It's probably well known in some quarters that for any integer N > 0 that's the sum of 2 square numbers, the torus T = R^2/Z^2 with the induced metric -- aka the "square torus" -- can be tiled by N congruent square tiles. (Note that such N include all square numbers.) It's a blast to work out how this goes, so I won't spoil your fun. I'm not sure how to prove the converse -- or certain that it's true (though this seems likely). Can someone prove the converse: "If T can be tiled by N congruent squares, then N is the sum of 2 square numbers." . . . . . . or come up with a counterexample ??? --Dan