# Magic Cubes - The Road to Perfect

This page is intended as a supplement to my primary perfect magic cubes page, which was written before I obtained a lot of additional material on magic cubes.

• Frost referred to this general type (pantriagonal, diagonal, pandiagonal and (Hendricks) perfect) as Nasik. [1]
• Barnard referred to it as perfect and perfectly magic. [2]
• Planck refined Frost's definition of nasik to mean only hypercubes where all lines sum correctly [3]
• Rosser and Walker referred to the perfect cube as Diabolic. [4]
• Boyer and Trump refer to the diagonal cube as perfect (higher classes as perfect with enhancements). [5]
• Nakamura refers to the perfect cube (my definition) as Pan-2,3-agonal, thus avoiding the entire controversy over the definition of perfect. [6]
• I am now promoting the use of the term nasik in place of Hendricks term perfect. [7]

[1] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103
[2] F. A. P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir, 1888, pp 207-270
[3] C. Planck, Theory of Paths Nasik,
[4] B. Rosser and R. J. Walker, 1939, A continuation of The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered 729 – 753.
[5] Christian Boyer's Multimagic Web site at www.multimagie.com/index.htm
[6] Mitsutoshi Nakamura's Web site at http://homepage2.nifty.com/googol/magcube/en/
[7] A quotation from Planck's paper is here.

As mentioned above, this page is intended as a supplement to my primary perfect magic cubes page with an emphasis on example cubes of a variety of types and orders, with a minimum of descriptive text.

I am including the dates of publication along with the cubes I show here. However, please do not conclude that these are the first published of that particular type of cube. For example, Rev. A. H. Frost published examples of all these types in 1866 and 1878! Others, in many cases, also have published prior to the date of the cubes I show. The cubes I choose for examples on this page was determined by variety, and a desire not to use a cube that had been shown on another page on this site.

As a review, I will include a brief description of the features of a particular type of cube when I first show that type.

### Order 3

The lowest possible order for a magic cube. I show 2 non-normal cubes from Dr. Planck (1917).

### Order 4

The lowest possible order for a pantriagonal magic cube. I show 2 simple cubes (1922 and 1956 ) and a pantriagonal cube (1980).

### Order 5

The lowest possible order for a pantriagonal associated magic cube. I show the one where Hendricks introduces the term pantriagonal(1972). Also a simple associated cube. (1899)

### Order 6

I show a simple cube (1917), a semi-pantriagonal cube (1922), and a diagonal cube (2003).

### Order 7

The lowest possible order for a pandiagonal magic cube. I show Langman's (1962).

### Order 8

The lowest possible order for a perfect magic cube. Shown here is Myer's (1970, called perfect) and Benson and Jacoby's true perfect cube (1988). See an article on nasik, the suggested alternative to the confusing term perfect.

### Order 9

The lowest possible order for an associated perfect  magic cube. I show one (1977), and also a pantriagonal magic cube (2000).

### Order 10

A simple J. R. Hendricks order 10 cube with inlaid order 5 semi-magic cubes in each octant.

### Order 11

An order 11 perfect magic cube from instructions published in 1976.

Order-3

The simple magic cube
There are (3m2)+4 lines that sum to m(m3+1)/2. These are the rows, columns, pillars and the 4 triagonals. Note that no diagonals are required to sum correctly, although some may. Order-3 is the smallest possible magic cube. It exists in all orders greater then 2.

The associated magic cube
All pairs of numbers that are diametrically equidistant on each side of the center point of the cube sum to m3 + 1. The minimum order for associated Nasik type cubes (pantriagonal, pandiagonal, and perfect) is one more then if the that type of cube is not associated.

Dr. C. Planck – simple - 1905
Because all four simple associated magic cubes of order 3 all already shown on other pages on this site, I will not repeat them here. Instead, I will start this tour through the types of magic cubes, from simple to perfect, with two unorthodox examples.

These two order 3 cubes are taken from Dr. Planck’s order 3 Octahedron (Tesseract) so do not use consecutive numbers. [1]
The first cube is one of the four central cubes. It is fully magic, but does not use consecutive numbers, so is not considered normal. It is associated, as are all central hyperplanes in a higher dimension odd order hypercube. For the same reason, the 3 central squares in this cube are associated magic.
Because this cube is associated magic, it is also semi-pantriagonal (even though it is not normal). This can be confirmed by adding up each of the four opposite short triagonals. We find that 6 + 46 + 36 + 76 = 164, 36 + 38 + 44 + 46 = 164, 38 + 14 + 68 + 44 = 164, and 6 + 68 + 14 + 76 = 164. In each case, when we subtract the center cell value, the answer is123, the magic sum for this cube.

```Top           Middle        Bottom            Top           Middle        Bottom
65   6  52    33  79  11    25  38  60        34  74  15    65   6  52    24  43  56
36  73  14    19  41  63    68   9  46        23  45  55    36  73  14    64   5  54
22  44  57    71   3  49    30  76  17        66   4  53    22  44  57    35  75  13
A Central cube                                An outside cube```
Now look at the second cube. It is one of the outside cubes from his order 3 tesseract. It is not magic by present standards because only 1 of the 4 triagonals is correct. This condition for the tesseract is consistent with the definition for a magic cube, which does not require the squares within it have correct diagonals.
These two cubes, as part of a greater hypercube, have some numbers in common. In this case, one of the three horizontal planes is common to both cubes.

[1] C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby, 1905 (This is from fig. 11, page 14)
Also in W. S. Andrews, Magic Squares and Cubes, page 367, fig. 688

Order-4

Pantriagonal magic cube
As well as the normal requirements of a simple magic cube , the additional requirement is that all pantriagonals sum correctly. There are 7m2 lines that sum to the constant. This cube is analogous to a pandiagonal magic square, but instead of being able to move a row or column to the opposite side, you may move a plane. Order-4 is the smallest possible pantriagonal magic cube, and this class exists in all orders greater then 3.

Weidemann – simple - 1922
This simple magic cube was constructed by I. Weidemann in 1922. It is not associated and has no special features, except it is semi-pantriagonal and is the equivalent to the Group IV magic squares of order 4.

