Getting stuck on difficult problems. First, a little background: I hope to go to graduate school in mathematics, but for financial reasons I will be unable to go back to school any sooner than the fall of 2016. However, since I feel quite sure I would like to undertake further studies in math, and because I really enjoy the subject, I have been self-studying on my own in my free time.
The issue I am having is this: When I tackle a problem, I am very determined, and I don't mind spending a long time working on any given problem. But, sometimes I get stuck on a difficult problem, and I work on it for days, only to end up frustrated by not finding a solution. Then, I find it difficult to skip that problem and move on to other problems, because the skipped problem keeps nagging at me. I worked nearly all the problems in Blue Rudin, chapters 1-8, but there are a few problems I never got that are still bothering me. Sometimes, I think there must be some fact that I didn't absorb well enough for that particular problem, and it seems like a problem I "should" be able to solve.
Does anyone else feel frustrated or dismayed when there is a single problem in a chapter's exercises that you just can't seem to get no matter how long you work on it? During my undergrad math studies, I did very well, so I don't think it is that I have a lack of ability. Does this sort of thing even happen to great mathematicians? I feel that I should somehow come to terms with this experience, as I am sure it will keep coming up as I continue to study math. I would very much like to hear others' thoughts and experiences with this sort of thing. 
 A: I think I know how you feel. I have had the frustrating feelings with unsolved problems. I am an university math student and have encountered a lot of problems that I couldn't solve. Sometimes I've spend days thinking about them. I have also had the problem that sometimes I don't want to ask for help, because I feel like this is my problem to solve and getting help would be cheating. 
I can't say for sure if great mathematicians feel this way sometimes, but I would imagine they do. 
Even though I myself don't ask for help often, I think that if you have given a specific problem a lot of time without results, you should ask for help. (for exemple, on this site) Whether or not you ask for help, sitting alone and looking at a specific problem can get counter productive after a while. It would simply be better for you math education if you concentrated either on new stuff or different problems. Hope this helped a bit :)
A: Look at the video here. If you do not speak French, use the subtitles. The state of being stuck is essential to getting your mind moving. Remember, rewiring your brain hurts!!
On a related note (again pointing to Connes), he says:
"I like to work on this problem because it is a test of myself that I cannot escape from. It’s not like you build a new theory, and then you can think you are the greatest. In mathematics, there is no better way to progress than to be confronted with a problem that you cannot solve. If you work on a problem that you can solve, it means that it’s not the right problem. Fighting with a very hard problem is a much better way to build a mental picture than when you are working on an easy problem. When the mind is blocked, it has much more power to build and to conceive. I see a problem like that as a gift."
This is in reference to working on the Riemann Hypothesis. Of course, Connes is more ready for that problem than the rest of us, but the idea holds weight.
