The celebrated Four-Vertex Theorem says that the number of vertices of a plane curve must be at least 4. Must it be, more specifically, an even number at least 4?
Here's why that seems to me that it should be the case: Suppose you have a closed curve $\gamma(s)$ parametrized by arclength, with length $L$. Assume WLOG that you parametrize such that your starting and ending point is not a stationary point of curvature (if no such point exists, you're looking at a circle...let's consider $\infty$ to be an even number). Since $\kappa(s)$ is smooth and periodic, its derivative at $0$ and at $L$ must be the same, and that derivative isn't zero. So $\kappa'(s)$ must change sign evenly many times on $[0,L]$, hence evenly many vertices.
QED, maybe? The thing that makes me suspicious is that, in discussion of the Four-Vertex Theorem, I've not seen this result stated. Did I miss something?