Question: Is it acceptable to leave a hint as an answer?

For the purposes of this question, let me attempt to define a "hint":

A hint is a short, generally terse, comment which does not provide an answer to the question asked, but instead gives a running start on the problem which, if followed correctly, will lead to a solution. For example, if someone asks how to evaluate $$ \int_{0}^{1} x \mathrm{e}^{x^2}\,\mathrm{d}x, $$ a hint might consist of the answer "Try the change of variables $u = x^2$." A proper hint (rather than an off-the-cuff suggestion for what one might try next) should ultimately lead to a complete solution. (Hat tip to quid for articulating this fairly well).

I am asking this question in order to seek some kind of consensus. There has been discussion in the past, but I don't see a clear consensus (and maybe there isn't one!), the earlier discussions seem rather out of date, and the questions seem to be more about leaving hints-as-answer vs hints-as-comments. So as not to muddy the issue, I want to focus very narrowly on whether or not it should be okay to leave a hint as an answer.

To summarize my impression of earlier discussions:

  • An early discussion of hints-as-answer seems to land on the side of "Yes, this is okay." However, I think that the underlying assumption of the discussion here is that it is always okay to leave a hint, and the question is where is the right place to do so (comment vs answer). I also note that at least one answer makes reference to the now burninated homework tag. This suggests that some new discussion might be productive.
  • More recently, it was suggested in this topic that we make a change in the culture of MSE to focus more on hint giving. This proposal seems to have been received negatively, but that doesn't really address whether or not providing hints-as-answers should be okay.
  • Answers to this question suggest that it is acceptable to leave hints-as-answers, but it seems that this is suggested as a way to bypass the limitation that low reputation accounts cannot comment.
  • Possibly relevant (there is some discussion, but not much): hints for homework questions (reloaded)
  • Should there be a policy about hints.
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    What constitutes a hint? I'd have thought what @quid describes in a comment to the "no" answer would be a hint, but it seems they disagree. – Theoretical Economist Aug 22 at 16:31
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    @TheoreticalEconomist in my mind a "hint" is some piece of information that can help to come up with a solution but does not provide one, or does not even clearly suggest what the solution is. – quid Aug 22 at 16:32
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    "does not even clearly suggest what the solution is" -- that, to me, is not even a hint, and most likely just a bad answer. Admittedly, I might also have a different interpretation of the word "clear" than you do. Defining a hint might be a struggle. @quid – Theoretical Economist Aug 22 at 16:37
  • The best of hints are more difficult to master than full solutions. However, for those who can use hints skillfully, I find such hints as far more valuable than a full solution. So I doubt you are going to get a straight-out "always right/always wrong consensus." Lacking any definition of what a "hint" constitutes, I think this was a ill-proposed question. – amWhy Aug 22 at 16:44
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    If by "hint", Xander, you mean: "Anything short of a fully worked out step by step solution to the very particular problem asked on a specific date, say 0x/0y/abcd, for user @username", I'd have to answer "it is acceptable". – amWhy Aug 22 at 16:54
  • That "hint" you added is about at the threshold of what I'd accept. A bit more would be better but alright. However, if it would only say "Use a change of variables." I'd say that's too vague for an answer and should be a comment (at best). – quid Aug 22 at 20:13
  • @quid I chose that as the example precisely because I myself find it to be pretty borderline. If I saw it in a review queue, I would probably recommend deletion. – Xander Henderson Aug 22 at 20:16
  • @Xander Why did you walk back (delete) the second most upvoted answer you first supplied: Yes, it is acceptable (which I also upvoted, with the understanding "Yes it is sometimes, but not often, acceptable.") – amWhy Aug 22 at 20:53
  • @amWhy The upvotes on the "yes" answer basically matched the downvotes on the "no" answer. It seemed redundant, and the "No" answer had more comments. – Xander Henderson Aug 22 at 20:56
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    @Xander Please keep this question above board: transparent. It is fine for you to edit your question (making clear what the update is), or to provide an alternative answer, with an explanation why you are doing so. But don't delete an option that 5 user upvoted: Yes. It is acceptable to leave hints as answers. (10K users can confirm this). That's just plain misrepresentation. You may not be happy with those upvotes voicing the opinions of five users (we can't know how many would have voted so, had you not deleted the option), but bite the bullet and let users' votes stand. – amWhy Aug 22 at 20:56
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    @amWhy I don't see how any action that I have taken is not transparent. The two answers I provided were "yes" and "no", but I realize (in retrospect) that this was redundant. An upvote on the yes answer is, so far as I can tell, entirely equivalent to a downvote on the no answer. The two answers were redundant. In retrospect, the right thing to do probably would have been to take a stance in the question itself, and let those votes speak. – Xander Henderson Aug 22 at 21:00
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    As I say to many askers on main: don't edit a question so as to create a moving target of a question, without leaving the original, and marking changes. Else current answers become irrelevant. – amWhy Aug 22 at 21:01
  • No, it is not redundant, because I meant yes, when I cast a now meaningless vote to to the deletion of the answer as "sometimes yes". And on NO I voted "yes, because sometimes no". Now the only representation of my vote is taken by you to mean, NO, never. So you have essentially, tilted the results in your favor. – amWhy Aug 22 at 21:04
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    @amWhy, really both of Xander Henderson's answers could be deleted. The question is tagged discussion: an answer which doesn't make any arguments for its position is inappropriate. – Peter Taylor Aug 23 at 8:09
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    @user170039 Your first link isn't even a hint. In fact, it's a complete answer, whether you understand it or not. – amWhy Aug 23 at 22:07

