Simple Magic Square

Simple Magic Square

In recreational mathematics, a magic square is an arrangement of numbers (usually integers) in a square grid, where the numbers in each row, and in each column, and the numbers that run diagonally in both directions, all add up to the same number. A magic square has the same number of rows as it has columns, and in conventional math notation, "n" stands for the number of rows (and columns) it has. Thus, a magic square always contains n2 numbers, and its size (the number of rows it has) is described as being "of order n". A magic square that contains the integers from 1 to n2 is called a normal magic square. (The term "magic square" is also sometimes used to refer to any of various types of word squares.)

It is possible to construct a normal magic square of any size except 2 x 2 (that is, where n = 2), although the solution to a magic square where n = 1 is trivial, since it consists simply of a single cell containing the number 1. The smallest nontrivial case, shown below, is a 3 x 3 grid (that is, a magic square of order 3).

The constant that is the sum of every row, column and diagonal is called the magic constant or magic sum, M. Every normal magic square has a unique constant determined solely by the value of n, which can be calculated using this formula:

For example, if n = 3, the formula says M = /2, which simplifies to 15. For normal magic squares of order n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260. (See sequence A006003 in the OEIS)