```Top               II                III               bottom
1  60  56  13    64   5   9  52    62   7  11  50     3  58  54  15
48  21  25  36    17  44  40  29    19  42  38  31    46  23  27  34
32  37  41  20    33  28  24  45    35  26  22  47    30  39  43  18
49  12   8  61    16  53  57   4    14  55  59   2    51  10   6  63```
Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922, p. 60
Translated title is Magic squares and other plane and solid magic figures.

Meloc – simple – associated - 1956
This is a different aspect of a cube shown in Encyclopedia Brittanica 1911 edition. It is simple magic and is associated. The editor refers to this as a ‘Jupiter’ cube because it is order 4. (I do not know the significance of the word ‘meloc’.)

```Top               II                III               bottom
64   2   3  61    33  31  30  36    17  47  46  20    16  50  51  13
5  59  58   8    28  38  39  25    44  22  23  41    53  11  10  56
9  55  54  12    24  42  43  21    40  26  27  37    57   7   6  60
52  14  15  49    45  19  18  48    29  35  34  32     4  62  63   1```
From a mimeographed magic newsletter called Treasury of Folklore – Fantasies in Figures, Mathematic Mysteries and Magic, edited by Stanley J. Coleman, 1956.

Hendricks - pantriagonal - not associated - 1980
Order 4 is the smallest order that can have a pantriagonal magic cube. John Hendricks published this one in 1980 [1] and established that there are 160 basic cubes of this type. Each of these can appear in 48 aspects (as can any order magic cube). He published an earlier pantriagonal magic cube in 1972 [2] when he introduced the term ‘pan-3-agonal’ (later modified to pantriagonal) for this type of magic cube.

```Top               II                III               bottom
48   7  57  18    10  33  31  56    51  28  38  13    21  62   4  43
54  29  35  12    20  59   5  46    41   2  64  23    15  40  26  49
27  52  14  37    61  22  44   3     8  47  17  58    34   9  55  32
1  42  24  63    39  16  50  25    30  53  11  36    60  19  45   6```
[1] John R. Hendricks, the Pan-3-agonal Magic Cube of Order 4, JRM 13:4, 1980-81, pp274-281
[2] John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972, pp 205-206

Order-5

The Diagonal magic cube exists in all orders from order-5 onward. I show one here.

Hendricks - pantriagonal – associated - 1972
This cube is pantriagonal and associated. It has no other special features except, of course, the 3 central planes are associated magic squares (because the cube is associated). [1][2]

This is the smallest order pantriagonal that can also be associated. Dr. Planck stated [3] that the smallest Nasik order in k dimensions is always 2k , (or 2k + 1 if we require association).
I believe this applies to pantriagonal, pandiagonal and perfect cubes because Planck cited Rev. Frost [4][5], who produced cubes of all three of these types and lumped them all together as Nasik cubes!

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

[1] John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972, pp 205-206
[2] John R. Hendricks, Magic Square Course, self-published, 1991, page 360.
[3] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, page 366 .
[4] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103
[5] A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123

Schubert – simple - semi-pantriagonal - associated - 1899
The center orthogonal plane in each orientation is magic (a feature of associated magic cubes). 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.
Because all the pantriagonals in 2 of the 4 directions sum correctly, perhaps we could call this a semi-pantriagonal cube?
No! Because that is not the traditional definition. However, this cube is equivalent to the definition for a semi-pandiagonal magic square. Opposite short triagonals (one of 4 such pairs is shown here in red) plus the center cell (green) sum to S. Also, shown here in blue, one of 4 opposite long diagonal pairs minus the center cell sums to S. I have more details on a semi-pantriagonal page.

```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 Essays and Recreations, Open Court, 1899, page 62.

Order-6

Simple, Pantriagonal, and Diagonal cubes can exist in order-6 (the same as order-5).

This is a simple magic cube but has the unique feature that if the cube is divided into 27 2x2x2 cubelets, the six faces of each cubelet and 2 of the 6 diagonal planes will each sum the same value. These 27 values step from 382 to 486. For example, the top left 2x2x2 cubelet:

```The six faces:                   Two diagonal planes
4    4    4  193  139   85       4   85
85   85  139  112  166  166     166  139
166  112   58   31   31   31      31   58
139  193  193   58   58  112     193  112
394  394  394  394  394  394     394  394

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

W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960 (1917), page 197 (The Monist, ,20,1910, pp299-303).

Diagonal type magic cubes
As well as the normal requirements of a simple magic cube, the additional requirement is that all orthogonal planes of the cube are simple magic squares.
This is the popular definition of the 'Perfect' magic cube. The smallest diagonal magic cube possible is order 5.

Trump - diagonal - 2003
This cube contains 18 orthogonal simple magic squares of order 6. Because all planar diagonals sum correct, it as a diagonal magic cube. The six oblique squares are also simple magic.
Until I received this cube from Walter in late summer of 2003, I had seen only two order 8 cubes and an order 12 cube of this type!
Walter sent me an order 7 and Christian Boyer sent an order 9 diagonal cube at about the same time.

```I - Top                          II                             III
109  143   76  123   88  112    137   48  157   68  158   83    103  101  159   36  119  133
87  156   49  170   63  126    155    2  198   27  207   62    162  196  201    8   29   55
140  174   52  150   53   82     34  187  212   13   22  183     46  206   28  197    3  171
75   66  182   51  139  138    147   32    1  208  193   70    163   15  191   18  210   54
136   40  148   65  176   86     44  213   23  186   12  173     93   17   14  211  192  124
104   72  144   92  132  107    134  169   60  149   59   80     84  116   58  181   98  114
IV                               V                               VI - Bottom
90   56   57  184  175   89    102  178  117   95   38  121    110  125   85  145   73  113
118   21   16  209  188   99     50  215   19  190   10  167     79   61  168   47  154  142
180   11  189   20  214   37    120   30    5  204  195   97    131   43  165   67  164   81
64  202   26  199    7  153    111  185  216    9   24  106     91  151   35  166   78  130
71  200  203    6   25  146    172    4  194   31  205   45    135  177   69  152   41   77
128  161  160   33   42  127     96   39  100  122  179  115    105   94  129   74  141  108```
ADDENDUM: Walter Trump discovered an order 5 cube of this type on Nov. 12, 2003. See it on my order 5 page.
NOTE that Trump and Boyer refer to this type of magic cube as perfect!

More from Walter himself at http://www.trump.de/magic-squares/magic-cubes/cubes-1.html.

Order-7

Other cube types
Due to space restraints, I will not show examples of a simple, a pantriagonal, and a diagonal magic cube of this order.

Pandiagonal magic cube
As well as the normal requirements of a simple magic cube, the additional requirement is that all orthogonal planes of the cube are pandiagonal magic squares. The 6 oblique squares are simple magic (1, 2, or 3 may be pandiagonal magic). [1]
There are at least (9m2)+4 lines that sum correctly. This is one of the original definitions of a ‘perfect magic cube’.
Order-7 is the smallest possible pandiagonal magic cube.

I have examined 13 order 7 pandiagonal magic cubes and all but 1 are associated.

Langman - pandiagonal -1962 [2]
This is a pandiagonal magic cube because all 21 planar square arrays are pandiagonal magic. All 6 diagonal square arrays are simple magic (3 of the six have all the pandiagonals in one direction correct).
All main triagonals are correct (of course) but all the pantriagonals are correct only for one of the 4 sets (i.e. only 1 of 4 directions). This cube has sometimes been referred to as perfect. It was published in 1962.

Rev. A. H. Frost published this order 7 pandiagonal magic cube in 1866 [3]

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

[1]  J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9
[2]  Harry Langman, Ph. D., Play Mathematics, Hafner Publ. 1962, p. 75-76.
[3]  A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103

Order-8

All 6 classes of magic cubes can exist, starting with order-8.

Diagonal magic cubes
As well as the normal requirements of a simple magic cube, the additional requirement is that all orthogonal planes of the cube are simple magic squares.
In Sept. 2003 I obtained orders 6 (shown earlier) and 7 diagonal cubes from Walter Trump and an order 9 from Christian Boyer.
On Nov. 13, 2003, Trump and Boyer announced the discovery of an order 5 cube of this type.

This order-8 cube was called perfect by an early definition (M. Gardner and others) because all squares are magic. It was constructed by R. L. Myers in 1970
All 30 squares are simple magic i.e. rows, columns and main diagonals of all orthogonal and oblique square arrays sum correctly. Corner sums of all order-5 sub-cubes sum to the constant. All orthogonal squares are simple magic so it is a diagonal cube. There are 3m2 + 6m + 4 lines that sum correctly.

```Top                                       II
19  497  255  285  432   78  324  162    134  360  106  396  313  219  469   55
303  205  451   33  148  370  128  414    442   92  342  184    5  487  233  267
336  174  420   66  243  273   31  509    473   59  309  215  102  392  138  364
116  402  160  382  463   45  291  193    229  263    9  491  346  188  438   88
486    8  266  236   89  443  181  343    371  145  415  125  208  302   36  450
218  316   54  472  357  135  393  107     79  429  163  321  500   18  288  254
185  347   85  439  262  232  490   12     48  462  196  290  403  113  383  157
389  103  361  139   58  476  214  312    276  242  512   30  175  333   67  417
III                                       IV
306  212  478   64  141  367   97  387    423   69  331  169   28  506  248  278
14  496  226  260  433   83  349  191    155  377  119  405  296  198  460   42
109  399  129  355  466   52  318  224    252  282   24  502  327  165  427   73
337  179  445   95  238  272    2  484    456   38  300  202  123  409  151  373
199  293   43  457  380  154  408  118     82  436  190  352  493   15  257  227
507   25  279  245   72  422  172  330    366  144  386  100  209  307   61  479
412  122  376  150   39  453  203  297    269  239  481    3  178  340   94  448
168  326   76  426  283  249  503   21     49  467  221  319  398  112  354  132
V                                         VI
381  159  401  115  194  292   46  464    492   10  264  230   87  437  187  345
65  419  173  335  510   32  274  244    216  310   60  474  363  137  391  101
34  452  206  304  413  127  369  147    183  341   91  441  268  234  488    6
286  256  498   20  161  323   77  431    395  105  359  133   56  470  220  314
140  362  104  390  311  213  475   57     29  511  241  275  418   68  334  176
440   86  348  186   11  489  231  261    289  195  461   47  158  384  114  404
471   53  315  217  108  394  136  358    322  164  430   80  253  287   17  499
235  265    7  485  344  182  444   90    126  416  146  372  449   35  301  207
VII                                       Bottom
96  446  180  338  483    1  271  237    201  299   37  455  374  152  410  124
356  130  400  110  223  317   51  465    501   23  281  251   74  428  166  328
259  225  495   13  192  350   84  434    406  120  378  156   41  459  197  295
63  477  211  305  388   98  368  142    170  332   70  424  277  247  505   27
425   75  325  167   22  504  250  284    320  222  468   50  131  353  111  397
149  375  121  411  298  204  454   40      4  482  240  270  447   93  339  177
246  280   26  508  329  171  421   71     99  385  143  365  480   62  308  210
458   44  294  200  117  407  153  379    351  189  435   81  228  258   16  494```
Martin Gardner, Time Travel and Other Mathematical Bewilderments, 1988, page 222
John R. Hendricks, Magic Square Course, self-published, 1991, (page 405).
W. H. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, 0-486-24140-8, pp 43-60
Rudolf Ondrejka, Letter to the Editor, Journal of Recreational Mathematics, 20:3, 1988, pp207-209

Pantriagonal Diagonal
A magic cube that is a combination Pantriagonal and Diagonal cube. All main and broken triagonals must sum correctly, In addition, it will contain 3m order m simple magic squares in the orthogonal planes, and 6 order m pandiagonal magic squares in the oblique planes.There are 7m2 + 6m lines that sum correctly.

