More 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

605    VIII

 9 11  8  6

16  7 10  1

 5 14  3 12

 4  2 13 15

409    VIII

16 14  1  3

 9  2 15  8

 4  1  6 13

 5  7 12 10

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

655   X

 5  9 16  4

 3 15 10  6

14  8  1 11

12  2  7 13

652   X

13  1  8 12

11  7  2 14

 6 16  9  3

 4 10 15  5

#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)

655   X

 4 16  9  5

 6 10 15  3

11  1  8 14

13  7  2 12

complement
of 652 (above)

652   X

12  8  1 13

14  2  7 11

 3  9 16  6

 5 15 10  4

complement
of 655 (above)

#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

209    XII

16 15  1  2

 4  3 13 14

 5 10  7 12

 9  6 11  8

12    VI

 1  4 13 16

 8 14  3  9

15  5 12  2

10 11  6  7

218    XII

 9 12  5  8

16  6 11  1

 7 13  4 10

 2  3 14 15  

Congruent mod(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

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.