Magic Cubes - Order 5
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Hugel - 1876 - associated magic cubeTheodor 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% 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.
Andrews 1908 - semi-pan- associatedThis 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. 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. Czepa - 1918 - semi-pan - associatedThis 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.
Weidemann - 1922 - simple - not assoc.This cube is the only order 5 simple cube I’ve seen
to date 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.
Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922, page 55.
Leeflang - 1978 - semi-pan - associatedThis 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.
B&J - 1981 - pantriagonal associatedA 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.
Soni - 2001 - pantriagonal - not assoc.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.
Abhinav Soni HyperMagicCube.exe program, obtainable from his Geocities magic cubes site. (Sorry. No longer available)
Collison - 1990 - semi-pan - associatedALMOST 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.
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. Trump/Boyer Diagonal Order 5On 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.
Walter discusses this cube
here. Two days later, I received another email from Walter
announcing an order 7 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. Little did he realize that the computer next door
(his son’s) had already found a solution two hours earlier!
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