For short, I will reduce this unwieldy name to PantriagDiag. This is number 4 in what is now 6 classes of magic cubes. So far, very little is known of this class of cube. The only ones constructed so far (both by Nakamura) are order 8 (not associated and associated).

This cube was discovered by Mitsutoshi Nakamura and named by him in 2004. So there are now 6 classes of magic cubes: Simple, Pantriagonal, Diagonal, Pantriagonal Diagonal (PantriagDiag), Pandiagonal, and Perfect.

Perfect (nasik) magic cube
A perfect magic cube is a combination pantriagonal and pandiagonal magic cube. Because of the combination of the two previous conditions, all 6 oblique squares are pandiagonal magic as well.
In a perfect magic cube there are 9m pandiagonal magic squares. That is, all 3m orthogonal planes, the 6 oblique planes, and the 6m-1 broken planes parallel to the oblique planes [1].
There are 13m2 lines that sum correctly. Order-8 is the smallest possible perfect magic cube.

Corners of all cubes (within the main cube) of orders 2, 3, 4, 5, 6, 7 and 8 (including wraparound) also sum correctly to 2052. Barnard (1888) confirms this. [2]
All order 8 perfect cubes I have seen also have another feature called complete. [3] Is this also common to all order 8 perfect cubes? To all 8x perfect cubes?

The Benson Jacoby Perfect Magic Cube - 1981
This is the 3rd known published order 8 perfect magic cube (Barnard 1888, Planck 1905). [2][7]
Benson and Jacoby referred to this cube as pandiagonal perfect. It appeared in octal [4] and in decimal in [5].

All 24 planar squares are pandiagonal magic as are also the 6 oblique squares and the seven broken squares parallel to each of these [1][6] for a total of 72 pandiagonal magic squares. The 256 pantriagonals also all sum correctly.

Corners of all orders 2, 3, 4, 5, 6, 7 and 8 (including wraparound) sub-cubes also sum correctly to 2052 making this cube complete. Also, many other shapes of parallelopipeds.

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

```219  386  305   85  230  447  272  108
182  471  360   60  139  490  345    5
99  250  393  301   94  199  440  276
14  175  480  324   51  146  481  381
283   66  241  405  294  127  208  428
374   23  168  508  331   42  153  453
419  314   73  237  414  263  120  212
462  367   32  132  499  338   33  189```

[1] B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement, a bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4. All papers are very technical. There are NO diagrams. One sect is: A continuation of The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered 729 – 753, (diabolic cubes pp 736-753).
[2] F. A. P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir, 1888, pp 207-270
[3] Complete = Every pantriagonal contains m/2 complement pairs spaced m/2 apart. Kanji Setsuda’s Compact (composite) and Complete magic Cubes Web page is http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
[4] W. H. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, 0-486-24140-8.
[5] R. Ondrejka, The Most Perfect (8x8x8) Magic Cube? (Letter to the Editor), JRM 20:3, 1988, pp207-209
[6] F. Liao, T. Katayama, K. Takaba, Technical Report 99021, School of Informatics, Kyoto University, 1999. Available on the Internet at http://www.amp.i.kyoto-u.ac.jp/tecrep/TR1999.html
[7] C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby, 1905 (This is fig. 11, page 14)

Order-9

Order 9 is the smallest possible for a perfect magic cube that is also associated. [1]

Soni Order-9 Pantriagonal Magic Cube - 2001?

This must be classed as a pantriagonal magic cube, although all 9 horizontal and all 9 vertical back to front planes are pandiagonal magic squares. The vertical squares parallel to the sides have incorrect diagonals, so are not magic. The 6 oblique squares are also pandiagonal magic.
Also, all 3x3 arrays of the horizontal planes and the vertical planes parallel with the front face (including wrap-around), sum to the constant.

I generated this cube from Abhinav Soni ‘s program, HyperMagic Cubes.exe. [2]

