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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.
Christian Boyer, France, then demonstrated two order 5
simple magic cubes. When the sums were considered mod 5, these cubes were
perfect! In other words, each cube contained 9m order 5 pandiagonal
magic squares!
Aale de Winkel, The Netherlands, then showed an order 5
pantriagonal magic cube. If all sums were considered modulo 5, this cube was
also perfect! All 21 squares contained in these last 3 cubes are
pandiagonal magic
when considered mod 5.)
I will show Trump’s mod 31 and de Winkel's mod 5 cubes in detail. The other cubes will just be listed.
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 perfect! |
de Winkel mod 5 cube |
This cube is pantriagonal magic. When the sums are considered mod 5, the cube is perfect! |
| For simpler computations, Walter
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. |
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Modulo-2 From Walter Trump 2003-07-28 modulo-02.xls My Cube_5-Trump-6.xls
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
Modulo-3 From Walter Trump 2003-06-21 S-concentric-5.xls. (My Cube_5-TrumpBordered.xls)
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 o f 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
Modulo-10 From Walter Trump 2003-07-30 modulo-10.xls. (My Cube_5-Trump-7.xls)
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
Modulo-62 From Walter Trump 2003-09-05 modulo-62.xls. (My Cube_5-Trump-3.xls)
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
| 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'. These cube contain 3m simple
magic squares (mod x). The following Boyer and de Winkel cubes contain 9m pandiagonal magic squares (mod 5)! |
Christian Boyer, Cube_5-22d.xls, June 10th 2003 (My Cube_5-Boyer-1.xls)
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
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_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. Because all lines of 5 numbers sum to a value divisible by 5, this cube is '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. |
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Order 5 Simple Magic Cube 2003 Associated
This cube is in my files as Cube_5-Aale-2.xls. It is a simple magic cube, but 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 '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
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Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 15, 2009
Copyright © 2003 by Harvey D. Heinz