I'm doing self study on a couple of topics in mathematics, such as real analysis, abstract algebra, and linear algebra. From time to time, there are always a couple of exercises which I found too difficult to solve. I spend quite some time to think about them. When I fail, I google to find the solutions. Most of the time, I get the solutions.
However, there are some downsides to this attitude. First, I would spend too much time on a single problem. So I feel that my progress is a little bit too slow. Other than that, when I read a solution, I don't get the real understanding of the problem. When I read a proof, my brain is working mechanically to check every statement, so I don't know what exactly is going on.
I start to wonder whether I'm doing this correctly. What I want to ask here is, what should I do when I encounter difficult exercises? Should I think about them myself until I get the answers? Or should I skip the difficult ones and move on to the next chapters, then go back after I gain more understanding?