# magic square

magic square
a square containing integers arranged in an equal number of rows and columns so that the sum of the integers in any row, column, or diagonal is the same.
[1695-1705]

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puzzle
square matrix often divided into cells, filled with numbers or letters in particular arrangements that were once thought to have special, magical properties. Originally used as religious symbols, they later became protective charms or tools for divination; and finally, when the original meanings were lost, people considered them mere curiosities or puzzles—except for some Western mathematicians who continue to study them as problems in number theory.

The most familiar lettered square in the Western world is the well-known SATOR square, composed of the words SATOR, AREPO, TENET, OPERA, and ROTAS. Arranged both vertically and horizontally, the meaningless phrase reads through the centre TENET, thus forming the two arms of a hidden cross. Examples of this square from the 1st century AD were found in the ruins of Pompeii, and it was still employed during the 19th century in Europe and the United States for fancied protection against fire, sickness, and other disasters.

Otherwise, numbered squares have always been far more significant, particularly in China (where they may have originated), the Arab world, and India.

In the arithmetical magic squares, the numbers are generally placed in separate cells and arranged so that each column, every row, and the two main diagonals can produce the same sum, called the constant. A standard magic square of any given number contains the sequence of natural numbers from 1 to the square of that number. Thus, the magic square of 3 contains the numbers 1 to 9. If these nine numbers are simply listed in three rows or three columns, they form the natural square of 3. A natural square has no “magical” properties, but one is often made as a first step in constructing a proper magic square. When these nine numbers in the 3 × 3 frame are rearranged so that they can produce a constant sum of 15, they constitute the magic square of 3.

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Universalium. 2010.

### Look at other dictionaries:

• Magic square — Magic Mag ic, Magical Mag ic*al, a. [L. magicus, Gr. ?, fr. ?: cf. F. magique. See {Magi}.] 1. Pertaining to the hidden wisdom supposed to be possessed by the Magi; relating to the occult powers of nature, and the producing of effects by their… …   The Collaborative International Dictionary of English

• Magic square — Square Square (skw[^a]r), n. [OF. esquarre, esquierre, F. [ e]querre a carpenter s square (cf. It. squadra), fr. (assumed) LL. exquadrare to make square; L. ex + quadrus a square, fr. quattuor four. See {Four}, and cf. {Quadrant}, {Squad},… …   The Collaborative International Dictionary of English

• magic square — n. a table of numbers with the same amount of rows and columns, in which the sum of each column, row, or main diagonal is always equal to the same number …   English World dictionary

• Magic square — 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.[1] A normal magic… …   Wikipedia

• magic square — noun a) A palindromic square word arrangement, usually in the form of a magic amulet. Probably the best known magic square is the 5 times;5 square consisting of the words ROTAS OPERA TENET AREPO SATOR. b) An n by n arrangement of n numbers such… …   Wiktionary

• magic square — square divided into smaller numbered squares so that that every row and column adds up to the same total …   English contemporary dictionary

• magic square — mag′ic square′ n. math. a square containing integers arranged in an equal number of rows and columns so that the sum of the integers in any row, column, or diagonal is the same • Etymology: 1695–1705 …   From formal English to slang

• magic square — noun a square matrix of n rows and columns; the first n^2 integers are arranged in the cells of the matrix in such a way that the sum of any row or column or diagonal is the same • Hypernyms: ↑square matrix …   Useful english dictionary

• magic square — noun Date: circa 1704 a square containing a number of integers arranged so that the sum of the numbers is the same in each row, column, and main diagonal and often in some or all of the other diagonals …   New Collegiate Dictionary

• magic square — noun a square divided into smaller squares each containing a number, such that the figures in each vertical, horizontal, and diagonal row add up to the same value …   English new terms dictionary