A bordered magic square is a magic square, which remains magic when its borders are removed. Let us take the depicted magic square of order 8. When we remove its border, we will get a square of order 6.

64

4

9

54

63

3

10

53

60

15

16

47

48

49

20

5

7

44

22

42

41

25

21

58

51

33

37

29

30

28

38

14

6

32

34

35

36

31

27

59

8

26

40

24

23

43

39

57

52

45

46

18

17

19

50

13

12

61

56

11

2

62

55

1

15

16

47

48

49

20

44

22

42

41

25

21

33

37

29

30

28

38

32

34

35

36

31

27

26

40

24

23

43

39

45

46

18

17

19

50

This square isn't normalized anymore, because it doesn't contain the numbers 1, 2, … , n^{2} anymore. But still each row, each column and both diagonals sum S=195.

A second condition for bordered magic squares demands that the numbers of the border enclose the numbers of the inner square. Satisfiying this condition means that the numbers 1, … ,2(n−1) and n^{2}−2(n−1) + 1 , … , n^{2} must form the border. All other numbers must be elements of the inner square.

Let's take a bordered magic square of order n=6 as an example. The inner square is of order n=4, and must be formed of 4^{2}=16 numbers. On the other side, there are

6^{2} − 4^{2}= 36 − 16 = 20

numbers for the border, which have to enclose the inner elements. So, ten numbers are less and ten numbers are greater than the inner numbers.

lower numbers of the border:

1 … 10

inner numbers:

11 … 26

upper numbers of the border:

27 … 36

One example of such an arrangement is shown in the following magic square of order 6: