Other Magic Squares
Seymour S Block, Santiago A Tavares · 2009
Abstract In a sequence {1, 2, ... , n2} of consecutive numbers, where n is the order of a magic square, any pair of numbers is said to be complementary if their sum equals the sum of the first and last number of the sequence, that is, n2 + 1. For example, in the case of magic squares of order 5, the pairs (12, 14) and (6, 20) are complementary with respect to the sequence {1, 2, ... , 52}, where the first is 1 and the last is 25. This implies n2 + 1 = 52 + 1 = 26; therefore, 12 + 14 = 6 + 20 = 52 + 1. A magic square is associated if the numbers on each two cells symmetric with respect to the center are complementary. If the square is even, it must be symmetric with respect to the center of the square.