What are some interesting mathematical things you have learned while grading student work (or marking, if you prefer)?
It is final exams time here, so if anyone can help cast a more positive light on the grading experience, it would be most welcome.
Answers can be things that students wrote, or inspired by something a student wrote, or just something we learned during the grading process in some way. For example, clever proofs that students came up with; nice counterexamples or insights; interesting new questions inspired while grading; even just something you looked up to find out if a student's work was valid. However, for an answer to be interesting, it should be something beyond just a different way to solve a problem.