Magic square

In recreational mathematics and combinatorial design, a magic square is a n × n {displaystyle n imes n} square grid (where n is the number of cells on each side) filled with distinct positive integers in the range 1 , 2 , . . . , n 2 {displaystyle 1,2,...,n^{2}} such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the magic constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n. A magic square in a musical composition is not a block of numbers -- it is a generating principle, to be learned and known intimately, perceived inwardly as a multi-dimensional projection into that vast (chaotic!) area of the internal ear -- the space/time crucible -- where music is conceived. ... Projected onto the page, a magic square is a dead, black conglomeration of digits; tune in, and one hears a powerful, orbiting dynamo of musical images, glowing with numen and lumen. In recreational mathematics and combinatorial design, a magic square is a n × n {displaystyle n imes n} square grid (where n is the number of cells on each side) filled with distinct positive integers in the range 1 , 2 , . . . , n 2 {displaystyle 1,2,...,n^{2}} such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the magic constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n. The mathematical study of magic squares typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. The magic squares are generally classified according to their order n as: odd if n is odd, evenly even (also referred to as 'doubly even') if n = 4k (e.g. 4, 8, 12, and so on), oddly even (also known as 'singly even') if n = 4k + 2 (e.g. 6, 10, 14, and so on). This classification is based on different techniques required to construct odd, evenly even, and oddly even squares. Beside this, depending on further properties, magic squares are also classified as associative magic squares, pandiagonal magic squares, most-perfect magic squares, and so on. More challengingly, attempts have also been made to classify all the magic squares of a given order as transformations of a smaller set of squares. Except for n ≤ 5, the enumeration of higher order magic squares is still an open challenge. The enumeration of most-perfect magic squares of any order was only accomplished in the late 20th century. In regard to magic sum, the problem of magic squares only requires the sum of each row, column and diagonal to be equal, it does not require the sum to be a particular value. Thus, although magic squares may contain negative integers, they are just variations by adding or multiplying a negative number to every positive integer in the original square. Magic squares composed of integers 1 , 2 , . . . , n 2 {displaystyle 1,2,...,n^{2}} are also called normal magic squares, in the sense that there are non-normal magic squares whose integers are not restricted to 1 , 2 , . . . , n 2 {displaystyle 1,2,...,n^{2}} . However, in some places, 'magic squares' is used as a general term to cover both the normal and non-normal ones, especially when non-normal ones are under discussion. Moreover, the term 'magic squares' is sometimes also used to refer to various types of word squares.