An important, relatively frequent special form of magic squares are symmetrical squares, where all pairs of cells diametrically equidistant from the center have the same sum n^{2} + 1. These number pairs are said to be complementary.

46

19

41

14

29

2

24

17

39

12

34

7

22

44

37

10

32

5

27

49

15

8

30

3

25

47

20

42

35

1

23

45

18

40

13

6

28

43

16

38

11

33

26

48

21

36

9

31

4

Symmetrical magic squares have been intensively investigated. Some important results are given here:

The only magic square of the third order is symmetrical.

The center cell of an odd symmetrical square is always equal to the middle number of the series.

There are 48 symmetrical squares of fourth order.

The minimum order for a magic square that is both symmetrical and pandiagonal, is n=5.

There is no symmetrical square of single-even order.

Each symmetrical magic square is also semi-pandiagonal. But the converse is not true: not every semi-pandiagonal square is also symmetrical.