Compile short list of links to model questions (by type)

Question

This thread is opened for a compilation of exemplary questions of various types, that can be linked as models or advice on how-to-post.

Here "exemplary" primarily means questions that are structurally excellent in ways that can be repeated in other postings, such as motivation of the question, strategic use of bounties, informative discussion of sources, or demonstration of un-obvious difficulties in the question. The specific mathematics within the questions is not as important, though there have been some postings that are memorable simply for the very high quality of mathematics brought to the questions and answers. If the good structural features can be articulated, they can be imitated.

The hope is that examples could replace the presently most common response -- negative criticism, whether direct or implied -- to questions of similar type that lack the positive attributes. There is at least one recurring category of posting (you know which one) where a change from negative to positive replies would be an ecological improvement to the site.

Answer 1

An [original] problem was contributed from the poster's [research]. Persistent questioning, further analysis and encouragement in comments, and strategic offer of bounties that kept the question visible, led to a progression of increasingly detailed answers, culminating in a magisterial resolution (and possibly new, probably publishable, mathematics):

Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$

Answer 2

This [general-topology] question is my all time favorite on the site. The question was great, and Ryan Budney's answer was FANTASTIC! This question is going to lead to a talk given at my local science cafe group, after speaking with Ryan this past week about his excellent answer.

I think it shows how a real life situation can be synthesized into an excellent question, and people can make multiple approaches to answering, even when the asker doesn't know the specific mathematics necessary to understand the complete solutions.

Answer 3

This [homework] question got a warm reception when the poster explained his partial progress, which contained several promising ideas, and noticed that his argument did not use one of the hypotheses in the problem.

Convergence of $a_{0} = 0, a_{n}=f(a_{n-1})$ when $|f'(x)|\leq \frac{5}{6}$

After solving the problem (within 20 minutes) with help from the answer, the poster updated the question to include the completed line of attack. This sets the stage for others to provide further information about the problem beyond the solution.