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Contents
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A brief
introduction this page and it's contents. |
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The 880
order-4 magic squares may be classified into 12 groups. |
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A method for
indexing and enumerating. Explanation of sequence patterns. |
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Examples and
a more in-depth discussion. |
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A method of
transforming one magic square into another one. Any order. |
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Several of
these methods will always produce a square in the same or a different
group. |
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Shown here is
one of the 3 sets of 16 magic squares related by cyclic row and column
changes. |
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15 methods to
transform one order-4 associated magic square into another one. |
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I and III by
swapping quadrants and rows. Between groups with like number of magic
squares by row and column permutation |
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Supportive Pages |
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Includes a
table listing and comparing 32 different order-4 transformations. |
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A summary of
more then 45 order-4 transformations. Also included on this page - 12
binary digit transformations. |
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His base-4
digit manipulation transformations. Also a 4 magic square loop. |
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Dudeney group
patterns. Groups I, II, III, XI and XII in magic square format.. All
880 magic squares in index order, in a tabular list format. |
Introduction
On this page I will discuss ways of transforming one magic square into
another one. Also I will discuss various patterns and the use of patterns
for classification.
The magic squares discussed will all be order-4 because;
1. The only order for which all magic squares are known (except for the
one order-3).
2. There are a suitably large number of magic squares to work with.
3. The small size makes it quicker to form the magic squares and easier to
see the patterns.
Methods described here may be extended to work with higher orders.
The transformations discussed will be from pure magic squares (those
using numbers from 1 to 16) to pure magic squares.
However, it is worth mentioning here that any magic square may be
converted to another magic square (not pure) simply by applying any
constant to each number in the square using any arithmetic operator. In
many cases, if a constant is applied to the units digit of each number,
and a different constant applied to the ten's digit of each number, the
result will also be a magic square.
The transformation principles may be extended also to work with higher
dimensions (cubes, tesseracts, etc).
Two sources for the complete set of order-4 magic squares are:
W. H. Benson & O. Jacoby, New Recreations With Magic Squares, Dover
Publications, Inc, 1976, 0-486-23236-0
Matsumi
Suzuki's Magic Squares web page are now found in the MathForuml
(Note however that Suzuki's list is not normalized or in index order.)
Since this page was originally written, I have written and ran a program
to find all order-4 magic squares.
The various lists may be found from my Order-4
Lists page.
Order-4 Groups
There are 880 basic magic squares of order-4. The complete set was
published in 1693. (Frénicle de Bessey, Des Quarrez Magiques. Acad. R.
des Sciences).
They were classified into 12 groups by H. E. Dudeney and first published
as such in The Queen, Jan. 15, 1910. It (the classes) appeared
later in Amusements in Mathematics, 1917 published by Thomas Nelson &
Sons, Ltd.
Group I |
Group II |
Group III |
Group IV |
Group V |
Group VI |
Group VII |
Group VIII |
Group IX |
Group X |
Group XI |
Group XII |
The 12 groups
The 12 groups are classified by the patterns formed by the 8 complement
pairs.
A complement pair is two numbers that together sum to n2 + 1.
For order-4 that number is 17.
| Group |
Characteristic |
# of basic magic
squares |
| I |
Pandiagonal |
48 |
| II |
Semi-pandiagonal |
48 |
| III |
Semi-pandiagonal & Associative |
48 |
| IV |
Semi-pandiagonal |
96 |
| V |
Semi-pandiagonal |
96 |
| VI |
Semi-pandiagonal |
96 |
| VI |
Simple |
208 |
| VII |
Simple |
56 |
| VIII |
Simple |
56 |
| IX |
Simple |
56 |
| X |
Simple |
56 |
| XI |
Simple |
8 |
| XII |
Simple |
8 |
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Some comments
- Group VI has some semi-pandiagonal and some simple magic squares.
- All groups are vertically symmetrical and all except XI and XII are
horizontally symmetrical.
- Group III has all complement pairs symmetrical around the center of
the magic square.
Be aware that for some of the normalized magic squares and in some of
the transformations to follow, the Dudeney pattern will be rotated from
that shown here.
Similar patterns may be formed for higher orders as well, although
Dudeney only published those for order-4. |
Normalized position
& Magic lines
Any magic square can appear in 8 different aspects due to rotations and
reflections. For enumeration purposes, one of these 8 squares must be
designated as the fundamental or basic one. The other 7 are often
referred to as disguised versions of this one.
