It's fine to work on any problem, so long as it generates interesting mathematics along the way - even if you don't solve it at the end of the day.

I loved doing problems in school.

I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about.

Fermat said he had a proof.

I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.

Just because we can't find a solution it doesn't mean that there isn't one.

Mathematics... is a bit like discovering oil. ... But mathematics has one great advantage over oil, in that no one has yet ... found a way that you can keep using the same oil forever.