Magic Cubes - Groups
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Magic Cubes - Groups
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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. 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
* 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.
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. 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. 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. 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 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. 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.
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,
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. 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. 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. 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.
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 Transform a Group IV magic square to a Group V magic
square by swapping rows and columns 2 and 3.
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.
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.
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). 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. 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
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.
Group ??? - Simple magic cube
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. So far I have found no magic cubes that correspond to
magic square groups 7 to 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. 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. Pattern 1 (Group I)
Pattern 2 (Group II)
Pattern 3 (Group III)
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