A. H. Frost's Magic Cubes

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Quite late in my research on magic cubes, I happily came into possession of copies of Rev. A. H. Frost's early papers on magic cubes. These papers quickly upset many of my preconceptions regarding magic cubes. The earliest previous publication of cubes (that I am aware of) that were magic by present standards (rows, columns, pillars, and triagonals all correct) was an order 3 cube by Hugel in 1876 and order 4 and 5 cubes published in 1899 by Schubert. The first publication of a perfect magic cube was by Ian Howard in a JRM paper in 1976 (he gave instructions on building an order 11).
ADDENDUM: F. Barnard published a lengthy paper in 1888 which included 3 perfect magic cubes and mentioned the Frankenstein cube (not perfect) of 1875. Arnoux constructed an order 17 perfect cube in 1887, but his paper was not rediscovered until 2003! [6] [7]

A well-researched and written Biography of Dr. Frost by Christian Boyer is on his Multimagic pages.

On this page I will present all of Frost’s cubes. They will appear in the same order and the same orientation, exactly as they are in his papers.
Following are the references to the papers I have been referring to. There must be a preceding paper, which I do not have. In [3] page 49, Frost says “By the method adopted in No. XXV, 1865, of this journal…” , and again in [4] page 118, he says “The method employed for squares of the form 4n in No. 25, 1865, of this Journal…”. That paper presumably deals with nasik magic squares, but I have not been able to locate it.

About the term Nasik
Dr. Frost coined the term Nasik for a magic square in which both diagonals and all parallel diagonal pairs sum to the magic constant. This type of magic square would later be referred to as Pandiagonal or Perfect. In fact it is a Perfect Magic Hypercube of dimension 2. Later, Dr. C. Planck would extend the definition of Nasik to Perfect Magic Hypercubes of any dimension. [8]

The term Nasik is derived from the town Nassick, in Western India , where Frost was stationed

[1] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103
He describes a method of constructing magic cubes and shows an order 7 pandiagonal and an order 8 pantriagonal magic cube. Frost was presumably still in India at this time, because the paper was submitted by his brother, Rev. Percival Frost

[2] A. H. Frost, Supplementary Note on Magic Cubes. Quarterly Journal of Mathematics, 8, 1867, p 74
A short description of a coordinated set of 7 cubes, details and illustrations of which were published in [4] and [5]

[3] A. H. Frost, On the General Properties of Nasik Squares, QJM 15, 1878, pp 34-49
Construction of pandiagonal magic squares. This is the origin of the term nasik, for what later become popularly known as pandiagonal magic squares.

[4] A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.
He shows two order 3 and order 4 cubes, and one each of orders 7 and 9, with method of construction. These cubes (in order) are not magic, disguised order 3, not magic, pantriagonal, pantriagonal and perfect.
He briefly discusses order 5 but shows no complete order 5 cube.

[5] A. H. Frost, Description of Plates 3 to 9, QJM 15, 1878, pp 366-368 plus plates 3 to 9.
Illustrations of a group of 7 interrelated order 7 cubes.
All the cubes shown on this Web page appear in either [1] or [4].

[6] F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4,1888, pp. 207-270.
A lengthy footnote describes the construction of G. Frankenstein's diagonal type order 8 cube of 1875.

[7] See my pages on Arnoux cubes and patterns.

[8] The Theory of Path Nasiks, by C. Planck, M.A., M.R.C.S., Private printing, 1905. See a length quote here.

The Magic Cubes

Order-7 Pandiagonal -- 1866 22 pandiagonal and 5 simple magic squares
Order-8 Pantriagonal -- 1866 no magic squares. Corners of all order 3, 5, and 7 cubes sum correctly
Order-3 not magic     -- 1878 rows, columns, and pillars o.k. but only 2 triagonals are correct
Order-3 Associated   -- 1878 disguised version of No 2 basic cube
Order-4 not magic     -- 1878 4 pandiagonal magic squares. Triagonals are incorrect
Order-4 Pantriagonal -- 1878 no special features except compact
Order-7 Pandiagonal  -- 1878 21 pandiagonal and 6 simple magic squares
Order-9 Perfect (nasik) -- 1878 33 pandiagonal and 6 simple magic squares
A presentation to Rev.  Frost Picture of a silver cup he received from native friends in India

Order-7 Pandiagonal --1866

This pandiagonal associated magic cube contains 22 pandiagonal and 5 simple magic squares.
It is almost certainly the first magic cube of modern definition to be published. To be as large as order 7 and as feature rich is astounding! Until W. S. Andrews defined the minimum requirements for a magic cube 42 years later, many such cubes did not have the triagonals (space diagonals) summing correctly. In fact, many did not even have the rows, columns, or pillars summing correctly, but were concerned only with the total for each complete planar array.