Sometimes the question explicitly only asks for hints.

Rather more often, a hint will actually be more useful to the asker. This is often the case for "homework" questions (when they deserve to be answered at all).

With many exercise-type questions, nobody (including the problem setter) actually cares about the answer, but the point of the exercise is to give the student a chance to acquire familiarity with definitions and techniques by thinking about them themself. Giving a full answer would rob the asker of the chance to learn; but a hint pointing in a fruitful direction might save them from being stuck.

For example, I remember a question asking whether various cartesian products of closed and half-open intervals, equipped with the lexicographic ordering, have the supremum property. That sounds like a good exercise for getting some feel for how the definitions of least upper bounds and/or lexicographic orders work -- but I'm pretty sure that in reality nobody cares whether those particular sets have least upper bounds. Giving a complete answer would have killed the value of the question without contributing anything useful to anyone.

Here is another example, posted during this discussion. Humanity is not made even a tiny bit richer by having the particular equation the answer concluded with posted on this site; and the asker has been deprived of any chance to practice things for himself. Posting just a hint that such-and-such two vectors need to be perpendicular would have been much better for everyone.

  • You seem to conflate "hints" and "terse answers." Also for the first phrase, well, yes. That's an obvious exception. (If you want you should point out it should be exclude in the question, to preface your answer with it creates confusion.) – quid Aug 22 at 16:37
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    @quid: What makes you think I confuse "hints" and "terse answers"? When I say hint, I mean hint. – Henning Makholm Aug 22 at 16:38
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    The example you give. – quid Aug 22 at 16:39
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    @quid: What about that example makes you think I mean something different from hint when I say hint? I think it is a typical example of a question where giving a hint is much better for everyone than giving an answer, however terse, to the exercise. – Henning Makholm Aug 22 at 16:40
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    A terse answer would not spell out the solution for each set either. – quid Aug 22 at 16:41
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    I think we do indeed need an option of "Sometimes giving a hint is more valuable than answering fully." – amWhy Aug 22 at 16:42
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    @quid: I think you're conflating "terse answer" with "hint". If the question is "does such-and-such set have the supremum property?" then an answer that does not give (or clearly imply) a "yes" or "no" is a hint, not a "terse answer". – Henning Makholm Aug 22 at 16:42
  • There is actually a spectrum. Maybe "terse answer" is not the ideal description of what I mean. This notwithstanding there are very different types of answer-posts that are marked-up as "hint" on the site. I think some are alright (and would not really need the marker "hint"), while others are not. – quid Aug 22 at 16:46
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    Of course, @quid. Just as there are some answers not marked as a hint which are not alright. – amWhy Aug 22 at 16:48
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    @quid: Well, yes. If "hint" is restricted to mean one of Doctor Germanguy's answers that consist only of an algebraic identity of dubious relevance to the question, then I'd agree that those should not be acceptable. But that's not because they're hints. – Henning Makholm Aug 22 at 16:49
  • In the high time of "hints" we also saw hints that said "Use induction on $n$" and just that. – quid Aug 22 at 16:51
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    @quid: "Providing a clear direction" is in my understanding the quintessential feature of a hint. (A purported "hint" that doesn't even implicitly promise that the hint will work is not a hint, just random musings). – Henning Makholm Aug 22 at 17:02
