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.


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:
36  2  3  7  32  31 
4  26  13  12  23  33 
9  15  20  21  18  28 
27  19  16  17  22  10 
29  14  25  24  11  8 
6  35  34  30  5  1 