```Top                                            II
635  169  291   68  331  696  473  493  129     44  379  672  449  541  105  611  217  267
485  496  114  647  172  276   80  334  681    639  164  292   72  326  697  477  488  130
27  362  706  432  524  139  594  200  301    480  500  115  642  176  277   75  338  682
615  230  250   48  392  655  453  554   88     24  368  703  429  530  136  591  206  298
438  566   91  600  242  253   33  404  658    616  234  245   49  396  650  454  558   83
58  351  686  463  513  119  625  189  281    439  561   95  601  237  257   34  399  662
574  210  311    7  372  716  412  534  149     55  348  692  460  510  125  622  186  287
415  519  161  577  195  323   10  357  728    569  211  315    2  373  720  407  535  153
38  382  675  443  544  108  605  220  270    419  520  156  581  196  318   14  358  723
III                                            IV
423  515  157  585  191  319   18  353  724    570  212  313    3  374  718  408  536  151
39  383  673  444  545  106  606  221  268    420  521  154  582  197  316   15  359  721
636  170  289   69  332  694  474  494  127     40  387  668  445  549  101  607  225  263
481  504  110  643  180  272   76  342  677    637  165  293   70  327  698  475  489  131
25  363  707  430  525  140  592  201  302    478  501  116  640  177  278   73  339  683
613  231  251   46  393  656  451  555   89     20  364  711  425  526  144  587  202  306
434  562   99  596  238  261   29  400  666    617  232  246   50  394  651  455  556   84
59  349  687  464  511  120  626  187  282    440  559   96  602  235  258   35  397  663
575  208  312    8  370  717  413  532  150     63  344  688  468  506  121  630  182  283
V                                              VI
60  350  685  465  512  118  627  188  280    436  567   92  598  243  254   31  405  659
571  216  308    4  378  713  409  540  146     61  345  689  466  507  122  628  183  284
421  516  158  583  192  320   16  354  725    568  213  314    1  375  719  406  537  152
37  384  674  442  546  107  604  222  269    416  517  162  578  193  324   11  355  729
632  166  297   65  328  702  470  490  135     41  385  669  446  547  102  608  223  264
482  502  111  644  178  273   77  340  678    638  163  294   71  325  699  476  487  132
26  361  708  431  523  141  593  199  303    486  497  112  648  173  274   81  335  679
621  227  247   54  389  652  459  551   85     21  365  709  426  527  142  588  203  304
435  563   97  597  239  259   30  401  664    618  233  244   51  395  649  456  557   82
VII                                            VIII
619  228  248   52  390  653  457  552   86     19  366  710  424  528  143  586  204  305
433  564   98  595  240  260   28  402  665    614  229  252   47  391  657  452  553   90
56  346  693  461  508  126  623  184  288    437  565   93  599  241  255   32  403  660
572  214  309    5  376  714  410  538  147     62  343  690  467  505  123  629  181  285
422  514  159  584  190  321   17  352  726    576  209  310    9  371  715  414  533  148
45  380  670  450  542  103  612  218  265    417  518  160  579  194  322   12  356  727
633  167  295   66  329  700  471  491  133     42  386  667  447  548  100  609  224  262
483  503  109  645  179  271   78  341  676    634  171  290   67  333  695  472  495  128
22  369  704  427  531  137  589  207  299    484  498  113  646  174  275   79  336  680
Bottom
479  499  117  641  175  279   74  337  684
23  367  705  428  529  138  590  205  300
620  226  249   53  388  654  458  550   87
441  560   94  603  236  256   36  398  661
57  347  691  462  509  124  624  185  286
573  215  307    6  377  712  411  539  145
418  522  155  580  198  317   13  360  722
43  381  671  448  543  104  610  219  266
631  168  296   64  330  701  469  492  134 ```
Seimiya Order 9 'Golden' Magic Cube - Associated – Perfect - 1977

This cube [3] uses the numbers from 0 to 728 (instead of 1 to 729) so the magic constant is 3276 (instead of 3285).
It is perfect, so 27 orthogonal square arrays, 6 oblique arrays, and 48 oblique broken arrays are pandiagonal magic. It is the smallest possible order for a cube that is both perfect and associated. [1]

A.H. Frost published a Perfect order 9 magic cube in 1878. However, it did not use consecutive numbers.
The first normal perfect magic cube seems to be the order 17 constructed by Gabriel Arnoux in 1887.

```Top                                            II
0  706  521  617  267  397  475  119  174    402  469  124  173    3  702  517  620  266
530  581  249  379  439  128  183   72  715    133  182   75  711  526  584  248  384  433
321  388  448   92  165   54  679  539  590     57  675  535  593  320  393  442   97  164
457  101  237   63  688  503  572  303  352    499  575  302  357  451  106  236   66  684
219   27  697  512  644  312  361  421   83    311  366  415   88  218   30  693  508  647
661  494  626  276  370  430  155  228   36    424  160  227   39  657  490  629  275  375
635  285  334  412  137  192   45  670  566    191   48  666  562  638  284  339  406  142
343  484  146  201    9  652  548  599  294    648  544  602  293  348  478  151  200   12
110  210   18  724  557  608  258  325  466    611  257  330  460  115  209   21  720  553
III                                            IV
705  513  616  269  401  474  118  178    2    473  123  172    7  704  516  612  265  404
580  251  383  438  127  187   74  714  522    181   79  713  525  576  247  386  437  132
392  447   91  169   56  678  531  589  323    677  534  585  319  395  446   96  163   61
100  241   65  687  495  571  305  356  456    567  301  359  455  105  235   70  686  498
29  696  504  643  314  365  420   82  223    368  419   87  217   34  695  507  639  310
486  625  278  374  429  154  232   38  660    159  226   43  659  489  621  274  377  428
287  338  411  136  196   47  669  558  634     52  668  561  630  283  341  410  141  190
483  145  205   11  651  540  598  296  347    543  594  292  350  482  150  199   16  650
214   20  723  549  607  260  329  465  109    256  332  464  114  208   25  722  552  603
V                                              VI
515  615  261  400  476  122  177    1  709    125  176    6  703  520  614  264  396  472
243  382  440  131  186   73  718  524  579     78  712  529  578  246  378  436  134  185
449   95  168   55  682  533  588  315  391    538  587  318  387  445   98  167   60  676
240   64  691  497  570  297  355  458  104    300  351  454  107  239   69  685  502  569
700  506  642  306  364  422   86  222   28    418   89  221   33  694  511  641  309  360
624  270  373  431  158  231   37  664  488    230   42  658  493  623  273  369  427  161
337  413  140  195   46  673  560  633  279    667  565  632  282  333  409  143  194   51
149  204   10  655  542  597  288  346  485    596  291  342  481  152  203   15  649  547
19  727  551  606  252  328  467  113  213    324  463  116  212   24  721  556  605  255
VII                                            VIII
619  263  399  468  121  179    5  708  514    175    8  707  519  613  268  398  471  117
381  432  130  188   77  717  523  583  245    716  528  577  250  380  435  126  184   80
94  170   59  681  532  592  317  390  441    586  322  389  444   90  166   62  680  537
68  690  496  574  299  354  450  103  242    353  453   99  238   71  689  501  568  304
505  646  308  363  414   85  224   32  699     81  220   35  698  510  640  313  362  417
272  372  423  157  233   41  663  487  628     44  662  492  622  277  371  426  153  229
405  139  197   50  672  559  637  281  336    564  631  286  335  408  135  193   53  671
206   14  654  541  601  290  345  477  148    295  344  480  144  202   17  653  546  595
726  550  610  254  327  459  112  215   23    462  108  211   26  725  555  604  259  326
Bottom
262  403  470  120  171    4  710  518  618
434  129  180   76  719  527  582  244  385
162   58  683  536  591  316  394  443   93
692  500  573  298  358  452  102  234   67
645  307  367  416   84  216   31  701  509
376  425  156  225   40  665  491  627  271
138  189   49  674  563  636  280  340  407
13  656  545  600  289  349  479  147  198
554  609  253  331  461  111  207   22  728 ```
[1] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, page 366.
[2] Abhinav Soni HyperMagicCube.exe program. Obtainable from soni_abhinav@yahoo.com
[3] Seimiya, Mathematical Sciences (Japanese) Magazine Dec. 1977, p. 45-47

Order-10

Order-10 PandiagDiag, Pandiagonal, and Nasik (perfect) cubes are impossible.
Simple, Pantriagonal, and Diagonal magic cubes are all possible for this order.

J. R. Hendricks Millennium 10 x 10 x 10 simple magic cube - 2000

This is a simple magic cube with all rows, columns, pillars, and 4 main triagonals correct. It has an order-5 semi-magic cube in each octant. In each case, 1 of the 4 triagonals of the inlaid cube is incorrect.

The two diagonals for each of the 30 planar squares sum to 4130 and 5880. It follows that the 4 oblique squares with incorrect row totals have 5 rows each that total 4130 and 5 rows that total 5880. The two oblique squares with incorrect column totals have 5 columns each with each of these totals. This cube contains no magic squares.

This cube was privately distributed by John Hendricks in December 2000 as a ‘Millennium’ magic cube. Following this listing, I will include one of the inlaid semi-magic order 5 cubes.

