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).
Other articles related to "magic square, magic squares, magic, square, squares":
... Certain extra restrictions can be imposed on magic squares ... not only the main diagonals but also the broken diagonals sum to the magic constant, the result is a panmagic square ... If raising each number to certain powers yields another magic square, the result is a bimagic, a trimagic, or, in general, a multimagic square ...
... and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa ...
... also brought to France from his Siamese travels a very simple method for creating n-odd magic squares, known as the "Siamese method" or the "de la Loubère method", which apparently was initially ...
... The resulting square is also known as a mystic square ... In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with ... A construction of a magic square of order 4 Go left to right through the square counting and filling in on the diagonals only ...
... An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value ...
Famous quotes containing the words square and/or magic:
“I would say it was the coffin of a midget
Or a square baby
Were there not such a din in it.”
—Sylvia Plath (19321963)
“Self-esteem is the real magic wand that can form a childs future. A childs self-esteem affects every area of her existence, from friends she chooses, to how well she does academically in school, to what kind of job she gets, to even the person she chooses to marry.”
—Stephanie Martson (20th century)