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This page will show a comparison between Dudeney Groups I, II and III order 4 magic squares and their equivalent in order 4 magic cubes. Compared will be both the complement pair pattern and the count of basic squares/cubes.
I will then show examples of cubes equivalent to the groups IV, V and VI semi-pandiagonal magic squares. And finally a complimentary pair cube pattern that has no equivalence in magic squares. Comparisons of the other six groups must await further developments.

A special thanks to Walter Trump of Germany who performed most of the preliminary investigations on the relationship of order 4 cubes to Dudeney patterns. Hopefully this page will serve as inspiration for others to explore this subject further.

For more information on magic square groups, refer to my pages
Order-4 magic squares and Transformations and Patterns

Order-4 Magic Squares Order-4 Magic Cubes

Group I - Pandiagonal *

Group I - Pantriagonal *

Group II - Bent diagonals -  Semi-pandiagonal

Group II - Bent triagonals - Semi-pantriagonal

Group III - Associated - Semi-pandiagonal

Group III - Associated - Semi-pantriagonal

How many each of groups I, II and III?

An investigation of order 4 magic cubes by Walter Trump.

Group IV - Semi-pandiagonal

Group IV - Semi-pantriagonal

Group V - Semi-pandiagonal

Group V - Semi-pantriagonal

Group VI - Semi-pandiagonal

Group VI - Semi-pantriagonal

Group VI - Simple magic

Group VI - Simple magic

Group ??? Simple magic cube

This cube has no equivalent pattern in the magic squares.

Groups I - III Other Relations

The horizontal planes of these cubes have Group I - III patterns.

* Pantriagonal in magic cubes is the equivalent classification to pandiagonal in magic squares. Pandiagonal in magic cubes is a much higher classification! See my Perfect Cubes page for more information.
Pantriagonals are the broken triagonals parallel to the 4 main triagonals (space diagonals) of a magic cube.

 In 1910 H. E. Dudeney [1] described a method of classifying the 880 basic order 4 magic squares by using complementary pair diagrams.

Here I show the relationship between six of these twelve groups of order 4 magic squares and how there is an equivalence in magic cubes.
Walter Trump took advantage of that fact when he counted the number of Associated order 4 cubes. He immediately knew that the count was the same for the order 4 pantriagonal cubes and one group (the bent triagonals) of the order 4 semi-pantriagonal cubes.

The complement pair (Dudeney) diagram for the square is easy to comprehend at a glance. The diagram for the cube is more difficult because of the increased complexity. Walter Trump came up with the method used here of showing it.

 [1] Mentioned on page 120 of H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, 486-20473-1 (reprint of 1917), pp120,121.

 

 Group I - Pandiagonal/Pantriagonal

The magic square shown below is pandiagonal magic so is Group I.
Because all order 4 pandiagonal magic squares are ‘most-perfect, all 2 x 2 squares sum to S. Also, all pairs of integers distant ½n along a pandiagonal sum to T.

The image to the right of it is the complementary pair diagram. The two dots at the end of each line represent numbers that together sum to 17, which is equal to the sum of the first and last numbers in the series.
The larger image on the right is the equivalent diagram for an order 4 cube. In this case, the two numbers representing a complement pair are shown with the same letter.

By swapping rows and columns 1 and 4, the magic square is transformed to a disguised version of the following group II bent diagonal semi-pandiagonal magic square (next section).

A pandiagonal square

 1   10  15
12  13   3   6
 7   2  16   9
14  11   5   4

Colors indicate some
complement pairs.

 

A pantriagonal cube 

62   4  57   7    19  45  24  42    14  52   9  55    35  29  40  26
 1  63   6  60    48  18  43  21    49  15  54  12    32  34  27  37
56  10  51  13    25  39  30  36     8  58   3  61    41  23  46  20
11  53  16  50    38  28  33  31    59   5  64   2    22  44  17  47

This cube is pantriagonal and is equivalent to the Group I magic squares.
It may be transformed into the bent-triagonal semi-pantriagonal magic cube in the next section by exchanging horizontal planes 2 and 3, vertical planes parallel to front 2 and 3, and vertical planes parallel to sides 2 and 3. In other words, exchange the two members of each pair of outside planes.
the three pairs of inside planes could be exchanged instead (2 sets of inside lines for the square) but a different cube (square) in group II would be formed.
The transformation works in both directions, groups I to II and II to I.

NOTE: Unlike order 4 pandiagonal magic squares, not all order 4 pantriagonal magic cubes belong to Group I. See Guenter Stertenbrink’s Closed Knight Tour pantriagonal cube. In his cube, not all 2 x 2 squares sum to S. Also, not all pairs of integers distant ½n along a pantriagonal sum to T.