```      Top                                                 II
625  377  409  686  843  343  811  284  252  375    401  683  840  617  399  274  367  340  808  276
381  413  695  847  604  354  347  820  288  256    687  844  621  378  410  285  253  371  344  812
417  699  826  608  390  265  358  326  824  292    848  605  382  414  691  816  289  257  355  348
678  835  612  394  421  296  269  362  335  803    609  386  418  700  827  327  825  293  261  359
839  616  398  405  682  807  280  273  366  339    395  422  679  831  613  363  331  804  297  270
214  491  523  530  307  182  655  648  741  714    520  547  304  206  488  738  706  179  672  645
303  210  487  519  546  671  644  737  710  178    484  511  543  325  202  702  200  668  636  734
542  324  201  483  515  640  733  701  199  667    223  480  507  539  316  191  664  632  730  723
506  538  320  222  479  729  722  195  663  631    312  219  496  503  535  660  628  746  719  187
500  502  534  311  218  718  186  659  627  750    526  308  215  492  524  649  742  715  183  651
III                                                 IV
832  614  391  423  680  805  298  266  364  332    388  420  697  829  606  356  329  822  295  263
618  400  402  684  836  336  809  277  275  368    424  676  833  615  392  267  365  333  801  299
379  406  688  845  622  372  345  813  281  254    685  837  619  396  403  278  271  369  337  810
415  692  849  601  383  258  351  349  817  290    841  623  380  407  689  814  282  255  373  341
696  828  610  387  419  294  262  360  328  821    602  384  411  693  850  350  818  286  259  352
321  203  485  512  544  669  637  735  703  196    477  509  536  318  225  725  193  661  634  727
540  317  224  476  508  633  726  724  192  665    216  498  505  532  314  189  657  630  748  716
504  531  313  220  497  747  720  188  656  629    310  212  494  521  528  653  646  744  712  185
493  525  527  309  211  711  184  652  650  743    549  301  208  490  517  642  740  708  176  674
207  489  516  548  305  180  673  641  739  707    513  545  322  204  481  731  704  197  670  638
V                                                   VI
694  846  603  385  412  287  260  353  346  819     69  971  978   10   37  787  760  228  596  569
830  607  389  416  698  823  291  264  357  330    955  982   14   41   73  573  791  764  232  580
611  393  425  677  834  334  802  300  268  361    986   18   50   52  959  584  552  800  768  236
397  404  681  838  620  370  338  806  279  272     22   29   56  963  995  245  588  556  779  772
408  690  842  624  376  251  374  342  815  283     33   65  967  999    1  751  249  592  565  783
533  315  217  499  501  626  749  717  190  658    908  940   92  124  876  126  874  467  440  158
522  529  306  213  495  745  713  181  654  647    897  904  931   88  120  870  463  431  154  147
486  518  550  302  209  709  177  675  643  736    111  893  925  927   84  459  427  175  143  861
205  482  514  541  323  198  666  639  732  705     80  107  889  916  948  448  166  139  857  455
319  221  478  510  537  662  635  728  721  194    944   96  103  885  912  162  135  853  471  444
VII                                                 VIII
13   45   72  954  981  231  579  572  795  763    957  989   16   48   55  555  798  766  239  582
49   51  958  990   17  767  240  583  551  799    993   25   27   59  961  586  559  777  775  243
60  962  994   21   28  778  771  244  587  560      4   31   63  970  997  247  595  563  781  754
966  998    5   32   64  564  782  755  248  591     40   67  974  976    8  758  226  599  567  790
977    9   36   68  975  600  568  786  759  227     71  953  985   12   44  794  762  235  578  571
102  884  911  943  100  475  443  161  134  852    946   78  110  887  919  169  137  860  453  446
91  123  880  907  939  439  157  130  873  466    915  942   99  101  883  133  851  474  442  165
935   87  119  896  903  153  146  869  462  435    879  906  938   95  122  872  470  438  156  129
924  926   83  115  892  142  865  458  426  174    118  900  902  934   86  461  434  152  150  868
888  920  947   79  106  856  454  447  170  138     82  114  891  923  930  430  173  141  864  457
IX                                                  Bottom
26   58  965  992   24  774  242  590  558  776   1000    2   34   61  968  593  561  784  752  250
62  969  996    3   35  785  753  246  594  562      6   38   70  972  979  229  597  570  788  756
973  980    7   39   66  566  789  757  230  598     42   74  951  983   15  765  233  576  574  792
984   11   43   75  952  577  575  793  761  234     53  960  987   19   46  796  769  237  585  553
20   47   54  956  988  238  581  554  797  770    964  991   23   30   57  557  780  773  241  589
895  922  929   81  113  863  456  429  172  145     89  116  898  905  932  432  155  148  866  464
109  886  918  950   77  452  450  168  136  859    928   85  112  894  921  171  144  862  460  428
98  105  882  914  941  441  164  132  855  473    917  949   76  108  890  140  858  451  449  167
937   94  121  878  910  160  128  871  469  437    881  913  945   97  104  854  472  445  163  131
901  933   90  117  899  149  867  465  433  151    125  877  909  936   93  468  436  159  127  875```

