Somehow while writing the first two posts I managed to miss this clear and thorough write-up about the inscribed square problem by Mark Nielsen, a math professor at the University of Idaho. From his page one learns that indeed every polygon has an inscribed square. In fact, every curve which is not somehow horrifically fractal does have an inscribed square — this is a theorem of Walter Stromquist from 1989. Nielsen also includes many related theorems, references, exercises, and a few toothsome unsolved problems at the bottom of the page. Go look!