Math Help

Magic Squares are square grids with a special arrangement of numbers in them.  It is an arrangement of the numbers from 1 to n^2 (n-squared) in an nxn matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same. It is not hard to show that this sum must be n (n^2+1)/2. These numbers are unique because each row, column and diagonal results to the same number by adding them. Magic squares received their name because there are so many relationships between the sums of the numbers filling the squares. So for the example below, 15 is the magic number. Could you work this out just from knowing that the square uses the numbers from 1 to 9?

6
1
8
7
5
3
2
9
4

One more important point is to notice that the two numbers that are opposite each other across the centre number will add up to the same number. So in the square above,  8 + 2 = 10, 6 + 4 = 10, 1 + 9 = 10 and 3 + 7 = 10. Why is this?

It is interesting to note that subtracting a constant from each cell in a magic square does not change its essential property: namely, that the sum of any row, column, or principal diagonal is identical. For example, the following magic square is obtained by subtracting 5 from each cell in the magic square above:

1
-4
3
2
0
-2
-3
4

-1

An interesting property about this square is that all rows, columns, and principal diagonals sum to 0. Also, it is easy to visually recognize symmetry in the square.

A magic square with 3 rows and columns is called magic square of order  3, and a square with 4 rows and columns is magic square of 0rder 4 and so on. The “order” of a magic square indicates how many rows or columns it has.
So the numbers in the Magic Square are unique, but why are they called magic? It seems that from ancient times they were connected with the supernatural and magical world. The first magic square in history was created in China by a mathematician, probably sometime about 2200 BC and was called “Lo-Shu” Square. There’s a legend that says that the Emperor Yu saw this magic square on the back of a divine tortoise in the Yellow River. Lo Shu supposedly means “river map” and the story of the appearance of the turtle had to do with a sacrifice to the river god. An essentially mathematical construction is combined with the supernatural!