 Group II - Semi-pandiagonal/Semi-pantriagonal - Bent diagonals/triagonals

This is a group II square so is bent-diagonal and semi-pandiagonal. This semi-pantriagonal magic cube is also bent-triagonal. They were transformed from group I by exchanging outside lines (the square) and outside planes (the cube).

Notice that each of the opposite short diagonals, of both the square and the cube, sum to S/2. This is one type of semi-pantriagonal magic cube and appears only in even orders.

Red numbers are a bent-diagonal in the square, bent-triagonal in the cube.
Blue numbers are opposite short diagonals in the square, opposite short triagonals in the cube.

Bent-diagonal and semi-pandiagonal

 4  11   5  14
 6  13   3  12
 9   2  16   7
15   8  10   1

Red numbers are a bent-diagonal in the square, bent-triagonal in the cube.
Blue numbers are opposite short diagonals in the square, opposite short triagonals in the cube.

This is a bent-triagonal, semi-pantriagonal magic cube. It is equivalent to the group II magic squares.

47  44  17  22    31  28  33  38     2   5  64  59    50  53  16  11
37  34  27  32    21  18  43  48    12  15  54  49    60  63   6   1
20  23  46  41    36  39  30  25    61  58   3   8    13  10  51  56
26  29  40  35    42  45  24  19    55  52   9  14     7   4  57  62

These magic figures are converted to Group III by exchanging lines 1 and 3 for the square, planes 1 and 3 for the cube. The results, however, would be a different square and cube then those shown in the next section. They were generated directly from the group I square and cube in section 1 by exchanging lines and planes 3 and 4.

 Group III - Semi-pandiagonal/Semi-pantriagonal - Associated

Transform the group I pandiagonal square to the following group III associated semi-pandiagonal square by exchanging rows 3 and 4, and columns 3 and 4.

From the group I pantriagonal cube, exchange horizontal planes 3 and 4,
Exchange vertical planes parallel to front 3 and 4,
Exchange vertical planes parallel to sides 3 and 4, to obtain this Group III cube.

 

Associated and semi-pandiagonal.

 1   8  15  10
12  13   6   3
14  11   4   5
 7   2   9  16

Blue numbers are an opposite short diagonal pair. Other colors are complement pairs.

 1  63  60   6    48  18  21  43    32  34  37  27    49  15  12  54
62   4   7  57    19  45  42  24    35  29  26  40    14  52  55   9
56  10  13  51    25  39  36  30    41  23  20  46     8  58  61   3
11  53  50  16    38  28  31  33    22  44  47  17    59   5   2  64

This cube is also associated and semi-pantriagonal. This is equivalent to the Group III magic squares

Notice that each of the opposite short diagonals, of both the square and the cube, do not sum to S/2. The two together do sum to S, as required for a semi-pan square or cube.
For example, the square 12 + 8 = 20, 9 + 5 = 14, 20 + 14 = S.
This type of semi-pantriagonal (and semi-pandiagonal for squares) is by far the most common. Of a total of 880 order 4 squares, 384 of these are semi-pandiagonal, and only 48 of the 384 are bent –diagonal (i.e. group II). The other 336 are of this type. Semi-pandiagonal cubes may be any order except order 3.

This associated (group III) cube is the same as Andrews cube of 1908.

 How many each of groups I, II and III?

 The above 3 cubes and the transformations between them were found by Walter Trump, although he started with the associated cube and ended up with the pantriagonal one.

I rewrote them in this order to be more consistent with my Transform.htm (order 4 magic squares) page.
I produced the transformations on my Cube_4_Transform.xls spreadsheet.

Walter found that these first three groups were isomorphic. He physically counted the associated cubes of order 4 that contained the number 1 in the lower left corner, plus other restrictions, and arrived at the figure 69,489,200. He then multiplied this number by 64, to compensate for the restrictions he used (not the fact that 1 is 1/64 of the series). The conclusion: there are 4,447,308,800 different associated (center symmetric) magic cubes of order 4. Because of the isomorphism, there are identical numbers of pantriagonal and bent-triagonal order 4 magic cubes.
The 48 aspects of each cube due to rotations and reflections, was not included in the count, so each group has 4,447,308,800 x 48 apparently different cubes.

There are 48 basic magic squares for each of groups I, II and III. Each has 8 aspects due to rotations and reflections. 