Here we use Frénicle's # 175 as an example to illustrate how one
fundamental magic square can also appear in 7 disguised versions.
All 8 aspects are Dudeney group III so are associated magic squares.
Note that whether the magic square is rotated right or left is arbitrary.
Here I use right (clockwise) rotation.
H. may also be considered as reflected around the leading
diagonal and I. as being reflected around the right
diagonal .
I. is the magic square from Albrecht Dürer’s engraving,
Melancholia, 1514.
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The magic line pattern at J. is
constructed by following the magic square numbers of A.
in sequence. To the left is the same diagram with some of the areas
filled in. This may be done in various ways to produce artistic
designs.
As well as making pretty patterns, these are sometimes used for
classification purposes.
Jim Moran has a lot of material on these in The Wonders
of Magic Squares, Random House, 1982, 0-394-74798-4.
He refers to these as sequence designs. |
Normalized position
For enumerating and listing the magic squares of a given order, one of the
eight above positions must be designated the normal position. This
standard prevents confusion and permits easy comparison. Frénicle established
two simple rules to determine the standard position for order-4. These same
rules may be used for all other orders as well.
- The smallest number in any corner of the magic square must be in the top
left corner. If it is not, rotate the square until it is.
- The second number in the top row must be of lower then the first number in
the second row. These are the two numbers adjacent to the one in the top left
corner. Reflect the square to obtain this condition.
Notice that magic square A. (# 175 above) and # 727 (below)
satisfy these two conditions.
| I put a lot of emphasis on the use of index numbers. Otherwise, in
comparing lists or tables of magic squares published by different authors,
the casual observer is liable to think that the lists are not the same.
Magic squares obtained by transformations are usually not normalized. I
generally indicate them as disguised and give the index numbers. |
More on Magic Lines
Normally magic lines are drawn between the centers of the cells.
However, this sometimes results in a longer line hiding a shorter one.
Shown here are two versions of magic line diagrams for magic square #727
(Group VIII).
For interest, each is also shown with some areas filled in, forming a type
of abstract art.
In the first line diagram, notice that the top line from 12 to 13 is
covered up by the line from 4 to 5. In fact, it looks like the line goes
from 4 to 12 to 13 to 5.
Likewise the left line looks like it is going from 14 to 1 to 15.
These concerns are resolved by offsetting the necessary lines as I did
with the second line diagram.
Which method is actually used is a matter of personal preference. The
first method looks tidier because all points appear as on a regular grid
pattern. The second method gives a truer picture.
Complementing
Any order of magic square may be transformed to another magic square
simply by subtracting each number in turn from n2 + 1.
The resulting magic square is a disguised version of another magic square
belonging to the same group.
A.
#202 XI |
B.
Complement of A. |
C.
B. Normalized = #724 |
D.
Group XI |
E.
Line pat. for A. & B. |
The resulting magic line diagram will be exactly the same for the original
and the complement magic squares. You simply move around it in the opposite
direction. However, it requires rotation for the normalized version of the
complement.
This may be considered complementing the magic square. Robert Sery refers to the
process as complementary pair interchange (CPI).
When complementing some magic squares you end up with the original square
except that it is rotated or reflected. Any order of associated (group III)
magic square is of this type and the complement will be rotated 180 degrees. Any
magic square with a complementary pair pattern like that of group VI also has
this self-similar pattern with the complementary reflected either vertically or
horizontally, depending on the orientation of the pattern. See my
Self-similar page for more information on this
type of magic square.
Swapping rows and columns
Complementing each number in a magic square will ALWAYS produce a magic
square in the same group.
The following row and column swaps will also always produce an order-4
magic square.
Swap rows and columns 1 and 4
Swap rows and columns 2 and 3.
Change row and column orders to 2, 1, 4, 3.
Change row and column orders to 3, 1, 4, 2.
However, the resulting magic square is likely to belong to a different
group. See the bold lines in the Summary table.
Pandiagonal m. s.
Transformations
A. # 204 is the pandiagonal magic square
example used in this section.
B. is it's complement, # 744 (disguised). All pandiagonal
magic squares create new pandiagonals when complemented.
C. is the magic square obtained by adding 8 mod(16) to
each number. It also is a disguised # 744. However, complementing and
adding 8 mod(16) to each number sometimes give different magic squares.
See. # 117 whose complement is # 483 and +8 is # 646.
D. is obtained by exchanging the two outside columns and
also the two outside rows. However, it is no longer pandiagonal but a
group II, (# 63 disguised). Any magic squares can be transposed to another
with this method.
E. is the Dudeney pattern for all pandiagonal (group I)
magic squares.
F. is the magic line pattern for the example, # 204.
Other pandiagonal magic squares will have other patterns.
A. #204 group I
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B. #744 group I
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C. #744 group I
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D. #63 group II
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E. Type I
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F. for # 204
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Following is a set of 15 transformations that work only with
pandiagonal magic squares and result in other pandiagonal magic squares.
Here we move a row from the top to the bottom of the magic square, a column from
the left to the right, or both a row and a column.
All of these magic squares are disguised except # 469, and of course, the
original, # 204.
I have used leading zeros simply to allow simpler formatting.
formatting.
Rows/columns No
column change 1 col.
2 col. 3 col.
no row change Original # 204
# 109 #396 #294
01 14 11 08
14 11 08 01 11 08 01 14 08 01 14 11
15 04 05 10 04 05 10
15 05 10 15 04 10 15 04 05
06 09 16 03 09 16 03
06 16 03 06 09 03 06 09 16
12 07 02 13 07 02 13
12 02 13 12 07 13 12 07 02
1 row # 171
#107 #691 #788
15 04 05
10 04 05 10 15 05 10 15 04 10 15 04 05
06 09 16 03 09 16 03 06 16 03
06 09 03 06 09 16
12 07 02 13 07 02 13 12 02 13
12 07 13 12 07 02
01 14 11 08 14 11 08 01 11 08
01 14 08 01 14 11
2 rows # 560
#621 #744 #469
06 09 16 03 09 16 03 06 16 03
06 09 03 06 09 16
12 07 02 13 07 02 13 12 02 13
12 07 13 12 07 02
01 14 11 08 14 11 08 01 11 08
01 14 08 01 14 11
15 04 05 10 04 05 10 15 05 10
15 04 10 15 04 05
3 rows # 532
#839 #355 #292
12 07 02 13 07
02 13 12 02 13 12 07 13 12 07 02
01 14 11 08 14 11 08 01
11 08 01 14 08 01 14 11
15 04 05 10 04 05 10 15 05 10
15 04 10 15 04 05
06 09 16 03 09 16 03 06 16 03
06 09 03 06 09 16
This set contains 16 magic squares out of the 48 pandiagonal
magic squares of order-4. Thus all 48 pandiagonal magic squares can be derived
from 3 essentially different magic squares.
Any one of these 16 magic squares could be considered to be the origin of the
set of 16. If the lowest index number is taken as the starting square, then the
other two sets of 16 would start with #104 and # 116.
Associated m. s.
Transformations
Here are 15 ways to transform an associated magic square
to another associated magic square (there are many more).
The result is 10 different group III magic squares with 5 of these
appearing twice.
Complementing is not included here because, as mentioned previously, you
obtain only a disguised version of the original.
The original is normalized as are 2 of the resulting magic squares. The
other 13 are all disguised versions of their index number.
#126 III #124 Exchange #206 Exchange #183 Exchange rows
Original rows 1 and 4 columns 1 and 4 and columns 1 & 4
1 8 15 10 7 2 9 16 10 8 15 1 16 2 9 7
14 11 4 5 14 11 4 5 5 11 4 14 5 11 4 14
12 13 6 3 12 13 6 3 3 13 6 12 3 13 6 12
7 2 9 16 1 8 15 10 16 2 9 7 10 8 15 1
#206 Exchange #478 Exchange rows #308 Swap col. 1 & #632 Swap rows and
columns 2 and 3 1 & 2 with 3 & 4 2 with col. 3 & 4 col. 1 & 2, 3 & 4
1 15 8 10 12 13 6 3 15 10 1 8 6 3 12 13
14 4 11 5 7 2 9 16 4 5 14 11 9 16 7 2
12 6 13 3 1 8 15 10 6 3 12 13 15 10 1 8
7 9 2 16 14 11 4 5 9 16 7 2 4 5 14 11
#124 Exchange #789 Move quadrants #289 Move quadrants #632 Swap kitty-
rows 2 and 3 clockwise counter-clockwise corner quadrants
1 8 15 10 12 13 1 8 15 10 6 3 6 3 12 13
12 13 6 3 7 2 14 11 4 5 9 16 9 16 7 2
14 11 4 5 6 3 15 10 1 8 12 13 15 10 1 8
7 2 9 16 9 16 4 5 14 11 7 2 4 5 14 11
#183 Swap columns #478 swap rows #395 swap columns #741 Swap columns &
and rows 2 & 3 1 & 2, 3 & 4 1 & 2, 3 & 4 rows 1 & 2, 3 & 4
1 15 8 10 14 11 4 5 15 1 10 8 4 14 5 11
12 6 13 3 1 8 15 10 4 14 5 11 15 1 10 8
14 4 11 5 7 2 9 16 6 12 3 13 9 7 16 2
7 9 2 16 12 13 6 3 9 7 16 2 6 12 3 13
Transformations between Groups
In almost all cases, the resulting magic square will be a
disguised version of the index number shown.
