The original post does not specify the nature of the problem. The problem might be something objective like scientific research. Or something deeply personal li

The original post does not specify the nature of the problem. The problem might be something objective like scientific research. Or something deeply personal like an illness of a friend or family. Depending on the problem, the approach will of course be completely different. In case of scientific research also, it depends on the situation. A PhD student's approach to a difficult problem with apparantly no solution would be and should be different from someone who does not have a deliverable. If you are a student, you need to deliver some result for a degree. So after some time it would be better to forget about a seemingly unsolvable problem and focus on a new problem which you can solve in perhaps a better way. A senior researcher however in my opinion should persistently try to solve the harder problems. As I feel it is their job. Coming back to different problems than scientific research, I can only think of one approach. Ask the question, how important is this problem to you and your life? If it is as important as the the well being of you, or your close ones, then you have no choice but to focus and fight to find a solution.