
110   11   42   73   79     49   55   81  112   18     88  119   25   26   57
48   54   85  111   17     87  118   24   30   56      1   32   63   94  125
86  117   23   29   60      5   31   62   93  124     69  100  101    7   38
4   35   61   92  123     68   99  105    6   37    107   13   44   75   76
IV                          V
89  120   21   27   58      3   34   65   91  122
2   33   64   95  121     66   97  103    9   40
70   96  102    8   39    109   15   41   72   78
108   14   45   71   77     47   53   84  115   16
46   52   83  114   20     90  116   22   28   59```

W. S. Andrews  Magic Squares & Cubes  Open Court  1908
W. S. Andrews  Magic Squares and Cubes  2nd edition  Dover Publ. 1960 p. 73 (First published in 1917)

### Worthington order-6 1910

All 6 outside planes are simple magic squares [1]. No other planar squares have any correct diagonals.

According to Leeflang [2] this was the first even order magic cube (by modern definition) to be published. Not so.
The Firth order 6 cube was actually constructed before 1890.  Herman Schubert published orders 4 and 5 magic cubes in 1898.
Late note. Frost published an order 8 pandiagonal magic cube in 1866, and an order 4 pantriagonal magic cube in 1878 (along with a number of
other magic cubes, including an order 9 perfect cube.). Actually, others that are not already shown on this page are; Huber 0rder 4 1891, Planck
order 10 1894 and order 8 perfect in 1905. All 6 outside planes are simple magic squares [1]. No other planar squares have any correct diagonals.

```Top                              II                             III
106    8    7  212  209  109    166  130  129   32   30  164    163  135  136   25   27  165
199  116  113   16   12  195     37  152  148  137  143   34     36  145  149  144  138   39
196  114  115   11   15  200     33  151  150  142  140   35     40  146  147  139  141   38
21  203  202  103  100   22    128   41   47  157  154  124    121   48   42  156  159  125
17  205  208   99  104   18    126   46   44  155  153  127    123   43   45  158  160  122
112    5    6  210  211  107    161  131  133   28   31  167    168  134  132   29   26  162
IV                               V                               VI
55  192  191   83   81   49     50  185  186   86   88   56    111    1    2  213  216  108
93   60   57  176  174   91     92   61   64  169  171   94    194  117  120    9   13  198
89   62   63  172  175   90     96   59   58  173  170   95    197  119  118   14   10  193
182   74   77   70   65  183    179   79   76   67   72  178     20  206  207   98  101   19
180   75   73   68   71  184    181   78   80   69   66  177     24  204  201  102   97   23
52  188  190   82   85   54     53  189  187   87   84   51    105    4    3  215  214  110```
[1] W. S. Andrews  Magic Squares & Cubes  1960 (1917)  p 202 .
[2] John Worthington, A Magic Cube of Six, The Monist, 20, 1910, pp 303-309

 This page was originally posted December 2002 It was last updated February 11, 2013 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz