Assuming we have a regular language $L$, how can we prove that $L'= \{ xz \mid \exists y : xyz \in L \text{ and } |x|=|y|=|z|\}$ isn't necessarily regular.
So far I can't come up with much for how to solve this. I was thinking we could solve this by either generating an example where $L'$ isn't regular or we could use the pumping-lemma, but I'm unsure how to get started. This was the hardest one in my set of practice problems.
If we take $L = \{a^n \mid n \in \mathbb{N}\}$, then $L' = \{a^{2n} \mid n \in \mathbb{N}\}$, which is a perfectly fine regular language. Or even more trival, we could observe that $\emptyset' = \emptyset$.
As such, it does not hold that $L'$ is never regular for a regular language $L$. This means that answering the question will involve constructing a particular language $L$ for which you can prove that $L'$ is not regular (and such a proof could use the pumping lemma).
As a heuristic for how to attempt this, I would first look at some typical examples of non-regular languages, and try to find a way to express them as $L'$ for some regular language. If I don't succeed with that, I'd attempt to built $L$ by simultaneously ensuring that $L$ itself is regular, and by running the pumping lemma on $L'$ to prove that it is not.
