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I notice that many people put "hints" as answers. For example, let's say I ask the question, how would I solve $\sum_{i=0}^\infty \frac{1}{n(n+1)}$, and someone answers with, "HINT: It is a telescoping series". Would that be a valid answer? Should answers like these be downvoted or flagged as not an answer? I notice that answers like these don't really get downvotes and they might get the occasional upvote, so I am not really sure whether they are okay on the site or not.

Here is my question which received a "hint" answer.

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    $\begingroup$ The hinter makes the assumption that the asker is not hoping for something he can copy-paste into a homework assignment without absorbing any of it, but rather would like to be an active partner in the exploration of an interesting mathematical topic. The hinter encourages/permits the asker to experience the joy of solving the problem on his own, with just a little assist. Sounds good to me (as long as it's a good & useful hint). $\endgroup$ – Gerry Myerson Feb 25 at 22:25
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    $\begingroup$ And why not to ask for a clarification or expansion first instead of thinking on downvotes and flags... $\endgroup$ – Verónica Rmz. Feb 25 at 23:22
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    $\begingroup$ Hint answers are by far the best answers since it gives some informations and let OP make some efforts. And I agree with @veronika downvoting is not the only way to react to an answer. $\endgroup$ – Aryadeva Feb 25 at 23:23
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    $\begingroup$ @Aryadeva why not leave a comment rather than giving an answer that does not really answer the post? $\endgroup$ – quid Feb 26 at 2:05
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    $\begingroup$ @GerryMyerson sounds good for a comment, why put that into an answer box? $\endgroup$ – quid Feb 26 at 2:06
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    $\begingroup$ Hint answers need to have a bit more detail than saying it's a telescoping series. One line hints are best served as comments. I would expect hint answers to provide an outline of the solution leaving out details to be filled in by asker. $\endgroup$ – Paramanand Singh Feb 26 at 2:17
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    $\begingroup$ @quid, yes, a hint may well be better left as a comment than posted as an answer. But some hints are more elaborate than others. $\endgroup$ – Gerry Myerson Feb 26 at 3:00
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    $\begingroup$ @quid because it's an answer. A minimal one but still an answer. And you can edit an answer and more insights if OP needs more help. Something you can not do with comments. $\endgroup$ – Aryadeva Feb 26 at 5:19
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    $\begingroup$ @Aryadeva then you can post a second comment, which preserves the communication. It is as a matter of fact not really an answer. If it were, why would it be preceded by "hint"? It is a sign-post towards finding an answer. There is an actual but important difference between "hints" and terse answers. (Which sometimes is merely formal.) $\endgroup$ – quid Feb 26 at 8:47
  • $\begingroup$ @GerryMyerson fair enough but in which category would "Hint: It is a telescoping series." be? I'd guess in the latter. $\endgroup$ – quid Feb 26 at 8:50
  • $\begingroup$ In answering a similar Question here, I noted that for every post offering a "hint" answer, there is likely a post with a detailed solution. That's what occurred with your (main) Question. Both Answers were posted about an hour before you came here to meta Math.SE. $\endgroup$ – hardmath Feb 26 at 15:14
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The problem of "hints" as answers is complex as what constitutes a hint is not even quite clear, or differently not everything that is presented as a hint actually even is a hint (in my opinion at least). What we are actually often talking about in practice is content in answer posts which is preceded by Hint.

I think it would be a step in the right direction if this style of prepending Hint would be minimized, and the actual intent of the author would be articulated more precisely.

I wrote about this in an answer to a post linked to in the comments; I will not repeat everything.

In comments here it is mentioned that on a Hint answer one can ask for further details, this is true and some will be happy to engage with such comments. However, others will not at all be happy about such comments, because if they had wanted to give a more complete actual answer, they would have given one right away.

To put it pointedly some write Hint because they want to engage in comments, others write Hint because they do not want to engage in comments.

The situation could be resolved by being explicit about ones intent, rather then to prepend Hint and leave it to the readers imagination why exactly this word is there.

For the "hint" answer in the linked to post (not the one in the body of the meta post), I would say that in my opinion this is not really a hint, but it is a brief and to the point answer. In this case, instead of "Hint" I would simply write "Note that" and would consider it as a terse but arguably complete answer. The question there was basically "How to write this as a telescoping series" and the answer gave the way to rewrite the general term.

For the "hint" answer in the meta post it is a bit less clear. I think a bit more detail would be useful. Something like "Writing the general term in a different way, one can see that the series is a telescoping series. Invoking results on the convergence of telescoping series that you likely know, you can conclude easily."

For somebody that knows what a telescoping series is and has seen a handful of examples, the two are basically equivalent. The problem with the Hint is that for somebody that does not know or has never seen the notion of telescoping series it could become confusing. Of course, not knowing what a telescoping series is will make both version incomprehensible, but arguably one can expect the reader to look this up (one might help them with a link, but I would not consider this as an obligation). But now when they try to look it up, things might diverge. It's not inconceivable that looking up telescoping series one will find something like "A series is a telescoping series when it is of the form $\sum_n (a_n -a_{n-1})$ for some sequence $a_n$." Now what? This does not at all look like the series in question. Now, obviously by continuing to dig one will find what is going on, but at that point with the hint, one also just might give up on that and (wrongly) conclude that the hint was not applicable as it does not look like a telescoping series.

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    $\begingroup$ "I think it would be a step in the right direction if . . . the actual intent of the author would be articulated more precisely." This is generally my own practice. As a teacher, I often found students interpreted textbook problems accompanied by "Hints" as though "Hints" were solutions in code (i.e. "the answer is right in front of you, if only you parse this one clue"), which of course they aren't. It distracts some people. Just removing that word and presenting context/guidance seems to work better on average. But many Math.SE users of "Hint" are far better with hints than textbook authors. $\endgroup$ – leslie townes Feb 27 at 1:31

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