TITLE: How to know when to give up on a ${}$ math problem, which may not have a solution?
QUESTION [6 upvotes]: When you're working on problems from textbooks, you know that the problem has a solution, because the author claims that it has.
But when you're doing original research, you generally don't know whether what you're looking for exists. You might spend weeks on it, but there might simply not be a solution at all.
How do you decide to give up on such a problem?

REPLY [3 votes]: This is where the Art of mathematics takes over from pure logic.  You may 'know' intuitively there's an answer and spend your career looking for it.  You'll get either a Nobel Prize or nowhere!  Probably better to work on other stuff as well though.