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, … , n2 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 n2−2(n−1) + 1 , … , n2 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 42=16 numbers. On the other side, there are
62 − 42 = 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: