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, consisting 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).
Read more about Magic Square: History, Types and Construction, Related Problems
Famous quotes containing the words magic and/or square:
“The echo is, to some extent, an original sound, and therein is the magic and charm of it. It is not merely a repetition of what was worth repeating in the bell, but partly the voice of the wood; the same trivial words and notes sung by a wood-nymph.”
—Henry David Thoreau (1817–1862)
“If magistrates had true justice, and if physicians had the true art of healing, they would have no occasion for square caps; the majesty of these sciences would of itself be venerable enough. But having only imaginary knowledge, they must employ those silly tools that strike the imagination with which they have to deal; and thereby, in fact, they inspire respect.”
—Blaise Pascal (1623–1662)