Modulo Magic Cubes

In the summer of 2003, Walter Trump, Germany, produced a series of order 5 cubes that had an unusual feature.
The row, column, pillar, and triagonal sums had varying totals but all sums for a particular cube were divisible by the same number.

He provided three cubes that were simple magic in the accepted sense. However, they were diagonal magic mod 2, 3, or 10 as well, because all required sums, including the two diagonals of each of the 15 orthogonal squares, were divisible by 2, 3, or 10 respectively.

He also showed two cubes that were not magic in the normal sense because some orthogonal lines summed incorrectly. However, in these two cases, all lines were modulo 31 or 62 respectively.

These cubes were followed within a few days by similar contributions from Christian Boyer (France) and Aale de Winkel (The Netherlands).

Trump mod 31 cube

This cube is NOT magic in the normal sense, because the orthogonal planes do not all sum correctly. However, because all these lines, plus all 30 planar diagonals are correct mod 5), this cube is diagonal magic.

Trump mod 2, 3, 10, and 62 cubes

The first 3 cubes are simple magic. All are diagonal magic mod 2, 3, 10 or 62 respectively.
One of the order-5 cubes is a bordered (concentric) magic cube.

Boyer mod 5 cube

Christian Boyer sent me two simple magic cubes with exactly the same characteristics. Here I show only the first one. Because all 9m squares are pandiagonal magic (mod 5) these cubes are nasik perfect!

de Winkel mod 5 cube

This cube is pantriagonal magic. When the sums are considered mod 5, the cube is nasik perfect!

Trump mod 31 cube

 For simpler computations, these Trump's cubes all use the number range 0 to 124, so S = 310. This cube is taken from modulo-31.xls of Sept. 4/03. It contains 15 + 6 simple magic squares modulo 31.This cube is NOT a regular magic cube because some rows or columns do not sum correctly. However, all these sums (for the rows and columns of the orthogonal planes) are evenly divisible by 31. Therefore, It is diagonal magic modulo 31 because all 21 squares are simple magic mod 31. In the illustration, the blue sums (310) are the required sum for the normal magic cube. The red sums are evenly divisible by 31. The black sums are for the broken diagonals.

Trump mod 2, 3, 10, and 62 cubes

Modulo-2
This is a normal simple magic cube with 13 + 4 simple magic squares.
It is also diagonal magic modulo-2 with all 21 squares magic modulo 2!

```Top - I                     II                         III
34    6  115   35  120     84    3    7  105  111     78  108   96   23    5
85  123   26   45   31    110   54   49   55   42     15   56   76   57  106
8   91   20   94   97     81   74   73   72   10     80   63   62   61   44
95   37   25  112   41     22   58   64   59  107     18   67   48   68  109
88   53  124   24   21     13  121  117   19   40    119   16   28  101   46
IV                     Bottom - V
11  122   92   47   38    103   71    0  100   36
17   65   60   66  102     83   12   99   87   29
114   52   51   50   43     27   30  104   33  116
82   69   75   70   14     93   79   98    1   39
86    2   32   77  113      4  118    9   89   90```

From Walter Trump 2003-07-28 modulo-02.xls My Cube_5-Trump-6.xls

Modulo-3
This order 5 simple magic cube contains an order 3 magic cube surrounded by 6 orthogonal planes that are order 5 simple magic squares.
The three middle order 5 planes are also simple magic squares making a total of 9 order 5 magic squares. The three middle planes of the order 3 cube are also simple magic squares.
This magic cube is similar to a bordered magic square. The central order 3 magic cube consists of the numbers 50 to 76, with the lowest 49 and highest 49 numbers placed in the outside cells.
Bordered magic squares and cubes are also called concentric.

NOTE: An Inlaid magic cube does not have this limitation on which numbers appear in the borders!

Because the sums of each of the 30 planar diagonals is divisible by 3, all 15 orthogonal planes are magic modulo 3.
Therefore, this cube is diagonal magic modulo 3!

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

From Walter Trump 2003-06-21 S-concentric-5.xls. (My Cube_5-TrumpBordered.xls)

Modulo-10
This cube is a normal simple magic cube. The three interior orthogonal planes in each orientation are simple magic squares, because in each case the two diagonals sum correctly to 310.

However, because the sums of each diagonal of each of the other six planes is divisible by 10, the cube is diagonal magic modulo 10.

```Top - I                     II                         III
45   14   48   90  113     29  111  119   20   31    100   53  114    6   37
120   25   40   86   39     17   49   73   56  115      3   58   72   65  112
2   80   22  101  105     83   74   67   60   26    108   69   62   55   16
116   82   77    0   35     88   63   46   70   43     12   59   52   66  121
27  109  123   33   18     93   13    5  104   95     87   71   10  118   24
IV                          Bottom - V
30  117   28  103   32    106   15    1   91   97
81   54   78   61   36     89  124   47   42    8
98   64   57   50   41     19   23  102   44  122
9   68   51   75  107     85   38   84   99    4
92   7   96   21   94     11   110   76   34   79```

From Walter Trump 2003-07-30 modulo-10.xls. (My Cube_5-Trump-7.xls)

Modulo-62
This cube is not magic because some of the orthogonal rows or columns do not sum correctly.

