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Hugel - 1876 - associated magic cube |
Contains 10 pandiagonal and 5 simple magic squares. |
Andrews 1908 - semi-pan- associated |
Contains 6 simple and 5 pandiagonal magic squares. |
Czepa - 1918 - semi-pan - associated |
Contains 5 simple magic squares. |
Weidemann - 1922 - simple - not assoc. |
Contains 3 simple magic squares. |
Leeflang - 1978 - semi-pan - associated |
Contains 5 simple and 10 pandiagonal magic squares. |
B&J - 1981 - pantriagonal associated |
Contains 3 simple magic squares. |
Soni - 2001 - pantriagonal - not assoc. |
Contains 1 simple magic square. |
Collison - 1990 - semi-pan - associated |
Contains 3 simple magic squares. Almost bimagic! |
Trump/Boyer Diagonal Order 5 |
This cube contains 21 order 5 simple magic squares! |
Theodor Hugel's Order 5 cube of 1876 is associated magic. The 5 horizontal and 5 vertical planes parallel to the front of the cube are all pandiagonal magic. The central vertical plane parallel to the sides of the cube, and 4 of the 6 oblique planes are simple magic squares. All pantriagonals in one of the 4 directions is correct.
This cube must be classed as 'simple', but the magic ratio (highest possible
is 3m2 monagonals + 6m diagonals + 4 triagonals) is 92.7%
The panmagic ratio (highest possible is 3m2 monagonals + 6m2
pandiagonals + 4m2 pantriagonals) is 67.4%
I - Top II III 93 121 62 4 35 12 29 85 118 71 110 68 21 37 79 52 9 45 98 111 95 123 61 2 34 11 27 84 120 73 50 88 101 57 19 51 7 44 100 113 94 125 63 1 32 106 67 24 40 78 49 90 103 56 17 53 6 42 99 115 14 30 83 116 72 108 66 22 39 80 47 89 105 58 16 IV V - Bottom 46 87 104 60 18 54 10 43 96 112 109 70 23 36 77 48 86 102 59 20 13 26 82 119 75 107 69 25 38 76 92 124 65 3 31 15 28 81 117 74 55 8 41 97 114 91 122 64 5 33
T. Hugel, Das Problem der magishen Systeme, 1876, Verlag von A. H. Gottschick, 70pp.
This second cube by W. S. Andrews has
only 3 of the planar squares magic. They are the center square of each of the x,
y and z planes, and they are associated. This will always be the case if the
cube is an odd order and is associated. All 15 planar squares have ALL
pandiagonals in one direction correct. Three of the 6 oblique squares are also
simple magic.
All associated (center-symmetric) magic hypercubes are semi-pantriagonal.
I -Top II III 1 82 38 119 75 33 114 70 21 77 65 16 97 28 109 74 5 81 37 118 76 32 113 69 25 108 64 20 96 27 117 73 4 85 36 24 80 31 112 68 26 107 63 19 100 40 116 72 3 84 67 23 79 35 111 99 30 106 62 18 83 39 120 71 2 115 66 22 78 34 17 98 29 110 61 IV V - Bottom 92 48 104 60 11 124 55 6 87 43 15 91 47 103 59 42 123 54 10 86 58 14 95 46 102 90 41 122 53 9 101 57 13 94 50 8 89 45 121 52 49 105 56 12 93 51 7 88 44 125
W. S. Andrews, Magic
Squares & Cubes, Open Court, 1908, page 76.
W. S. Andrews, Magic Squares & Cubes, 2nd edition,
Dover Publ. 1960 (originally 1917), page 76.
This magic cube is a different aspect of the Schubert cube of 1899.
Because it is an odd order associated cube, the 3 central planar squares are magic. Two of the oblique squares are also simple magic. The 5 horizontal planes and 5 vertical planes parallel to the front have all diagonals in one direction correct.
