The paper, by Strashimir Popvassiliev, constructs for every positive integer n a simple closed plane curve with exactly n inscribed squares. (It’s an old conjecture of Toeplitz that every simple closed plane curve contains at least one inscribed square.) This seems to speak against philosophy, mentioned by Denne in her guest post here, that “the reason” every curve has at least one inscribed square is because every curve has an odd number of inscribed squares.
I’m not sure Popvassiliev’s example really contradicts this philosophy. Surely the squares should be counted with multiplicity, in the appropriate sense. With a more naive notion of “counting” you can’t expect parity conditions to hold. For instance, you certainly want to say that a straight line intersects a smooth closed curve an even number of times. Naively, you might complain that a tangent line to a circle intersects the circle only once! But of course, it really crosses the curve twice; it’s just that the two crossings are at the same point.