Thanks for all the suggestions! I also just realized that my nondeterministic solution of choosing random points and finding out whether they're in the circle works just as well with deterministic uniform point selection. In other words, counting cells inside a circle drawn on graph paper instead of counting darts thrown at a dart board. Let's see if I can write it as a formula instead of an algorithm: n n 4 ___ ___ --- \ \ pi ~ n*n * / / 1 if sqrt(x*x+y*y) < n --- --- x=0 y=0 Does that look about right? I'm surprised I didn't run across anything like that. Maybe I should add it to the wikipedia article for the benefit of readers with more elementary math backgrounds. I suppose it's not quite as nice as a series, since you have to decide when to stop before you begin. Perhaps we could get a run-until-you-get-board series if we did something tricky like progressively subdividing the coordinates rather than drawing bigger and bigger circles. I'm also curious whether a similar algorithm could be constructed for approximating sines. Maybe something doing recursive angle bisections as was suggested for the Archimedes approach? -J