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Most of my questions on math.SE come up in the course of personal investigations into some topic, and usually get asked after a fair bit of reflection and attempted work on the problem. Unlike questions from a textbook exercise, I have no idea a priori what sort of effort might be required to solve the question, and states of partial progress are very common.

As such, I'm frequently able to make substantial but less-than-complete progress on a question, and when I post, I'm looking for ways to extend my existing work to a full answer.

In such cases, it seems to me that there are two options:

  • Post a (rather long) question, giving both the problem at hand and a summary of my work thus far, with enough detail that relevant strategies and papers and such could be utilized by future answerers of the question.

  • Post the question alone, and then post a self-answer, giving whatever progress I have separately and leaving the option for future answers to fully resolve the question.

As an example of such a situation, take this question. I realized early on that there were different strengths of restriction I could impose on the problem, and as I found examples for multiple cases I included them in the draft of my question, but by the end the resulting post was quite long and involved, and parts of it feel rather more like an answer. On the other hand, with this question I posted an improved bound to the question as an answer rather than editing it in, given the separation in time between the two, and perhaps it should have been an edit to the question instead?

I can see some points for and against both options:

  • When parts of the content are written in a much more definitive and proof-based way, it makes sense to group them with the answers, for better separation of the relevant parts of the question and to allow users to address the two pieces separately in two different comment sections.

  • Posting the same content across two answers could be seen as a karma-farming attempt, by trying to get upvotes from users on both of the submissions at once, and should therefore be avoided.

  • Having the question show up on feeds as already answered could reduce its visibility (even if it is not accepted), because people assume that the question has already been wrapped up and so don't click on it in the first place. (Because the answer is not complete, unlike most self-answers, I want for the question to still get viewed by potential solvers.)

It is not clear to me how I ought to balance these considerations, and when to take each approach. Any guidance would be appreciated!

This meta post seems a bit related, though it deals with a rather different scenario.

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    $\begingroup$ Don't post a bare question and then a non-answer for which you want others' help to complete. Post a long question with your progress in the question, and ask for any assistance to complete it. You can group a question to separate the question from your work on it. Two posts aren't necessary to do this. Hence the first option is the best option with none of the drawbacks you seem to imply. Question / My Work and Progress, etc. They don't need to appear in separate posts. If you seek help, leave answer fields to those who can help. $\endgroup$ – amWhy Jan 18 at 3:57
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    $\begingroup$ @amWhy: The cases I'm talking about aren't "non-answers"; they're usually substantive progress on cases of the question which I'd have been happy to receive as an answer if someone else had posted them. For instance, the question might be of the form "How small of an $X$ can we find with property $P$?" and, disjoint from the problem statement and motivation, I have a long description of how to construct an $X$ of size $24$ and a proof that it is $P$. For questions of the form "here's an exercise and what I've attempted so far", I agree it doesn't make sense to post progress as an answer. $\endgroup$ – RavenclawPrefect Jan 18 at 5:03
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    $\begingroup$ +1 That helps to understand. Do you have an example of such a question/extensive progress? But if extensive, I see no problem with posting a question and "answer". But be sure to include more than just the statement of the problem in the question. Thanks for the detailed question and response. $\endgroup$ – amWhy Jan 18 at 5:08
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    $\begingroup$ If you have made a substantial (but not complete) progress and including it in question makes it too long (like tldr) then you may summarize your progress in the question by putting key ideas there and mention that details are available in separate answer. I don't think people mind of rep earned by both questions and answers. In fact self answering is also allowed if it is done in sincere manner. $\endgroup$ – Paramanand Singh Jan 19 at 2:18
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You write, "when I post, I'm looking for ways to extend my work into a full answer." In cases where this is important to you, then imo your OP here has a clear answer: post the partial progress as part of the question. It doesn't matter how long it gets. (But please use section headers / bold print / etc. to make it easier to read!)

The reason is that it is integral to the question you are asking. You are not just asking the original math question, you are asking for engagement with your partial answer to that question. If you instead post the partial progress as an answer, then you are not asking readers to engage with that partial progress at all.

On the other hand, if you only want answerers to engage with the original math question itself, and not your partial progress on it, then I can see a case on either side, and I don't think there's a clear-cut guideline. I think for me this decision is usually about timing: if I have my partial progress as of the date I ask the question, or soon thereafter, then it goes in the question (example). If I come to it much later, and especially if it's substantial (like at least 2/3 of a solution), then it's an answer (example [on MO]).

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a friend set up some web pages for me, mostly intended to hold pdf's rather than more links. My best question was my first on MO.

https://mathoverflow.net/questions/12486/integers-not-represented-by-2-x2-x-y-3-y2-z3-z

I was able to link to various write-ups there and relevant articles, already downloaded there. Very successful. Kevin Buzzard also gave the easier direction of my problem to a Master's student. I asked him for the student's thesis, so I got a pdf that largely explains necessary matters in an expanded manner.

Material now at particular directory http://zakuski.utsa.edu/~jagy/inhom.cgi

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