Self-Complementary Magic Squares

Abstract

A $magicsquare$ of order $n$ is an $n\times n$ array of integers from $1,2,\dots ,n^2$ such that the sum of the integers in each row, column, and diagonal is the same number. Two magic squares are $equivalent$ if one can be obtained from the other by rotation or reflection. The $complement$ of a magic square $M$ of order $n$ is obtained by replacing every entry $a$ with $n^2+1–a$, yielding another magic square. A magic square is $self-complementary$ if it is equivalent to its complement. In this paper, we prove a structural theorem characterizing self-complementary magic squares and present a method for constructing self-complementary magic squares of even order. Combining this construction with the structural theorem and known results on magic squares, we establish the existence of self-complementary magic squares of order $n$ for every $n\ge 3$.