This is a pandiagonal magic cube, so all 21 planar squares are pandiagonal magic. In addition 1 oblique square is also pandiagonal magic and the other 5 are simple magic. Four of the 5 simple magic squares have all pandiagonals in one direction summing correctly to 1204.

It is interesting that Frost calls this type of cube Nasik , which we all know is another term (coined by him) for pandiagonal magic squares. John Hendricks was unaware of Frost’s nasik cubes when he named this type of magic cube pandiagonal.

Frost also referred to the order 8 cube (following) in this first paper, as nasik.
The order 7 cube is nasik (pandiagonal) magic because all the planar squares are nasik magic. We now call Frost’s second cube (the order 8), pantriagonal magic because all pantriagonals sum correctly. Frost presumably classed it with the order 7, because all pandiagonals of the 6 oblique squares sum correctly. In a pandiagonal cube, it seems these 6 oblique squares will always be magic, in some combination of pandiagonal and simple. In a pantriagonal magic cube, it seems these 6 oblique squares are never magic, because rows and columns do not sum correctly!

As mentioned above, all 4 triagonals sum correct. All lines parallel to 2 of these triagonals (the broken triagonals) are correct for 2 of the 4 directions.

[1] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, page 100

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

Order-8 Pantriagonal --1866

This is the second of two cubes Frost presented in his paper on the Invention of Magic Cubes. The first cube (shown above) he showed was an order 7 pandiagonal magic cube. These two cubes are in QJM 7, 1866, pp 100, 101

This order 8 cube is called a pantriagonal magic cube because all pantriagonals sum correctly. This type of cube is possible starting with order 4! Frost also called it a Nasik (pandiagonal) magic cube, probably because the diagonals of the oblique squares are correct.

Pantriagonal magic cubes have a feature similar to pandiagonal magic squares:

  • Pandiagonal magic square: Any line (i.e. row or column) may be moved from one side of the square to the other without losing the magic feature.
  • Pantriagonal magic cube: Any plane may be moved from one side of the cube to the other without losing the magic feature.

Note that this particular pantriagonal magic cube has a bonus feature. All 8 corner cells of all cubical arrays within this cube that have sides of 3, 5 or 7 sum correctly to the constant. This includes wrapping around from 1 edge to the opposite edge. Corners of many other parrallelopiped arrays also sum correctly. This feature is fairly common with order 8 magic cubes and for perfect cubes of this order, it seems that all orders of the sub-cubes sum correctly.

This cube contains no magic squares. The 3 x 8 planar squares have all rows and columns correct. The diagonals of these squares are incorrect, but the 2 diagonals of each square are equal value and in 2 of the 3 cases form a pattern.

Diagonal sums of each planar square

Horizontal

Vertical
parallel to sides

Vertical
parallel to front

1796

1540

3028

2308

2564

1044

2308

2564

1012

1796

1540

3124

2308

2564

1076

1796

1540

3060

1796

1540

3092

2308

2564

980

This cube, as well as being pantriagonal magic, is called complete, because each pantriagonal contains 4 compliment pairs, with the members of each pair spaced 1/2m apart on the triagonal.

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

Order-3 not magic --1878

Rows, columns and pillars all sum correctly on all 12 planar squares so the square sums to 3 x 42 = 126. Two oblique squares have all rows summing correctly and 4 have all columns correct. Only 2 of the 4 triagonals are correct but one of these has all pantriagonals correct.

It is in QJM 15, 1878, page 97

18   23    1       20    7   15        4   12   26
 5   10   27       16   24    2       21    8   13
19    9   14        6   11   25       17   22    3

Order-3 Associated --1878

This cube has the exact same features as the preceding one, except all 4 triagonals are correct, making it a true magic cube.
It is a rotated version of basic cube #2, and is associated, as are all order 3 magic hypercubes. Because this cube is associated, the 3 central orthogonal planes are associated magic squares. This cube is in QJM 15, 1878, page 118

There are a total of 4 basic order 3 magic cubes. This is a rotated version of index # 2.