I - Top II III 95 21 52 108 39 64 120 46 77 8 33 89 20 71 102 104 35 86 17 73 98 4 60 111 42 67 123 29 85 11 13 69 125 26 82 107 38 94 25 51 76 7 63 119 50 47 78 9 65 116 16 72 103 34 90 115 41 97 3 59 56 112 43 99 5 30 81 12 68 124 24 55 106 37 93 IV V - Bottom 2 58 114 45 96 121 27 83 14 70 36 92 23 54 110 10 61 117 48 79 75 101 32 88 19 44 100 1 57 113 84 15 66 122 28 53 109 40 91 22 118 49 80 6 62 87 18 74 105 31
A. Czepa, Mathematische Spielereien (Mathematical Games), Union Deutsche, 1918, page 42.
This cube is the only order 5 simple cube I’ve seen that is not associated. 1 planar square (not a central plane) and 2 oblique squares are simple magic. All 10 planar squares and 4 oblique squares have all pandiagonals in 1 direction correct. 1 oblique square has all pandiagonals in both directions correct. All pantriagonals in 2 of the 4 directions are correct
I - Top II III 124 30 81 12 68 5 56 112 43 99 31 87 18 74 105 8 64 120 46 77 39 95 21 52 108 70 121 27 83 14 42 98 4 60 111 73 104 35 86 17 79 10 61 117 48 51 107 38 94 25 82 13 69 125 26 113 44 100 1 57 90 16 72 103 34 116 47 78 9 65 22 53 109 40 91 IV V 62 118 49 80 6 93 24 55 106 37 96 2 58 114 45 102 33 89 20 71 110 36 92 23 54 11 67 123 29 85 19 75 101 32 88 50 76 7 63 119 28 84 15 66 122 59 115 41 97 3
Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922, page 55.
This magic cube but is associated and so is semi-pantriagonal. All orthogonal planes in two directions are pandiagonal magic. Only the center plane in the third direction is magic (because the cube is associated), and it is not pandiagonal. Four of the oblique squares are simple magic. The 3 central orthogonal squares and the 4 oblique magic squares are associated, the other magic squares are not. The main triagonals are all magic so it qualifies as a magic cube. It is not pantriagonal magic because all the triagonals in only 1 of the 4 directions is correct.
Mention is made in this article by Leeflang, about the confusion over terminology for perfect magic cubes.
I - Top II III 87 118 24 30 56 66 97 103 9 40 50 51 82 113 19 5 31 62 93 124 109 15 41 72 78 88 119 25 26 57 68 99 105 6 37 47 53 84 115 16 1 32 63 94 125 106 12 43 74 80 90 116 22 28 59 69 100 101 7 38 49 55 81 112 18 3 34 65 91 122 107 13 44 75 76 IV V - Bottom 4 35 61 92 123 108 14 45 71 77 67 98 104 10 36 46 52 83 114 20 110 11 42 73 79 89 120 21 27 58 48 54 85 111 17 2 33 64 95 121 86 117 23 29 60 70 96 102 8 39
K. W. H. Leeflang, Magic Cubes of Prime Order, JRM 11:4, 1978-79, pp 241-257
A standard pantriagonal cube with no extra features except it is associated. Therefore the 3 central planes are associated magic squares. Both main diagonals of each planar square sum to the same (but not correct) value.
I -Top II III 110 86 67 48 4 14 120 96 52 33 43 24 105 81 62 89 70 46 2 108 118 99 55 31 12 22 103 84 65 41 68 49 5 106 87 97 53 34 15 116 101 82 63 44 25 47 3 109 90 66 51 32 13 119 100 85 61 42 23 104 1 107 88 69 50 35 11 117 98 54 64 45 21 102 83 IV V - Bottom 72 28 9 115 91 76 57 38 19 125 26 7 113 94 75 60 36 17 123 79 10 111 92 73 29 39 20 121 77 58 114 95 71 27 8 18 124 80 56 37 93 74 30 6 112 122 78 59 40 16
W. H. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, 0-486-24140-8, page 30.
This cube is not associated, and is the only one of the order-5 pantriagonal magic cubes I’ve seen that is not. Of course, any of the pantriagonal ones could be made non-associated simply by moving any exterior plane from one side of the cube to the other. One of the planar squares is simple magic. No other special features.
I - Top II III 111 49 82 20 53 29 87 25 58 116 92 5 63 121 34 54 112 50 83 16 117 30 88 21 59 35 93 1 64 122 17 55 113 46 84 60 118 26 89 22 123 31 94 2 65 85 18 51 114 47 23 56 119 27 90 61 124 32 95 3 48 81 19 52 115 86 24 57 120 28 4 62 125 33 91 IV V - Bottom 10 68 101 39 97 73 106 44 77 15 98 6 69 102 40 11 74 107 45 78 36 99 7 70 103 79 12 75 108 41 104 37 100 8 66 42 80 13 71 109 67 105 38 96 9 110 43 76 14 72
Abhinav Soni HyperMagicCube.exe program, obtainable from his magic cubes site.
ALMOST bimagic! The squares of the numbers do not form a magic cube. But the total of the 25 cells in each orthogonal plane (and 3 of the 6 oblique planes) sum to the same value (131775).
This cube is associated. It is not pantriagonal because only the pantriagonals in 3 of the 4 directions are correct. Three central orthogonal planes and 1 oblique plane are simple magic squares.
I - Top II III 27 66 85 124 13 58 97 111 5 44 89 103 17 31 75 65 79 118 7 46 91 110 24 38 52 122 11 30 69 83 98 112 1 45 59 104 18 32 71 90 10 49 63 77 116 106 25 39 53 92 12 26 70 84 123 43 57 96 115 4 19 33 72 86 105 50 64 78 117 6 51 95 109 23 37 IV V - Bottom 120 9 48 62 76 21 40 54 93 107 3 42 56 100 114 34 73 87 101 20 36 55 94 108 22 67 81 125 14 28 74 88 102 16 35 80 119 8 47 61 82 121 15 29 68 113 2 41 60 99
As a check of the semi-pantriagonal property, one of the four opposite short triagonals is 65 + 97 + 29 +61 + 63 = 315.
John R. Hendricks, Magic Square Course, self-published, 1991, page 411.
On September 1, 2003, I received from Walter Trump of Germany, by email, an order 6 diagonal magic cube.
A diagonal cube has the additional characteristic that all
planar arrays have diagonals that sum correctly. This means that a diagonal
magic cube has 3m orthogonal simple magic squares. I describe this cube
here.
Walter discusses this cube
here.
Until I received this cube, I had seen only two order 8 cubes, and one order 12
cube of this type.
Two days later, I received another email from Walter
announcing an order 7 cube of this type.
The same day (September 3, 2003), I received an email from Christian Boyer of
France with an order 9 cube of this type.
Walter then started searching in earnest for an order 5 cube of this type. Periodically he received encouragement from Christian, and also suggestions for improvements in the search routines. By early November there were five computers involved in the search; including Christian Boyer’s and one belonging to Walter’s son, who lived next door to him.
During this time, I am sorry to say, I was being quite
negative about the possibility of such a cube existing.
On November the 12, I received an email from Walter conceding that I might
indeed be correct and a solution to the order 5 diagonal (he called it ‘perfect')
cube was beginning to seem unlikely.
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Little did he realize that the computer next door
(his son’s) had already found a solution two hours earlier! As an amusing point of interest Walter mentioned on November 14 that his son, and Christian Boyer, had each found another solution while his computer was still searching! Congratulations Walter Trump and Christian Boyer on this important discovery. This order 5 cube is magic because all rows, columns, pillars, and the 4 triagonals sum correctly to 315. It is ‘diagonal’ magic because all 30 planar diagonals also sum to 315. This means that the 5 planes in each of the 3 orientations are simple magic squares. Because the rows and columns of the 6 oblique arrays sum to 315, these arrays are also order 5 simple magic squares. This cube is not associated i.e. center symmetric. However, the 3 central magic squares are. The discovery of this cube caused some interest in the mathematics
world.! Within two years, articles about it had appeared in over 25
publications. See a write-up on Christian Boyer's page at
www.multimagie.com/index.htm (click on Perfect Magic
Cubes). |
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Diagonal or Perfect? Both Christian and Walter refer to these cubes as
‘perfect’. This is an old, but commonly used definition for any magic cube
that was a little bit out of the ordinary. It was further popularized
by Martin Gardner in his Scientific American column in January 1976. Because of these problems, a new coordinated set of definitions
was developed during the 1990’s. |
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Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 14, 2009
Copyright © 2003 by Harvey D. Heinz
