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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:

  1. 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".
  2. I spent way to little time on the approach that would have led to the solution.
  3. 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.

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    This is off-topic for this site but to give my personal opinion... I experienced that same feeling a few years ago back when I was an undergraduate majoring in math. You just have to push through it and keep learning as much as you can. What you're experiencing is common amongst (pretty much) everyone studying math at advanced levels no matter what kind of prior background and education they have. Rarely do I ever see a problem and know how to do it right away, and even when I have some idea, it doesn't seem to work most of the time. – Accelerator Jul 22 '23 at 04:37
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    If you see a proof that's completely different from what you wrote, you shouldn't get discouraged about that because at least you wrote something rather than nothing at all. You don't need some superpowered and photographic memory to do well. And let's be real here: you're going to fail and get a whole bunch of your proofs wrong, but that's just part of learning math. That can be applied to virtually anything difficult like sports or academics. – Accelerator Jul 22 '23 at 05:00
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    You would know if using some so-and-so approach is useful or not when you try it out and see if it works. If it doesn't work, then try another approach. If that doesn't work, then try another one. Or maybe ask for help from a teacher, friend, or online. Ultimately, it's really about exposing yourself to the material as much as you can, so yes, making those connections comes with experience. – Accelerator Jul 22 '23 at 05:00
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    There is no real "math" answer to that, so I wouldn't be surprised if the question got summarily dismissed. That said, my random opinion... 1/2/ It is a positive that you see several possible approaches to begin with. It helps to ponder the problem for a little while before you try any of those, maybe something else comes to mind. Then, don't try to push any one approach all the way through. Stop when you hit the first snag, go back, try another one. Repeat as necessary. You gain better insight along the way, which will eventually help solve it. 3/ That comes with, and only with, practice. – dxiv Jul 22 '23 at 05:06
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    Finished proofs that people have made for others to read have had all those dead ends pruned away. That very much does not mean they weren't there when the person who wrote the proof worked on the proof. It's a huge part of doing math that you almost never see when math is presented. And I personally think that's a shame, because what you experience here can certainly be demotivating. I think it would help to see that it happens all the time, even to the most celebrated geniuses. – Arthur Jul 22 '23 at 05:29
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    One trick I learned is that if you cannot figure how to prove something is true, try to use the roadblocks you've encountered to ttry to prove it false instead. Either you will be successful, or else you will run into roadblocks this way too. Now try to use those roadblocks to figure out ways around the original roadblocks. Keep repeating. And remember, it took Rowan Hamilton, one of the finest mathematicians (and vandals) of his age, 20 years to figure out how to multiply triples together (thus inventing quaternions, and leading to vectors). Now we can guide you through it in minutes. – Paul Sinclair Jul 23 '23 at 18:07
  • Does this answer your question? Maximum amount of time to solve a problem and persistence – user1110341 Oct 09 '23 at 02:04

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