Hendricks – Top Right Back Octant

Following  is the top right back octant of the above 10x10x10 Millennium cube.
This is not a normal magic cube because the number series does not start at 1 and numbers are not consecutive.
All rows and columns of the orthogonal planes sum correctly as do all pandiagonals in one direction.
However, only 3 of the 4 main triagonals sum correctly, so this is only a semi (?) magic cube.
One orthogonal plane and 2 oblique planes are simple magic (some of the inlays have 2 orthogonal planes magic).

```      Top                        II                         III
343  811  284  252  375    274  367  340  808  276    805  298  266  364  332
354  347  820  288  256    285  253  371  344  812    336  809  277  275  368
265  358  326  824  292    816  289  257  355  348    372  345  813  281  254
296  269  362  335  803    327  825  293  261  359    258  351  349  817  290
807  280  273  366  339    363  331  804  297  270    294  262  360  328  821
IV                          V
356  329  822  295  263    287  260  353  346  819
267  365  333  801  299    823  291  264  357  330
278  271  369  337  810    334  802  300  268  361
814  282  255  373  341    370  338  806  279  272
350  818  286  259  352    251  374  342  815  283```

Order-11

Howard Order 11 Perfect Magic Cube 1976 (2002) not associated

This cube was constructed by myself using instructions published in Ian Howard’s Letters to the Editor, JRM. [1]

Howard mentions that order 11 is the smallest possible for a prime order perfect magic cube. (we know that perfect cubes can exist for non-prime orders as small as 8).
Rev. A. H. Frost mentioned the same bottom limit in 1878. [2] However, he publishes an order 9 perfect cube that does not use consecutive numbers.
Because this is a perfect cube, there are 9m pandiagonal magic squares in this cube. That is, 33 orthogonal, 6 oblique, and 10 broken squares parallel to each oblique one. There are a total of m2 each of rows, columns and pillars, 6m2 diagonals and 4m2 triagonals for a total of 1573 lines that sum to the constant 7326.

