In recreational mathematics, a magic square of order n is an arrangement of n² 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. A normal magic square contains the integers from 1 to n². The term "magic square" is also sometimes used to refer to any of various types of word square.

Normal 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 or magic sum, M. The magic constant of a normal magic square depends only on n and has the value

For normal magic squares of order n = 3, 4, 5, …, the magic constants are:

15, 34, 65, 111, 175, 260, … (sequence A006003 in OEIS).

See also:

• Science: Math: Recreations: Games and Puzzles (72)

• Mathpages: Solving Magic Squares
  mathpages.com
  An attempt to solve for all possible magic squares, a mathematical treatment.
• Magic Squares
  wou.edu
  Java applet generates odd ordered squares, with options to transformations by rotations and reflections. Source code provided.
• Magic Rectangles and Squares
  gpj.connectfree.co.uk
  Presents 2xn and 3xn rectangles, as well as squares of order 4,5 and 8. Generation of the last by knight's tour.