Summary: We count the $3 \times 3$ magic, semimagic, and magilatin squares, as functions either of the magic sum or of an upper bound on the entries in the square. Our results on magic and semimagic squares differ from previous ones, in that we require the entries in the square to be distinct from each other and we derive our results not by ad hoc reasoning, but from the general geometric and algebraic method of our paper "An enumerative geometry for magic and magilatin labellings". Here we illustrate that method with a detailed analysis of $3 \times 3$ squares.

05B15, 05A15, 52B20, 52C35

magic square, semimagic square, magic graph, Latin square, magilatin square, lattice-point counting, rational convex polytope, arrangement of hyperplanes