How much time is too much (to put into a single problem)?

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?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.