Between Group I and II by exchanging rows 2 and
3 and columns 2 and 3.
Each of the 48 group I magic squares transforms to one of the 48 group II.
In this and all following cases the transformation works in reverse, so in this
case, group II can also be transformed to group I
#171 I
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Group I
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Exchange rows 2 & 3
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and col. 2 & 3 = #57
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Group II
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The same result is obtained by swapping rows and columns 1 and 4 instead of 2
and 3. The same index number magic square is obtained but it is a reflected
version.
Between Group II and III by exchanging
rows 1 and 3 and columns 1 and 3.
Each of the 48 group II magic squares transforms to one of the 48 group
III.
#213 II
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Group II
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Exchange rows 1 & 3
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and col. 1 & 3 = # 808
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Group III
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The same result is obtained by swapping rows and columns 2 and 4 instead of 1
and 3. This time a different magic square is obtained. From #213 it would be
#361.
Between Group I and Group III using
quadrants to rows or rows to quadrants.
There are 48 magic squares of Group I and 48 of Group III. Each
of these may be transformed from one type to a magic square of the other
type by the following procedure. See A. and B. Use the same procedure but
start at a different number in the quadrants (or rows) to get three more
squares of the second type from the one original.
A. #183 III
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B.
#171 I
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Using #183 Group III for our original,
construct #171 Group I by taking the four numbers in turn from each
quadrant to form the four lines of # 171. C. and D. are the Dudeney
patterns for the two magic squares.
E. F. and G. illustrate how we can get three more pandiagonal magic
squares from the same associated magic square, by simply starting at a
different cell in the original. |
C. Type III
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D. Type I
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E. #204 I
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F. #532 I
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G. #560 I
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H. #171 I
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J. #183 III
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#808 III
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#698 III
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#361 III
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Those first 4 pandiagonal magic squares were obtained by
starting with quadrant a. of the original associated magic
square. By repeating the procedure but starting with quadrant b., then c. and
finally d., 12 more pandiagonal magic squares are obtained. In fact we end up
with the same set of 16 squares that were obtained by shifting rows and/or
columns (proceeding section).
The bottom row illustrates how by reversing the process, four
associated magic squares can be formed from one pandiagonal magic square.
Simply form the 4 quadrants of J. in turn from the 4 rows of H. to get back to
the original associated square. The other 3 squares are formed by starting each
quadrant from a different position in the row. And by starting with a different
quadrant we end up with 16 associated magic squares generated by 1 pandiagonal.
If we try the same procedure on any of the other 15 pandiagonal magic squares
obtained above, we obtained disguised versions of the same 16 associated magic
squares.
Another method (not shown). Exchange the last
two columns, then exchange the last two rows, to convert magic squares between
Group I and Group III.
Between Group IV and Groups V and VI
interchanging rows (and columns)
There are 96 magic squares of Group IV. Each of these may be
transformed to one of the 96 magic squares of Group V and to one of the 96
magic squares of Group VI. Here we use Group IV #251 as an example.
Note that this transformation works in both directions i.e. also from V or VI
to IV.
However, for Group VI, it works only on the 96 that are semi-pandiagonal. On
the 208 simple magic squares of order-VI, the result of this transformation is
only semi-magic.
Between Group VII and Groups VIII, IX and X
There are 56 magic squares of Group VII. Each of these may be
transformed to magic squares of Groups VIII, IX or X by simple permutation
of rows and columns.
Between Group XI and Group XII
There are 8 magic square each of Groups XI and XII. Magic squares
of one group may be transformed to magic squares of the other group by
exchanging rows 2 & 4 and columns 2 & 4.
#374 XI
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Group XI
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exchange rows
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& columns = # 209
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Group XII
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