More Order-4 Transformations

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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
Four_3.gif (1982 bytes)
+ 4 = #326             IV
Four_4.gif (1977 bytes)
+ 4 = #621             I
Four_5.gif (1972 bytes)
+ 4 = #467             IV
Four_6.gif (1971 bytes)
#469             I
Four_7.gif (1979 bytes)
- 4 = #93               IV
Four_8.gif (1972 bytes)
- 4 = #294             I
Four_9.gif (1976 bytes)
- 4 = #664             IV
Four_10.gif (1975 bytes)

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.

 

This page was originally posted June 2000
It was last updated October 19, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz