Question

In the magic square, each row, column and diagonal have the same sum. A magic square with an order of 5 is shown below. Notice that the number that is located in the center of the square is 13. Middle values exist only for the case where n is an odd number. What is the middle value for the magic square with an order of 7?

Answer 1

Each row/column adds to n^2(n^2+1)/2n = n(n^2+1)/2
with n=7, that is 175
The center column forms an arithmetic sequence such that
n/2 (2 + (n-1)d) = n(n^2+1)/2
or, d = n+1
Thus, the (n+1)/2 cell has the value 1 + (n+1)/2 (n) = 1 + n(n+1)/2
So the center column forms the sequence 1,9,17,25,...
the center square. 1 + 7^2/2 = 25

There are, of course, many other ways to form a magic square, where the cells have different values.