 2   24   16      15    7   20      25   11    6
18    1   23      19   14    9       5   27   10
22   17    3       8   21   13      12    4   26            

Order-4 not magic --1878

I obtained this cube originally from Christian Boyer (France) via email. He tipped me off as to its origin, resulting in my eventually obtaining copies of these Frost papers. It is in QJM 15, 1878, pp 116-117.

Horizontal planes are pandiagonal magic. Rows and columns of the other planes are correct. However, only 1 triagonal is correct so by present standards this is not a magic cube.

 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    

Order-4 Pantriagonal --1878

This cube is pantriagonal magic, non-associated. It is in QJM 15, 1878, page 119.

It contains no magic squares, but because each triagonal (and broken triagonal) contains two complement pairs with the members of each air spaced 1/2m apart, the cube is called complete. 

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

Frost presents a simple method I have not seen before, for constructing the above cube.
Put successive odd numbers from 1 to 63 in the cells of the planes as arranged here.
Add 65 to each number and divide by 2 to get the final numbers for the finished cube.

1

-3

-5

7

 

-17

19

21

-23

 

-33

35

37

-39

 

49

-51

-53

55

-9

11

13

-15

 

25

-27

-29

31

 

41

-43

-45

47

 

-57

59

61

-63

-37

39

33

-35

 

53

-55

-49

51

 

5

-7

-1

3

 

-21

23

17

-19

45

-47

-41

43

 

-61

63

57

-59

 

-13

15

9

-11

 

29

-31

-25

27

Order-7 Pandiagonal  --1878

This magic cube pandiagonal and associated. It contains 21 pandiagonal (the planar) and 6 simple (the oblique) magic squares. It is in QJM 15, 1878, pp 99-101

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

Order-9 Perfect (nasik) --1878

This must be the first perfect magic cube ever published. This at a time when others were producing magic(?) cubes with incorrect diagonals (and in some cases incorrect rows and columns.

In 1866 Frost published an order 7 pandiagonal and an order 8 pantriagonal magic cube. This cube is perfect (nasik) because it combines the features of both the pandiagonal and the pantriagonal cubes. The 3 x 9 planar squares are pandiagonal magic (pandiagonal cube), as are the 6 oblique squares. All 4m2 pantriagonals sum correctly (pantriagonal cube). Total summations to the constant 4005 is m2 rows, m2 columns, m2 pillars, 6m2 pandiagonals, and 4m2 pantriagonals. Total summations 13m2 . The diagonal magic cube which was originally called perfect, has 3m2 + 6m + 4 correct lines!

However, this cube is not normal, because it uses numbers from 1 to 889 instead of the consecutive numbers from 1 to 729. Frost points out [1, p 115] that the smallest order cube possible with consecutive numbers, using this method is order 11. Ian Howard mentioned the same thing when he published [2] his method for constructing the order 11 perfect magic cube 98 years later!
We now know that order 8 and order 9 normal perfect magic cubes do exist, presumably constructed using other methods.

This cube is also associated. Pairs of numbers diametrically equidistant to the center number of 445 sum to 2 times that number. Note also that 445 is one-half the sum of the first and last numbers used in the cube.
Note further, that because the cube is associated, the 3 central orthogonal magic squares are also associated. This feature is common to associated odd order magic hypercubes of any dimension. The lesser central hypercubes will always be magic and associated.

[1] Quarterly Journal of Mathematics 15 1878 pp 110-116
[2] Ian P. Howard, Pan-diagonal Associative Magic Cubes (Letter to the Editor), JRM 9:4, 1976, pp276-278.
 

Top                                              II 
824  606  733  149  258   85  362  471  517       55  382  461  577  814  626  703  139  248
473  519  828  605  732  141  257   84  366      131  247   54  386  463  579  818  625  702
 88  365  472  511  827  604  736  143  259      624  706  133  249  58   385  462  571  817
142  251   87  364  476  513  829  608  735      573  819  628  705  132  241   57  384  466
607  734  146  253   89  368  475  512  821      388  465  572  811  627  704  136  243   59
516  823  609  738  145  252   81  367  474      242   51  387  464  576  813  629  708  135
369  478  515  822  601  737  144  256   83      707  134  246   53  389  468  575  812  621
255   82  361  477  514  826  603  739  148      816  623  709  138  245   52  381  467  574
731  147  254   86  363  479  518  825  602      469  578  815  622  701  137  244   56  383
III                                              IV 
616  723  109  238   45  352  481  567  874      342  451  587  864  676  713  129  208   35
569  878  615  722  101  237   44  356  483      207   34  346  453  589  868  675  712  121
355  482  561  877  614  726  103  239   48      716  123  209   38  345  452  581  867  674
231   47  354  486  563  879  618  725  102      869  678  715  122  201   37  344  456  583
724  106  233   49  358  485  562  871  617      455  582  861  677  714  126  203   39  348
873  619  728  105  232   41  357  484  566       31  347  454  586  863  679  718  125  202
488  565  872  611  727  104  236   43  359      124  206   33  349  458  585  862  671  717
 42  351  487  564  876  613  729  108  235      673  719  128  205   32   341 457  584  866
107  234   46  353  489  568  875  612  721      588  865  672  711  127  204   36  343  459
V                                                VI 
773  119  228   5   332  441  557  884  666      431  547  854  686  763  179  218   25  302
888  665  772  111  227    4  336  443  559       24  306  433  549  858  685  762  171  217
442  551  887  664  776  113  229    8  335      173  219   28  305  432  541  857  684  766
  7  334  446  553  889  668  775  112  221      688  765  172  211   27   304 436  543  859
116  223    9  338  445  552  881  667  774      542  851  687  764  176  213   29  308  435
669  778  115  222    1  337  444  556  883      307  434  546  853  689  768  175  212   21
555  882  661  777  114  226    3  339  448      216   23  309  438  545  852  681  767  174
331  447  554  886  663  779  118  225    2      769  178  215   22  301  437  544  856  683
224    6  333  449  558  885  662  771  117      855  682  761  177  214   26  303  439  548
VII                                              VIII 
169  278   15  322  401  537  844  656  783      507  834  646  753  189  268   75  312  421
655  782  161  277   14  326  403  539  848      316  423  509  838  645  752  181  267   74
531  847  654  786  163  279   18  325  402      269   78  315  422  501  837  644  756  183
324  406  533  849  658  785  162  271   17      755  182  261   77  314  426  503  839  648
273   19  328  405  532  841  657  784  166      831  647  754  186  263   79  318  425  502
788  165  272   11  327  404  536  843  659      424  506  833  649  758  185  262   71  317
842  651  787  164  276   13  329  408  535       73  319  428  505  832  641  757  184  266
407  534  846  653  789  168  275   12  321      188  265   72  311  427  504  836  643  759
 16  323  409  538  845  652  781  167  274      642  751  187  264   76  313  429  508  835
IX
288  65   372  411  527  804  636  743  159
742  151  287   64  376  413  529  808  635
807  634  746  153  289   68  375  412  521
416  523  809  638  745  152  281   67  374
 69  378  415  522  801  637  744  156  283
155  282   61  377  414  526  803  639  748
631  747  154  286   63  379  418  525  802
524  806  633  749  158  285   62  371  417
373  419  528  805  632  741  157  284   66

A presentation to Rev. Frost

On Feb. 19/03 I received an email and this picture from Ms Carol LeBlanc. She graciously gave me permission to post the picture on this page. Thanks Carol. (click on picture to enlarge)

The inscription reads

Rev. A. H. Frost, MA, Missionary, C, M, S, Nassick Western India
Presented by his native friends".

The Reverend Andrew Hollingworth Frost (1820-1907), a mathematics wrangler of St. John's College, Cambridge, was a church missionary in Nasik, Bombay, India, from 1855 to 1867. [1], [2]

[1] Ollerenshaw and Brée, Most-Perfect Pandiagonal Magic Squares, IMA, 1998, 0-905091-06-X, footnote on page 6.
[2]
A well-researched and written Biography of Dr. Frost by Christian Boyer is on his Multimagic pages.

This page was originally posted December 2002
It was last updated March 10, 2010
Harvey Heinz   harveyheinz@shaw.ca
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