 Group IV - Semi-pandiagonal/Semi-pantriagonal

This group cannot be reached by row and column transformations from groups I to III. It is part of another isomorphic set of 3 groups, IV, V and VI, which may be transformed one to the other.
Each of these groups consist of 96 semi-pandiagonal magic squares.
Group VI also includes 208 simple magic squares that are not isomorphic with the the semi-pandiagonal squares. They simply have the same Dudeney pattern.

A semi-pandiagonal magic square

 1   14  15
16  13   3   2
 7   6  12   9
10  11   5   8

Blue shows an opposite short diagonal pair.
Other colors represent some complement pairs.

45   8  26  51    20  57  39  14    34  11  21  64    31  54  44   1
 3  42  56  29    62  23   9  36    16  37  59  18    49  28   6  47
60  17  15  38     5  48  50  27    55  30   4  41    10  35  61  24
22  63  33  12    43   2  32  53    25  52  46   7    40  13  19  58

This cube is semi-pantriagonal and is equivalent to the Group IV magic squares
Notice that each of the opposite short diagonals, of both the square and the cube, do not sum to S/2. The two together do sum to S, as required for a semi-pan square or cube. This is common to all groups III, IV, V, and VI semi-pan squares and cubes.

Transform a Group IV magic square to a Group V magic square by swapping rows and columns 2 and 3.
From a Group IV cube, exchange horizontal planes 2 and 3,
Exchange vertical planes parallel to front 2 and 3
Exchange vertical planes parallel to sides 2 and 3 to obtain this Group V cube.

 Group V - Semi-pandiagonal/Semi-pantriagonal

 This group V square and cube were obtained from the group IV square and cube by exchanging lines and planes 2 and 3.

A semi-pandiagonal magic square

 1  14   15
 7  12   6   9
16   13   2
10   11   8

Blue shows an opposite short diagonal pair.
Other colors represent some complement pairs.

45  26   8  51    34  21  11  64    20  39  57  14    31  44  54   1
60  15  17  38    55   4  30  41     5  50  48  27    10  61  35  24
 3  56  42  29    16  59  37  18    62   9  23  36    49   6  28  47
22  33  63  12    25  46  52   7    43  32   2  53    40  19  13  58

 Group VI - Semi-pandiagonal/Semi-pantriagonal

This square and cube was obtained from the Group IV objects by exchanges of lines (the square) and planes (the cube) 2 and 4.

A semi-pandiagonal magic square

 1  15  14   4
10   8   5  11
 7   9  12   6
16   2   3  13

Blue shows an opposite short diagonal pair.
Other colors represent some complement pairs.

45  51  26   8    31   1  44  54    34  64  21  11    20  14  39  57
22  12  33  63    40  58  19  13    25   7  46  52    43  53  32   2
60  38  15  17    10  24  61  35    55  41   4  30     5  27  50  48
 3  29  56  42    49  47   6  28    16  18  59  37    62  36   9  23

This cube is semi-pantriagonal, not associated (and not bent-triagonal).
This cube was originally adapted from G. Pfeffermann’s order 8 bimagic square in 1891 by A. Huber. [1]
Actually, because these transformations work in both directions, I was able to start with this cube, generate the group IV cube, then the group V cube.

I produced the transformations on my Cube_4_Transform-2.xls spreadsheet.

[1] Revue des Jeux, July 10, 1891, Paris (Brought to my attention by Christian Boyer)

Group VI - Simple magic

There are 208 simple magic squares in this group and 96 semi-pandiagonal squares.
Only the semi-pandiagonal squares can be obtained by row/column exchange from the groups 4 and 6 squares.
Likewise for the cubes. Only the semi-pantriagonal cubes can be obtained from plane exchanges for groups 4 and 5.

The group 6 simple cubes cannot be reached by a simple direct transformation from any other group. However, because both types of squares or cubes appear in the same group, the complementary pair pattern in both cases is the same .

Because there are exactly twice as many of each of the groups 4, 5 and 6 (semi-pandiagonal squares as there are for groups 1, 2, and 3, does that mean that there are 96/48 x 4,447,308,800 basic group VI, semi-pantriagonal, and
208/48 x 4,447,308,800 basic group VI, simple magic cubes?

A simple magic square

 1   2  15  16
12  14   3   5
13   7  10   4
 8  11   6   9

Blue shows an opposite short diagonal pair.
Other colors represent some complement pairs.

Notice that these complement pair diagrams are exactly the same as for the Group VI semi-pan squares and cubes.

