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).

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. 

The first 3 cubes are simple magic. All
are diagonal magic mod 2, 3, 10 or 62 respectively.
One of the order5 cubes is a bordered (concentric) magic 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! 

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 modulo31.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
Modulo2
This is a normal simple magic cube with 13 + 4 simple magic squares.
It is also diagonal magic modulo2 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 20030728 modulo02.xls My
Cube_5Trump6.xls
Modulo3
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 20030621
Sconcentric5.xls. (My Cube_5TrumpBordered.xls)
Modulo10
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 20030730 modulo10.xls.
(My Cube_5Trump7.xls)
Modulo62
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 20030905
modulo62.xls. (My Cube_5Trump3.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 twosegment 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_522d.xls, June 10th
2003 (My Cube_5Boyer1.xls)
de Winkel
mod 5 cube
Order 5 Pantriagonal Magic Cube 2003 Not
Associated I 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_5Aale1.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 4m^{2}
triagonals (1, 2, and 3segment) 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_5Aale2.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
