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:
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 |