Looking Away from the Temptations of the Solution Key This is quite a soft question and I believe that it is a very important one and one that many self-learners can relate to.
So I recently was going through a problem set in topology and I came across two or three problems that I could not solve for about three days. As a self-learner, you really have no one to go to other than the textbook and the internet. I found a fully in-depth solution key to all the questions in the text and looked it over. I found out that the answers were simple enough and all I needed to do was interpret a lemma in a way that it can be interpreted to the question. 
I told myself that I would not look at the answer key unless I really needed it because I wanted to answer the question by my own knowledge alone. However that did not last for long as I started to reluctantly peek at the solution key periodically. It became quite irritating and frustrating that I needed the solution to work out a problem set that I am suppose to be able to do after reading the material for it. 
The problem sets are supposed to 
prove your understanding of the subject matter and I think that if I look at the solutions, I am not understanding and should not move on.
So in terms of questions: 


*

*When is it OK to look at the solution key?

*Is there a way I can stray away for the solution key or learn a different way?

*If you were a self learner, what would you do in this situation? Keep sitting on the problem until you solve it or look at the solution key, understand and move on?

*What else should be addressed? Are there more relevant questions to ask here? Suggestions? 


Feel free to add to this question with more to further tailor and enhance this problem to address everything possible.
 A: I think it can come down to the mindset you have while working on the problem. It's easy to beat yourself up for not seeing the solution immediately when you know the person who wrote the book understands it simply, just try not to see it as a reflection of your potential in mathematics. All the great mathematicians of history have their struggles and difficulties largely forgotten with time so it seems like they understood these things intuitively without much effort or time. The thing that gets me through hard problems is remembering that it isn't the ability to see problems quickly that makes you great in this field, it's your ability to remain comfortable and confident when up to your nose in the unknown. So, next time you feel daunted like you'll never get it on your own, remember you are practicing a skill which is more important than remembering mathematical facts: not losing your determination on a difficult problem. 
That being said, when is it appropriate to look at the solutions? I say whenever, you feel the need. If the debate is between giving up and looking at the solution, the choice is clear. Just because you've seen a solution doesn't mean you can't ask any meaningful questions about the solution or use the solution to delve into another topic. I think a lot about how most of mathematics growing up is just remembering number facts and it's incredibly useful because it allows you to use them without having to question them. Seeing new solutions to new problems helps you tackle those problems in the future, it isn't always about have the right answer to a question right when it's asked. Looking at solutions is a good way to keep you on track and keep your interest piqued and by looking at the answers when you are emotionally invested in them you are more likely to remember them down the road anyway. 
