Magic square

In mathematics, a magic square (??) of order n is an arrangement of n² numbers in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n².

Related Topics:
Mathematics - Square - Integer

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Magic squares exist for all orders n ≥ 1 except n = 2, although the case n = 1 is trivial—it consists of a single cell containing the number 1. The smallest nontrivial case, shown below, is of order 3.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The constant sum in every row, column and diagonal is called the magic constant, M. The magic constant of a normal magic square depends only on n and has the value

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:M(n) = rac{n(n^2+1)}{2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For normal magic squares of order n = 3, 4, 5, …, the magic constants (sequence in OEIS) are:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:15, 34, 65, 111, 175, 260, …

~ ~ ~ ~ ~ ~ ~ ~ ~ ~