MAGIC SQUARE
mathematics,especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same.The "order" of the magic square is the number of integers along one side (n), and the constant sum is called the "magic constant". If the array includes just the positive integers {\displaystyle 1,2,...,n^{2}}, the magic square is said to be "normal". Some authors take "magic square" to mean "normal magic square
A magic square is a grid of numbers where the sums of numbers in every row, column, and diagonal are the same. This constant sum is called the magic constant or magic sum. Magic squares have been studied for thousands of years and are an important part of recreational part of recreational mathematics, with applications in art, mysticism, and even problem-solving techniques.
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The smallest (and unique up to rotation and reflection) non-trivial case of a magic square, order 3 |
Magic squares have a long history, dating back to at least 190 BCE in China. At various times they have acquired occult or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations.
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Iron plate with an order-6 magic square in Eastern Arabic numerals from China, dating to the Yuan Dynasty (1271–1365)
Magic square of primesCertain extra restrictions can be imposed on magic squares.
If raising each number to the nth power yields another magic square, the result is a bimagic (n = 2), a trimagic (n = 3), or, in general, a multimagic square.
A magic square in which the number of letters in the name of each number in the square generates another magic square is called an alphamagic square.
Magic square in popular culture
- Goethe's Faust, the witch's spell used to make a youth elixir for Faust, the Hexen-Einmal-Eins [de], has been interpreted as a construction of a magic square.
- The English composer Peter Maxwell Davies has used magic squares to structure many of his compositions. For example, his 1975 Ave Maris Stella uses the 9×9 magic square of Moon while his 1977 A Mirror of Whitening Light uses the 8×8 magic square of Mercury to create the entire set of notes and durations for the piece. His other works that employ magic squaresinclude TheLighthouse (1979), Resurrection (1987), Strathclyde Concerto No. 3 for Horn and Trumpet (1989), as well as many of his symphonies