Within a Dudeney group, some of the basic squares may be rotated 90 degrees from the complementary pair pattern. This has nothing to do with the fact that this particular square is simple. Presumably the same situation applies for the basic (normalized) cubes.

In these examples, I have paid no attention to whether the cube is normalized or not. I have simply matched it to the pattern analogous to the Dudeney pattern for the square. Also, after a square (or cube) is transformed from another normalized object, it will likely be rotated or reflected.

 1   5  61  63    40  15  44  31    25  50  21  34    64  60   4   2
14  58  52   6    18  55  27  30    47  10  38  35    51   7  13  59
62  43   9  16    46  37  11  36    19  28  54  29     3  22  56  49
53  24   8  45    26  23  48  33    39  42  17  32    12  41  57  20

This is a simple magic cube. Because 14 + 15 + 17 + 49, the opposite short triagonal indicated above, does not equal S = 130, the cube is not semi-pantriagonal. And because broken triagonals, such as 14 + 37 + 17 + 2 also do not equal S, the cube is not pantriagonal.

This is a plane symmetrical cube, one of 4 types of symmetry in order 4 magic cubes.
There are billions of cubes with this symmetry.
This is Walter Trumps # 4 from his Symmetrical Magic Cubes.htm paper of March 3, 2003.

Group ??? - Simple magic cube

This is Walter Trumps # 3 from his Symmetrical Magic Cubes.xls of Mar. 2, 2003.

It is diagonally symmetric, so no surprise that there is not an equivalent group among the order 4 magic squares, because they do not have such symmetry.

There is no equivalent magic square or Dudeney pattern.

There are probably a lot of order 4 cube patterns waiting to be discovered, that have no equivalent in the order 4 magic squares.

 1   4  62  63    23  31  44  32    42  40  18  30    64  55   6   5
10  58  45  17    25  53  36  16    34  12  38  46    61   7  11  51
59  54   8   9    47  27  13  43    21  29  52  28     3  20  57  50
60  14  15  41    35  19  37  39    33  49  22  26     2  48  56  24

Conclusion

In discussing the order 4 magic squares, I have referred to 'basic' and 'disguised' versions. These same terms may be used in reference to magic cubes, but do not have the same relevance. Defining one of the 8 aspects (due to rotation and reflect) of a square as 'basic' permits comparison of squares in a list.

A magic cube has 48 aspects with one of these defined as basic. However, no one has compiled a list of magic cubes and I doubt if anyone ever will. After all, Walter Trump has determined that there are 4,447,308,800 basic cubes for each of groups I, II and III while order 4 magic squares have 48 for each of these groups.
There are a total of 12 groups for these squares and a total of 880 basic squares. We now know that there are more (maybe many more) groups of order 4 magic cubes, so the total number of basic cubes of this order is very high!

So far I have found no magic cubes that correspond to magic square groups 7 to 12.
When one is found that fit within group 7, 8, 9, or 10, the other three will be available by swapping planes. Likewise for groups 11 and 12.

Groups I - III Other Relations

The following 3 cubes were sent to me by Walter Trump on May 6, 2003.

All four planes of each cube have the same complementary pair pattern.
The cube itself does NOT belong to that Dudeney group, however.

For example, the third group has all 4 horizontal planes with the Group III pattern which is the associated (center-symmetric) pattern. However, the cube itself is NOT associated.
All 3 cubes shown here are semi-pantriagonal and none of them are associated.
No cube of this type can belong to a Dudeney Group as the first seven cubes shown on this page do.

Pattern 1 (Group I)

 1  62  43  24    61   7  18  44    52  10  31  37    16  51  38  25
59   8  17  46     2  60  45  23    15  53  36  26    54   9  32  35
22  41  64   3    47  21   4  58    34  28  13  55    27  40  49  14
48  19   6  57    20  42  63   5    29  39  50  12    33  30  11  56

Pattern 2 (Group II)

 1  43  62  24    52  31  10  37    61  18   7  44    16  38  51  25
22  64  41   3    34  13  28  55    47   4  21  58    27  49  40  14
59  17   8  46    15  36  53  26     2  45  60  23    54  32   9  35
48   6  19  57    29  50  39  12    20  63  42   5    33  11  30  56

Pattern 3 (Group III)

 1  24  62  43    16  25  51  38    61  44   7  18    52  37  10  31
48  57  19   6    33  56  30  11    20   5  42  63    29  12  39  50
59  46   8  17    54  35   9  32     2  23  60  45    15  26  53  36
22   3  41  64    27  14  40  49    47  58  21   4    34  55  28  13

This page was originally posted March 2003
It was last updated October 19, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz