# Bordered magic squares

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, … , 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:

 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