```Top                                                       II
120  231  331  442  553  664  775  886  997 1108 1219    520  631  742  853 1085 1196 1307   87  198  298  409
261  372  604  715  815  926 1037 1148 1259   39  150    782  893 1004 1115 1226    6  238  349  460  571  682
523  634  745  856 1088 1199  1299  79  190  301  412   1055 1166 1266   46  157  268  379  490  722  833  944
785  896 1007 1118 1229    9  241  352  452  563  674   1317   97  208  319  419  530  641  752  863  974 1206
1047 1158 1269   49  160  271  382  493  725  836  936    127  359  470  581  692  803  903 1014 1125 1236   16
1320   89  200  311  422  533  644  755  866  977 1209    389  500  611  843  954 1065 1176 1287   56  167  278
130  362  473  573  684  795  906 1017 1128 1239   19    651  762  873  984 1095 1327  107  218  329  440  540
392  503  614  846  957 1057 1168 1279   59  170  281    924 1024 1135 1246   26  137  248  480  591  702  813
654  765  876  987 1098 1330  110  210  321  432  543   1186 1297   77  177  288  399  510  621  732  964 1075
916 1027 1138 1249   29  140  251  483  594  694  805    117  228  339  450  561  661  772  883  994 1105 1216
1178 1289   69  180  291  402  513  624  735  967 1078    258  369  601  712  823  934 1045 1145 1256   36  147
III                                                       IV
1052 1163 1274   54  165  265  376  487  719  830  941    132  353  464  575  686  797  908 1019 1130 1241   21
1314   94  205  316  427  538  649  749  860  971 1203    394  505  616  837  948 1059 1170 1281   61  172  283
124  356  467  578  689  800  911 1022 1133 1233   13    656  767  878  989 1100 1321  101  212  323  434  545
386  497  608  840  951 1062 1173 1284   64  175  286    918 1029 1140 1251   31  142  253  474  585  696  807
659  770  870  981 1092 1324  104  215  326  437  548   1180 1291   71  182  293  404  515  626  737  958 1069
921 1032 1143 1254   23  134  245  477  588  699  810    111  222  333  444  555  666  777  888  999 1110 1221
1183 1294   74  185  296  407  507  618  729  961 1072    263  374  595  706  817  928 1039 1150 1261   41  152
114  225  336  447  558  669  780  891  991 1102 1213    525  636  747  858 1079 1190 1301   81  192  303  414
255  366  598  709  820  931 1042 1153 1264   44  144    787  898 1009 1120 1231   11  232  343  454  565  676
528  628  739  850 1082 1193 1304   84  195  306  417   1049 1160 1271   51  162  273  384  495  716  827  938
790  901 1012 1112 1223    3  235  346  457  568  679   1311   91  202  313  424  535  646  757  868  979 1200
V                                                         VI
653  764  875  986 1097 1329  109  220  320  431  542   1185 1296   76  187  287  398  509  620  731  963 1074
915 1026 1137 1248   28  139  250  482  593  704  804    116  227  338  449  560  671  771  882  993 1104 1215
1188 1288   68  179  290  401  512  623  734  966 1077    257  368  600  711  822  933 1044 1155 1255   35  146
119  230  341  441  552  663  774  885  996 1107 1218    519  630  741  852 1084 1195 1306   86  197  308  408
260  371  603  714  825  925 1036 1147 1258   38  149    792  892 1003 1114 1225    5  237  348  459  570  681
522  633  744  855 1087 1198 1309   78  189  300  411   1054 1165 1276   45  156  267  378  489  721  832  943
784  895 1006 1117 1228    8  240  351  462  562  673   1316   96  207  318  429  529  640  751  862  973 1205
1046 1157 1268   48  159  270  381  492  724  835  946    126  358  469  580  691  802  913 1013 1124 1235   15
1319   99  199  310  421  532  643  754  865  976 1208    388  499  610  842  953 1064 1175 1286   66  166  277
129  361  472  583  683  794  905 1016 1127 1238   18    650  761  872  983 1094 1326  106  217  328  439  550
391  502  613  845  956 1067 1167 1278   58  169  280    923 1034 1134 1245   25  136  247  479  590  701  812
VII                                                       VIII
254  365  597  708  819  930 1041 1152 1263   43  154    786  897 1008 1119 1230   10  242  342  453  564  675
527  638  738  849 1081 1192 1303   83  194  305  416   1048 1159 1270   50  161  272  383  494  726  826  937
789  900 1011 1122 1222    2  234  345  456  567  678   1310   90  201  312  423  534  645  756  867  978 1210
1051 1162 1273   53  164  275  375  486  718  829  940    131  363  463  574  685  796  907 1018 1129 1240   20
1313   93  204  315  426  537  648  759  859  970 1202    393  504  615  847  947 1058 1169 1280   60  171  282
123  355  466  577  688  799  910 1021 1132 1243   12    655  766  877  988 1099 1331  100  211  322  433  544
396  496  607  839  950 1061 1172 1283   63  174  285    917 1028 1139 1250   30  141  252  484  584  695  806
658  769  880  980 1091 1323  103  214  325  436  547   1179 1290   70  181  292  403  514  625  736  968 1068
920 1031 1142 1253   33  133  244  476  587  698  809    121  221  332  443  554  665  776  887  998 1109 1220
1182 1293   73  184  295  406  517  617  728  960 1071    262  373  605  705  816  927 1038 1149 1260   40  151
113  224  335  446  557  668  779  890 1001 1101 1212    524  635  746  857 1089 1189 1300   80  191  302  413
IX                                                        X
1318   98  209  309  420  531  642  753  864  975 1207    387  498  609  841  952 1063 1174 1285   65  176  276
128  360  471  582  693  793  904 1015 1126 1237   17    660  760  871  982 1093 1325  105  216  327  438  549
390  501  612  844  955 1066 1177 1277   57  168  279    922 1033 1144 1244   24  135  246  478  589  700  811
652  763  874  985 1096 1328  108  219  330  430  541   1184 1295   75  186  297  397  508  619  730  962 1073
914 1025 1136 1247   27  138  249  481  592  703  814    115  226  337  448  559  670  781  881  992 1103 1214
1187 1298   67  178  289  400  511  622  733  965 1076    256  367  599  710  821  932 1043 1154 1265   34  145
118  229  340  451  551  662  773  884  995 1106 1217    518  629  740  851 1083 1194 1305   85  196  307  418
259  370  602  713  824  935 1035 1146 1257   37  148    791  902 1002 1113 1224    4  236  347  458  569  680
521  632  743  854 1086 1197 1308   88  188  299  410   1053 1164 1275   55  155  266  377  488  720  831  942
783  894 1005 1116 1227    7  239  350  461  572  672   1315   95  206  317  428  539  639  750  861  972 1204
1056 1156 1267   47  158  269  380  491  723  834  945    125  357  468  579  690  801  912 1023 1123 1234   14
XI -Bottom
919 1030 1141 1252   32  143  243  475  586  697  808
1181 1292   72  183  294  405  516  627  727  959 1070
112  223  334  445  556  667  778  889 1000 1111 1211
264  364  596  707  818  929 1040 1151 1262   42  153
526  637  748  848 1080 1191 1302   82  193  304  415
788  899 1010 1121 1232    1  233  344  455  566  677
1050 1161 1272   52  163  274  385  485  717  828  939
1312   92  203  314  425  536  647  758  869  969 1201
122  354  465  576  687  798  909 1020 1131 1242   22
395  506  606  838  949 1060 1171 1282   62  173  284
657  768  879  990 1090 1322  102  213  324  435  546 ```
[1] Constructed by H. Heinz from instructions in Ian P. Howard, Letters to the Editor, JRM,9:4, 1976-77, pp276-278
[2] A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.

 This page was originally posted December 2002 It was last updated December 03, 2012 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz