| This page is one of 4 on this subject
Please enjoy them all! |
| Swap
columns only |
Examples by swapping columns 1 & 2 and columns 3 & 4. |
| Swap rows only |
Examples by swapping rows 1 & 2 and rows 3 & 4. |
| Diagonals
to rows |
Works on any order-4 pandiagonal or semi-pandiagonal magic
square. |
| A
double magic square loop |
Sets of four order-4 magic squares where each number
differs by 4 (modulo 16). |
| ...Quadruplets |
A pair of magic squares related by +8. Their complements
are another loop. |
| ...Triplets |
A loop of three magic squares using +4 and +8 with +12
returning to the original. |
| ...Pairs |
Two different loops of two magic squares using CPI and +8. |
| Congruent
modulo 8 |
If # < 9 new number = # + 8, if # > 8 then new # = # - 8. |
| Intro to Order-4
Transforms. |
Back to the introduction page to this subject. (Also up
arrow above and at end). |
| Summary |
More transformations and a Table of over 45 order-4
transformations. |
Swap columns only
The second magic square of each pair is a result of swapping columns 1
and two and columns 3 and 4. In many cases this magic square is either
rotated or reflected from the normalized position.
Notice that the complementary pair diagrams are a great help in
visualizing row and column interchanges.
 |
#102 I #785 I
1 8 10 15 8 1 15 10
12 13 3 6 13 12 6 3
7 2 16 9 2 7 9 16
14 11 5 4 11 14 4 5
|
|
#22 II #592 II
1 4 14 15 4 1 15 14
13 16 2 3 16 13 3 2
12 9 7 6 9 12 6 7
8 5 11 10 5 8 10 11
|
 |
#120 III #487 III
1 8 14 11 8 1 11 14
12 13 7 2 13 12 2 7
15 10 4 5 10 15 5 4
6 3 9 16 3 6 16 9
|
|
#93 IV #664 IV
1 7 14 12 7 1 12 14
16 10 3 5 10 16 5 3
11 13 8 2 13 11 2 8
6 4 9 15 4 6 15 9
|
 |
#33 V #578 V
1 4 16 13 4 1 13 16
14 15 3 2 15 14 2 3
11 10 6 7 10 11 7 6
8 5 9 12 5 8 12 9
|
|
#48 VI-P #824 VI-P
1 6 11 16 6 1 16 11
12 15 2 5 15 12 5 2
8 3 14 9 3 8 9 14
13 10 7 4 10 13 4 7
|
| 96 of the group VI magic squares are
semi-pandiagonal. These 96 squares are linked in pairs by the above column
interchange (and also by the row interchange shown below). However,
there are another 208 magic squares of order-4 group VI that are only
simple magic squares. The reason they are group VI is because they have
the same complementary pair pattern as the 96 semi-pandiagonal group VI.
When the above transformation (and the equivalent one with rows, below) is
attempted, the result is not magic. |
|
#144 VI-S not magic
1 10 15 8 10 1 8 15
14 11 4 5 11 14 5 4
3 6 13 12 6 3 12 13
16 7 2 9 7 16 9 2
#144 is one of the 208
group VI that is not
semi-pandiagonal.
|
Swap rows only
The second magic square of each pair below is a result of swapping rows
1 and two and rows 3 and 4. In many cases this magic square is either
rotated or reflected from the normalized position.
This is a companion transformation to the one presented above and has the
same conditions. However, the partner magic square in each case is
different. To illustrate, I will use the same origin square for each
group.
|
#102 I #828 I
1 8 10 15 12 13 3 6
12 13 3 6 1 8 10 15
7 2 16 9 14 11 5 4
14 11 5 4 7 2 16 9
|
|
#22 II #446 II
1 4 14 15 13 16 2 3
13 16 2 3 1 4 14 15
12 9 7 6 8 5 11 10
8 5 11 10 12 9 7 6
|
|
#120 III #297 III
1 8 14 11 12 13 7 2
12 13 7 2 1 8 14 11
15 10 4 5 6 3 9 16
6 3 9 16 15 10 4 5
|
|
#93 IV #326 IV
1 7 14 12 16 10 3 5
16 10 3 5 1 7 14 12
11 13 8 2 6 4 9 15
6 4 9 15 11 13 8 2
|
|
#33 V #229 V
1 4 16 13 14 15 3 2
14 15 3 2 1 4 16 13
11 10 6 7 8 5 9 12
8 5 9 12 11 10 6 7
|
|
#48 VI-P #763 VI-P
1 6 11 16 12 15 2 5
12 15 2 5 1 6 11 16
8 3 14 9 13 10 7 4
13 10 7 4 8 3 14 9
|
Summery
These two transformations work for all 288 magic squares of groups I to V.
They also work for the 96 semi-pandiagonal magic squares of group VI.
In each case they produce a partner magic square of the same group.
They do not work for the 208 simple magic squares of group VI.
They also do not work for any of the 240 magic squares of groups VII to XII.
A third related transformation in which both the rows and columns are
exchanged will produce a magic square in the same group, for all 12 groups.
There are a number of other row and/or column interchanges that also transform
one magic square into another one. They are summarized in the table at the
bottom of this page.
Diagonals to rows
 |
This transformation works for all of group I to
Group VI-P (the semi-pandiagonal) magic squares. It also works for
48 of the 208 group VI-S simple magic squares.
If the diagonals are converted to columns instead of
rows, the same magic squares are produced but rotated.
|
 |
102 I 173 V
1 8 10 15 1 13 16 4
12 13 3 6 12 8 5 9
7 2 16 9 7 11 10 6
14 11 5 4 14 2 3 15
|
By converting diagonals to rows as per the above
diagram, the 48 group I are transformed to 48 group V. Note that
#173 shown here is not normalized. |
 |
32 V 201 I
1 4 16 13 1 15 10 8
14 15 3 2 14 4 5 11
7 6 10 1 7 9 16 2
12 9 5 8 12 6 3 13
|
Whether the transformation is to a group I or a
group V is determined by the orientation of the complement pairs of
the originating magic square. |
 |
173 V 103 V
1 12 7 14 1 8 10 15
13 8 11 2 13 12 6 3
16 5 10 3 16 9 7 2
4 9 6 15 4 5 11 14
|
When converting Group V magic squares by diagonals
to rows, 48 transform to the 48 group I, the other 48 transform to
another group V magic square of the same orientation. |
A double magic
square loop
This transformation is not universal. It works only on some order-4
magic squares.
The top row of magic squares is a loop obtained by adding 4 to each number of
the preceding square. Adding 4 to each number of the last square gets you back
to the first magic square, making a loop of four magic squares.
The second row of magic squares are, in each case, the complement of the square
above. They form another loop of four magic squares, this time by subtracting
four from each number.
As mentioned in the previous section, the complement belongs to the same group
as the original magic square. Also, as mentioned in the previous section, the
complement is a disguised version of the fundamental magic square for the index
number shown. (In the first loop, all but the first magic square (the original)
are disguised versions of the index # shown.)
#109 I
 |
+ 4 = |
#326 IV
 |
+ 4 = |
#621 I
 |
+ 4 = |
#467 IV
 |
#469 I
 |
- 4 = |
#93 IV
 |
- 4 = |
#294 I
 |
- 4 = |
#664 IV
 |
In a paper called Magic Squares of Order-4 and their Magic Square Loops
[1], Robert S. Sery investigates all
880 order-4 for such loops. He found:
| Octuplets |
6 |
|
pairs of loops of 4 magic squares each |
6 x 8 = |
48 |
| Quadruplets |
64 |
|
pairs of loops of 2 magic squares each |
64 x 4 = |
256 |
| Triplets |
16 |
|
loops of 3, no CPI involved |
16 x 3 = |
48 |
| Pairs |
232 |
|
loops of twins by CPI or by adding 8 |
232 x 2 = |
464 |
| Singles |
64 |
|
with no connections to others, all type VI |
64 x 1 = |
64 |
| Total |
|
|
|
|
880 |
The singles are all self-similar magic squares because the complement of each
is a reflected copy of itself.
Examples of the quadruplets, triplets and pairs follow.
1. Robert S. Sery, Magic squares of
order-4 and their Magic Square Loops, Journal of Recreational Mathematics, vol.
