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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
http://www.magic-squares.net/order4list.htm and
http://www.magic-squares.net/transform.htm
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.
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
8
10 15 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.
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 Red
numbers are a bent-diagonal in the square, bent-triagonal in the cube. |
 |
 |
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
This is a bent-triagonal, semi-pantriagonal magic cube. It is equivalent to the group II magic squares.
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.
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 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.
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.
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 the the
semi-pandiagonal squares. They simply have the same Dudeney pattern.
| A semi-pandiagonal magic square
1
4
14 15
Blue shows an
opposite short diagonal pair. |
 |
 |
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.
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
4
15
Blue shows an
opposite short diagonal pair. |
Note that sometimes a square
is rotated 90 degrees from the pattern.
|
 |
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
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
Blue shows an
opposite short diagonal pair. |
 |
 |
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)
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
Blue shows an
opposite short diagonal pair. |
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 ???
|
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
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.
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 |
|
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Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 14, 2009
Copyright © 2003 by Harvey D. Heinz
