A self-complementary magic square of order n is a magic square M whose entries consist of the first n² natural numbers such that when each entry i of M is replaced by n² + 1 — i, the resulting square is equivalent to M (under rotation or reflection). The purpose of this paper is twofold. For any given singly even natural number n, (i) we extend the method of Strachey to produce many magic squares (instead of just one) of order n, and (ii) by making a variation of this extended method, we construct many self-complementary magic squares of order n.