A Magic Square of order n is an arrangement of $n^2$ numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.
To construct Magic Squares of n-odd size, a method known as Siamese method is given on Wikipedia, the method is ::
starting from the central box of the first row with the number 1 (or
the first number of any arithmetic progression), the fundamental
movement for filling the boxes is diagonally up and right (↗), one step
at a time. When a move would leave the square, it is wrapped around to
the last row or first column, respectively.
If a filled box is encountered, one moves vertically down one box (↓)
instead, then continuing as before.
How does this method work?
Best Answer
There is a detailed discussion of "the Siamese method," with proofs, at this website. There's also a proof here.