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Definition: magic square from The Hutchinson Unabridged Encyclopedia with Atlas and Weather Guide

In mathematics, a square array of numbers in which the rows, columns, and diagonals add up to the same total. A simple example employing the numbers 1 to 9, with a total of 15, has a first row of 6, 7, 2, a second row of 1, 5, 9, and a third row of 8, 3, 4.

A pandiagonal magic square is one in which all the broken diagonals also add up to the magic constant.


Abracadabra – a Look at Magic Squares


magic squares

Summary Article: magic square
From Chambers Dictionary of the Unexplained

Usually, a square filled with rows of numbers so arranged that the total is the same whether the figures are added up horizontally, vertically or diagonally; often used in talismans.

Magic squares have fascinated humans for thousands of years, and are found in a number of cultures. The most common type of magic square is a grid of figures which give the same total (known as the magic constant) whether they are added up horizontally, vertically or diagonally, for example a square made up of the rows 3, 7 and 6; 9, 5 and 1; and 4, 3 and 8, which all add up to 15. A magic square of the order 3 has three numbers in each row and column, one of the order 4 has four numbers in each row, and so on, and numerical magic squares can be constructed using a mathematical formula. Chinese literature from around 2800 bc tells the legend of the Lo Shu square, a magic square of the order 3 which was said to have appeared on the back of a turtle which came out of the River Lo, and a magic square from a Greek text was used by 9th-century Arab astrologers in drawing up horoscopes. The earliest magic square of the order 4 was found in an Indian inscription dating from the 11th or 12th century.

Such squares are often used for magical purposes, such as ensuring a long life and warding off disease, since they are believed to have astrological significance and divine qualities. Stone and metal talismans engraved with magic squares have been found in Egypt and India, and Cornelius Agrippa, the 16th-century alchemist and philosopher, constructed seven magic squares of the orders 3 to 9 inclusive, each of which he associated with one of the seven heavenly bodies or ‘planets’ then known, for talismanic purposes.

Another type of numeric magic square is the multiplicative magic square, in which it is the product of the numbers in each line which is constant, rather than the sum. There are also alphabetic magic squares, which consist of a series of letters arranged in a square to spell certain words which appear in the same order both horizontally and vertically, the best-known of which is made up of the Latin words SATOR, AREPO, TENET, OPERA, ROTAS.

© Chambers Harrap Publishers Ltd 2007

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Full text Article magic square
The Penguin Dictionary of Mathematics

A square array of numbers in which the numbers in any row, column, or full diagonal have the same sum. This sum is called the magic constant ...

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