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As a self-studier, I have no due dates and no time pressure. This is partly a good thing since I do have a life outside of my pursuit of mathematics. However, the lack of pressure is also a bad thing, as I often do not progress as quickly as I would like.

I am very often faced with a dilemma: how much time should I spend on a given problem? I am sure there is something to be said for seeing a problem through to the end and pushing and struggling until one solves it. However, there are times that I ask for a solution or a hint here, and upon seeing it feel certain that I would not have seen what I would have needed to in order to solve that problem, simply due to lack of experience.

So what I want to know is:

How much time is too much time to put into a single problem? At what point should one move on, so as not to get bogged down on one problem too much? How does one balance 'giving up' (or let's call it "moving on/prioritizing one's time") with spending so much time on single problems as to impede progress?

Thanks!

EDIT: This seems to be a somewhat popular question, and honestly I'm surprised it hasn't received any answers yet. This matter seems to be highly nontrivial, and especially so for those doing research and trying to prove things that may not even be true! I'm sure mathematicians have developed some sort of strategy for these situations. I'd love to hear some. Anyone?

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    $\begingroup$ Just to have an idea of the order of magnitude, roughly how much time do you spend per problem now? Seconds? Months? Also, what topic(s) are you studying? $\endgroup$
    – ShreevatsaR
    Nov 8, 2012 at 4:32
  • $\begingroup$ I'll work on problems on a topic, usually skip around a little, put some thought into them at times throughout the day... Then for the problems that entice me the most I'll try to put a solid hour or two into them, moving on at that point if I haven't made any significant progress (and often posting the question here to obtain a hint or solution). Right now I'm working on point-set topology. $\endgroup$
    – Alex Petzke
    Nov 8, 2012 at 5:24
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    $\begingroup$ @AlexPetzke: Dear Alex, you may find this link useful: math.stackexchange.com/questions/111819/… $\endgroup$
    – Student
    Nov 12, 2012 at 4:53
  • $\begingroup$ That was helpful, thank you. $\endgroup$
    – Alex Petzke
    Nov 12, 2012 at 16:40

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Obviously, there is no good answer to your question measured in minutes, hours, days, etc. I am also a self-learner of math, and I have found a few heuristics useful:

  1. Have you ever taken a test, and managed to solve everything besides one diabolical problem? You may have come close to the solution, but just couldn't get the algebra to work in the end. Or can't remember the one simplifying assumption you had to make in order to finish a proof. As soon as the exam is over you open up your textbook and kick yourself for not remembering the simple fact that could have solved it for you. These are the kind of problems you never forget -- whenever I get that feeling that I need to know the answer, I'm generally not going to forget it (even if I look it up).
  2. Make sure you are "prepared" to be working on the problems you choose. I tutored a group of high school students a couple years ago, and was helping them with their algebra II homework. We were doing working on factoring polynomials, and I noticed that they would pull out their calculators to do single digit multiplication problems as they came up. This makes most people cringe because algebra is an abstraction of arithmetic. Why should you study a certain kind of problem solving machinery if you don't even know what its purpose is? This doesn't just apply to basic subjects. If you can't do it in 1 dimension, you certainly can't do it in $n$ dimensions.
  3. A good progression for proving mathematical facts is as follows: convince yourself, convince a friend, then convince a skeptic. If you are struggling to write a formal proof (one that even a skeptic would have to accept), ask yourself if you are even convinced of a fact. Can you instantly produce an example of the theorem you are proving? Can you instantly produce an example where the theorem doesn't hold, and identify which of the hypotheses is missing?
  4. Know which problems are worth working all the way through, and which ones only warrant sketching a solution. The problems I always do are the ones that mirror central theorems or examples from the chapter -- you really know you understand a line of reasoning if you can apply it. I also find it helpful to work through at least one of the extra-involved problems at the end of the chapter, (you know, the one with three stars and a warning for how hard it is). Most books have these, and they usually involve a pretty tangential problem, but they really make you get your hands dirty. Pick up a book you think you have mastered, and look at some of these problems.

These aren't so much criteria for how much time to put in to any single problem, but hopefully they will advise you (as they have for me) on where your time is best spent.

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I would consider having more than one problem 'on the go' at a time. Similar to the concept often repeated at school before exams, "if you get stuck on a problem, move on and come back to it later". There are two clear benefits I can see to this approach:

  1. You find the answer to the problem you're stuck on by solving a new problem.
  2. Over the passage of time, the answer to your problem finds its way into your brain through the ether.

Decide what a comfortable number of problems to have 'on the go' at a time is (maybe something like 5) and if you find that in a week, you can create no solutions to these then it's probably time to start getting some hints for yourself.

Disclaimer: Most of this is personal opinion though I do I have what I would consider good experience at working at problem solving as I worked in academic research for a number of years and now work as a software developer.

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  • $\begingroup$ Thanks for the answer. I like this idea - I do it somewhat already, but could probably emphasize it more. There are often times that I mean to go back to a problem, but never do. It would probably be better to have those pre-decided problems, and decide to get help after a week if I'm not making progress. $\endgroup$
    – Alex Petzke
    Jan 10, 2013 at 21:22
  • $\begingroup$ Thank you - I really like your answer. I am also having trouble with this and certainly going to give this a try. $\endgroup$
    – xcrypt
    Jan 24, 2013 at 17:10
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A rule of thumb I found useful in many situations is to ask yourself if you have any new ideas for a possible solution.

If yes, then usually it is worth it to try to work through the ideas, see if they work, see where they fail and why, that sort of thing. If you are able to identify a weaker version of the problem, it is usually worth having a go at it as well. Even if this does not in fact get you to the solution, this usually brings you closer to understanding of the problem.

It can unfortunately also happen that the answer is no (at least, it happens to me quite often). In this case, it is usually reasonable to abandon the problem, at least for the time being. If you catch yourself looking at ideas you have already looked at (without much progress), or can't thing of a new idea to try for a significant period of time, then I would treat this as a warning sign.

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  • $\begingroup$ Thanks, that sounds like good advice. Something else I've thought about recently is that I should make sure I really exhaust each idea I have, and make sure I give it enough of a chance. But this question applies at the idea level too: when should one give up on an idea and try another? $\endgroup$
    – Alex Petzke
    Mar 30, 2013 at 1:58
  • $\begingroup$ This was a very underrated post. Great explanation. $\endgroup$
    – AnotherPerson
    Oct 5, 2015 at 6:35
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I have one problem that I have been working on for over ten years now. I work on it when a new idea occurs to me or when I get the desire to review my notes. I really can't leave it, but I am seeing other problems on the side.

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I think you can pursue a problem as long as you want, and as long as you can afford it. But the latter is a crucial point, which may be a reason for asking this question. Getting stuck can also mean that there is no more time for other successful work. There are some persons who do not mind such disadvantages, or persons who are extremely brilliant and sometimes do not want to accept the money when they obtain a fields medal for the solution of a famous problem (after having worked more than ten years on this problem).

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My answer would be for you to ask yourself a few questions before moving on. I am not saying give up on the project or problem. Sometimes our brains need a rest and a change of scenery for a bit. Sometimes 5 min works, we are all different. Also I would like to add this is something that works for me, not everyone will feel the same way BUT it's worth a shot.

Question 1: Do you feel like you keep moving and getting no where?

Question 2: Make a list of all parts of the problem or process and guesstimate the time. (BUT I DO realize this is not possible in all situations).

Question 3: this is really an action not a question. Get up and take 5 minute break. Get a cup of water, or whatever, look out the window. Move your body around if you can. Main thing is to take your eyes off the current project.

Question 4: Put this problem or project aside and start or finish another problem.

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