Every time I'm trying to solve a tough math problem it always seems like I'm overthinking things after like 3-5 (and in one case like 4 days) hours of work (with a few breaks of course) and several sheets full of math, and I get very discouraged when I see the 2-paragraph answer (or even a single equation). It seems like every time this happens it's because of a few things:
- I spent way too much time on a single approach that won't work. It's always the case that this approach definitely feels like the right approach. By this, I mean that I can see several different (non-obvious, so it's probably not designed to trick you into doing the obvious thing) connections between the given and the solution, but turns out that they are "fake".
- I spent way to little time on the approach that would have led to the solution.
- I don't see the connection between the right things.
For the third problem, I feel like it's because I don't know the content like the back of my hand. Sometimes the solution requires connections between ideas that were discussed in the chapter, but that I didn't realize would led to the solution. But I don't have a super-memory or infinite time to remember all of these things. Even my notes which are supposed to only highlight the important stuff, don't help that much.
How can you know that an approach won't work? Does making these connections come with experience? etc.