However, sums of all orthogonal lines and all diagonals (and the four main triagonals) are divisible by 62, which qualifies this cube as a diagonal magic
cube modulo 62!
Reminder: If all 3m planar squares of a cube are simple magic (some may be pandiagonal magic), it is classed as a diagonal magic cube.

```Top - I                     II                         III
33   24  119   39   95     25   44   23  116   40    118   45    2   30  115
21  121   86   35   47     90   49   73   56   42    111   58   72   65    4
10   32  106   53  109     97   74   67   60   12     76   69   62   55   48
17   31  123   96   43     14   63   46   70  117    120   59   52   66   13
105  102    0   87   16     84   80  101    8   99      9   79  122   94    6
IV                          Bottom - V
26  113  104   88   41    108   22  124   37   19
7   54   78   61  110     81   28    1   93  107
112   64   57   50   27     15   71   18   92  114
82   68   51   75   34     77   89   38    3  103
83   11   20   36   98     29  100    5   85   91```

From Walter Trump 2003-09-05 modulo-62.xls. (My Cube_5-Trump-3.xls)

 Notice that both Boyer and Trump refer to these cubes as 'perfect magic' mod x. However, by the finer definitions of Hendricks, they are called 'diagonal magic' (mod x). These cube contain 3m simple magic squares (mod x). The following Boyer and de Winkel cubes contain 9m pandiagonal magic squares (mod 5)! Hendricks called this type perfect. To avoid confusion, they are now referred to as nasik (or nasik perfect) as per C Planck's 1905 definition.

Boyer mod 5 cube

This is one of two cubes Christian supplied me with. Both are associated and have exactly the same characteristics. These cubes use the numbers from 0 to 124
Their 25 rows, 25 columns, 25 pillars, and 4 triagonals are magic so this qualifies as a simple magic cube.
Because 22 diagonals and many broken diagonals also sum correctly, this cube contains 10 planar pandiagonal & 1 simple magic squares, and 4 oblique simple magic squares.

Because the 8 remaining diagonals, all broken diagonals, and all pantriagonals have sums that are multiples of 5, these cubes are perfect magic, modulo 5!
When considered modulo 5, these cubes each contain 3m orthogonal, 6 oblique, and 6m – 6 two-segment oblique pandiagonal order 5 magic squares!

```Top - I                     II                         III
70   91  112    8   29      70   91  112    8   29     113    9   25   71   92
107    3   49   65   86     107    3   49   65   86      45   66   87  108    4
44   60   81  102   23      44   60   81  102   23      82  103   24   40   61
76  122   18   39   55      76  122   18   39   55      19   35   56   77  123
13   34   50   96  117      13   34   50   96  117      51   97  118   14   30
IV                          Bottom - V
94  110    6   27   73       7   28   74   90  111
1   47   68   89  105      69   85  106    2   48
63   84  100   21   42     101   22   43   64   80
120   16   37   58   79      38   59   75  121   17
32   53   99  115   11      95  116   12   33   54```

Christian Boyer, Cube_5-22d.xls, June 10th 2003            (My Cube_5-Boyer-1.xls)

de Winkel mod 5 cube

 Order 5 Pantriagonal Magic Cube 2003 Not AssociatedI have modified this order 5 cube which Aale de Winkel denotes as KJ{[0,0,0].<2,0,2>,<2,2,0>,<0,-1,0>} from the format from my Cube_5-Aale-1.xls to match that of the first Trump cube, shown at the top of this page. This cube is a normal magic cube, because all rows, columns and pillars, as well as the 4 main triagonals, sum correctly. In this case, S = 315 (instead of 310) because, unlike the cubes by Trump and Boyer, shown above, this cube uses the more popular series 1 to 53. The cube contains no magic squares because no planar squares have correct diagonals and no oblique squares have both all rows and all columns summing correctly. It is a pantriagonal magic cube because all 4m2  triagonals (1, 2, and 3-segment) sum correctly. It is not associated. Because all lines of 5 numbers sum to a value divisible by 5, this cube is nasik perfect modulo 5. It contains 9m order 5 pandiagonal magic squares modulo 5. In the illustration, the red sums are the required sum for the normal magic cube. The blue sums are not required to be correct (for a simple magic cube). The black sums are for the broken diagonals. Click to enlarge

This cube is in my files as Cube_5-Aale-2.xls. It is a simple magic cube, but is associated. Because the cube is associated, the 3 central planes are associated magic squares.

Because all lines of 5 numbers sum to a value divisible by 5, this cube is nasik perfect modulo 5.
It contains 9m order 5 pandiagonal magic squares modulo 5.

```Top - I                     II                          III
40   16  122   78   59      72   28    9  115   91      84   65   41   22  103
121   77   58   39   20       8  114   95   71   27      45   21  102   83   64
57   38   19  125   76      94   75   26    7  113     101   82   63   44   25
18  124   80   56   37      30    6  112   93   74      62   43   24  105   81
79   60   36   17  123     111   92   73   29   10      23  104   85   61   42
IV                          bottom - V
116   97   53   34   15       3  109   90   66   47
52   33   14  120   96      89   70   46    2  108
13  119  100   51   32      50    1  107   88   69
99   55   31   12  118     106   87   68   49    5
35   11  117   98   54      67   48    4  110   86```
 This page was originally posted November 2003 It was last updated February 19, 2010 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz