Properties
self-complement

8	17	1	15	24
14	23	7	16	5
20	4	13	22	6
21	10	19	3	12
2	11	25	9	18

14	6	25	2	18
16	15	7	23	4
5	17	13	9	21
22	3	19	11	10
8	24	1	20	12

symmetrical

23	7	16	5	14
1	15	24	8	17
20	4	13	22	6
9	18	2	11	25
12	21	10	19	3

not symmetrical

11	22	19	10	3
2	14	25	18	6
9	5	13	21	17
20	8	1	12	24
23	16	7	4	15

example n=8

20	23	1	9	12
21	4	7	15	18
2	10	13	16	24
8	11	19	22	5
14	17	25	3	6

example n=9

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Self-complement magic squares

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.