Magic Cubes - Order 5

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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!

Hugel - 1876 - associated magic cube

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 3m
2 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.

Andrews 1908 - semi-pan- associated

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.

Czepa - 1918 - semi-pan - associated

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.

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.     

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.

Leeflang - 1978 - semi-pan - associated

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

B&J - 1981 - pantriagonal associated

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.

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.      

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 Geocities magic cubes site. (Sorry. No longer available)

Collison - 1990 - semi-pan - associated

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.

Trump/Boyer Diagonal Order 5

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. 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.

Little did he realize that the computer next door (his son’s) had already found a solution two hours earlier!
He presented the cube shown here by email on November 13, 2003.

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).
 

Click to enlarge

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.
This term is very ambiguous because no differentiation is made for the type of magic squares (or even whether or not the oblique squares are considered.

Because of these problems, a new coordinated set of definitions was developed during the 1990’s.
I am using a term from this set when I refer to this cube as ‘diagonal’.

For more information see my Perfect and Perfect-2 pages.

This page was originally posted December 2002
It was last updated October 16, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz