Here's a problem from Rolfsen's Knots and Links that has me scratching my head:
Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to draw a counterexample, try holding two pencils at once.]"
Here's the "chord theorem": If $C$ is a line segment of length $|C|$ with endpoints in a path-connected subspace $X \subset \mathbb{R}^2$, then for each $n \in \mathbb{Z}_{>0}$ there is a line segment parallel to $C$ of length $|C|/n$.
Any hints would be greatly appreciated.
My thoughts: I can't figure out how to make the "pencil" hint work. I'm currently trying to prove that if $\alpha \in \mathbb{R}_{>0}$ is not of the form $1/n$, then there exists a continuous function $f:[0,1]\to \mathbb{R}$ such that $f(0)=f(1)=0$ and $f(x) \neq f(x-\alpha)$ for all $x \in [\alpha,1]$. Then the graph of this function will be a "counterexample".
Update: With the help of everybody's comments below, I've gotten most of the way to an answer, posted below.
Thanks to the comments regarding Rolfsen's "pencil" hint, I've had some luck drawing a sort of "modified sine wave". For values of $\alpha$ that aren't of the form $2/n$, one can use $f(x)=\sin(2\pi x/\alpha)- x \sin(2\pi/\alpha)$, seen below for $\alpha\approx .37$. Now I just need to cover the case of $2/n$ with $n$ odd. Here's how we know it works for $\alpha \neq 2/n$ with $n \in \mathbb{Z}$:
Suppose $f(x)=f(x-\alpha)$. Then we have \begin{align} \sin\left(\frac{2\pi x}{\alpha}\right)-x \sin\left(\frac{2\pi}{\alpha}\right) &= \sin\left(\frac{ 2\pi(x-\alpha)}{\alpha}\right)-(x-\alpha)\sin\left(\frac{2\pi}{\alpha}\right) \\ &= \sin\left(\frac{ 2\pi x}{\alpha}\right)-x\sin\left(\frac{2\pi}{\alpha}\right)+\alpha\sin\left(\frac{2\pi}{\alpha}\right). \\ \end{align} It follows that $\alpha \sin(2\pi/\alpha)=0$, hence $2\pi/\alpha = n \pi$ and thus $\alpha=2/n$ for $n \in \mathbb{Z}$.