MAGIC SQUARE

                             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.

       

 

  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.

                          

 (Albrecht Dürer, 1514) includes an order 4 square with magic sum 3

                History               

                     Iron plate with an order-6 magic square in Eastern Arabic numerals from China, dating to the Yuan Dynasty (1271–1365)

Ancient China: One of the earliest references is the "Lo Shu Square," a 3x3 magic square associated with Chinese mythology around 2200 BCE.

India and Persia: Magic squares were used in astrology and religious practices.

Europe: Magic squares became popular in the Renaissance through the work of mathematicians and artists like Albrecht Dürer.

Classification of magic squares

Euler diagram of the requirements of some types of 4×4 magic squares. Cells of the same colour sum to the magic constant.  In 4×4 most-perfect magic squares, any 2 cells that are 2 cells diagonally apart (including wraparound) sum to half the magic constant, hence any 2 such pairs also sum to the magic constant.

Magic square of primes

     

Certain 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

Application of magic square

1. Art and Design: Artists have used magic squares in compositions for aesthetic balance.

2. Recreational Mathematics: Creating and solving magic squares is a popular puzzle activity.

3. Mathematical Research: Studying magic squares provides insights into combinatorics and number theory.

4. Symbolism and Mysticism: Magic squares have been used in talismans and religious contexts for their supposed magical properties