A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, …, arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constant….see equation 1 above.

If every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a “normal” magic square.

The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order-4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer’s magic square. Magic squares of order 3 through 8 are shown above.

The magic constant for an th order general magic square starting with an integer and with entries in an increasing arithmetic series with difference between terms is shown in the second equation above.

See: math-art.blogspot.co.nz and mathworld.wolfram.com