29(4) 274-281, 1998
(Dated 1998 but only released in May, 2000)
...Quadruplets
8 VIII
1 3 16 14
8 15 2 9
13 6 11 4
12 10 5 7
|
+ 8 = |
605 VIII
9 11 8 6
16 7 10 1
5 14 3 12
4 2 13 15
|
Adding 8 (modulo 16) to each cell of
index #8 magic square produces a disguised version of index # 605
and, of course adding 8 to each cell of this version of #605
reproduces #8.
The complements of these two magic squares produce #409 and # 242 which
together form a similar loop.
There are 64 sets such as this consisting of two loops of two . |
409 VIII
16 14 1 3
9 2 15 8
4 1 6 13
5 7 12 10
|
+ 8 = |
242 VIII
8 6 9 11
1 10 7 16
12 3 14 5
13 15 4 2
|
...Triplets
This loop consists of 3 magic squares before looping back to the
starting square. It uses three different addition constants (modulo 16 of
course). This is one of 16 such sets of three.
Note that the paired loop consists of the same 3 magic squares but in a
different orientation.
Also, in the second loop, the positions of squares 655 and 652 are
reversed.
#38 VI
1 5 12 16
15 11 6 2
10 4 13 7
8 14 3 9
|
+ 4 = |
655 X
5 9 16 4
3 15 10 6
14 8 1 11
12 2 7 13
|
+ 8 = |
652 X
13 1 8 12
11 7 2 14
6 16 9 3
4 10 15 5
|
+ 12 = |
#38 VI
1 5 12 16
15 11 6 2
10 4 13 7
8 14 3 9
|
#38 VI
16 12 5 1
2 6 11 15
7 13 4 10
9 3 14 8
complement
of 38 (above)
|
+ 4 = |
655 X
4 16 9 5
6 10 15 3
11 1 8 14
13 7 2 12
complement
of 652 (above)
|
+ 8 = |
652 X
12 8 1 13
14 2 7 11
3 9 16 6
5 15 10 4
complement
of 655 (above)
|
+ 12 = |
#38 VI
16 12 5 1
2 6 11 15
7 13 4 10
9 3 14 8
complement
of 38 (above)
|
Because the complement of # 38 is a reflected version of itself,
#38 is a self-similar magic square (Suzuki).
...Pairs
#3 XII
1 2 16 15
13 14 4 3
12 7 10 5
8 11 6 9
|
CPI = |
209 XII
16 15 1 2
4 3 13 14
5 10 7 12
9 6 11 8
|
Unlike the other loop examples presented, this is
not a pair of loops but individual loops of two magic squares.
This top loop is accomplished by complementing the #3 magic square
to obtain the #209 square.
(CPI refers to Complementary Pair Interchange)
|
12 VI
1 4 13 16
8 14 3 9
15 5 12 2
10 11 6 7
|
+ 8 = |
218 XII
9 12 5 8
16 6 11 1
7 13 4 10
2 3 14 15
|
Here the constant 8 has been added to each number in
magic square # 12 to obtain the #218 square.
In each case the same operation loops back to the original magic
square.
There are 232 pairs of magic squares that make such loops. |
Congruent
modulo 8
If you replace each number in the order-4 magic square with its
congruent mod (8) number, a new magic square of the same group is
obtained. In effect, if the number in the original magic square is less
then 9, the new number is the number plus 8. If the number in the original
magic square is greater then 8, the new number is the number minus 8.
This works for all magic squares belong to Groups I, II, III, IV, V, VI-P,
VII and VIII.
For Groups VI-S, IX, X, XI and XII, the rows and columns sum correctly, but the
diagonals are incorrect.
Short-cut number swap
| Original decimal number |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
| After congruent mod (8) swap |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
Three examples
Original Transform Original Transform Original Transform
104 I 623 I 478 III 632 III 148 V 720 V
01 08 10 15 09 16 02 07 03 06 13 12 11 14 05 04 01 10 16 07 09 02 08 15
14 11 05 04 06 03 13 12 16 09 02 07 08 01 10 15 15 08 02 09 07 16 10 01
07 02 16 09 15 10 08 01 10 15 08 01 02 07 16 09 04 11 13 06 12 03 05 14
12 13 03 06 04 05 11 14 05 04 11 14 13 12 03 06 14 05 03 12 06 13 11 04
This method was suggested by Yvan Saint-Pierre in an e-mail dated Sept. 8, 2000.
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