The following square of order 4 is magic with magic sum S=34.

1

2

15

16

12

14

3

5

13

7

10

4

8

11

6

9

Now each number z is replaced by its complement

which results in a new magic square.

1

2

15

16

12

14

3

5

13

7

10

4

8

11

6

9

16

15

2

1

5

3

14

12

4

10

7

13

9

6

11

8

Under certain conditions the resulting magic square will be a reflected or rotated copy of the original magic square. These squares are called self-complement or self-similar magic squares.