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    @HenningMakholm this is a question and answer site, not a tutoring service. Answers should be readily usable. "Brevity is acceptable, but fuller explanations are better." This is the official stance and not my personal opinion. – quid Aug 22 at 18:19
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    @quid: And an answer that takes away the asker's opportunity to learn from the exercise are the opposite of usable -- readily or not. – Henning Makholm Aug 22 at 18:24
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    @quid When "Can someone help me? I'm not clear where to begin" or "Can anyone help me? I just need a suggestion to start", and someone wants "to help them with a suggestion to start", that is a valid answer to the given question. When a user (and many users use this daily) says "Please help me", there is no policy on this site as to how to best help. You're assuming that anything short of a fully worked out, particular solution to a particular, isolated problem is not helping. Show me where it is defined on this site or on MSE, what is helpful? – amWhy Aug 22 at 18:54

Apparently there is considerable divergence regarding what a hint is. So, here is what I expect from the content of an answer-post, whether it is prefaced "hint" or not is maybe not crucial. (Obviously there are questions to which this does not fully apply or for which it does not make sense at all, say a reference request, it is for your typical question that asks about a mathematical problem without particular restrictions on the answer.)

  1. An answer-post comes with the (implicit) promise what is described or suggested actually works. If it says "Try to do {this}." Then doing {this} should actually work. An answer post is not for hopeful recommendations what might work.

  2. Related to this, the answerer actually should be able to give a complete and detailed answer. (If somebody believes they have a good suggestion that could work, but are not sure about it, they can leave it as a comment.)

  3. The post should contain something that directly allows some progress, and is described in an unambiguous way. In particular, answer-posts in the form of rhetorical questions should be avoided. It is annoying for everybody involved when those get flagged up as Not An Answer and then get "if you have a new answer" comments from review. (Yes, it is sometimes more the flaggers fault, but it can be tricky sometimes, and we want to make life easy for everybody.) The very same information can usually be conveyed in a form that does not lead to that confusion. Moreover, rhetorical questions especially in writing are not really an efficient way to get somebody to think about the problem. Often they'll rather think about what the asker likely expects.

To avoid confusion, this is my personal opinion quickly stashed down, I am not in the habit of removing content that goes against these preferences.

That said, as this discussions shows "hints" means all kinds of things. Independent of the actual content of the answer post somebody creates, I would kindly like to invite everybody when they write "HINT:" at the start of the post to pause and think what they actually want to convey by this and if this could not be conveyed more directly.

Some things that seems to be common use cases:

  • "Hint" because it is an outline of the method but does not carry out or finish routine calculations or spell out all details. In my opinion, that's not really a hint but an outline of a solution and should be described as such if at all (actually mostly no actual explanation is needed, usually people get what is intended).

  • "Hint" because it only does one or a few of similar problems. That's a partial answer not a hint.

  • "Hint" because it stops at some point. That's a partial answer not a hint. That's alright. It's good though to say that it is incomplete though. This is not always clear to everybody.

  • "Hint" because it gives the crucial idea but does not say much in addition. This one is an actual "hint" in my book. Still I'd say on the one hand it can often be useful to say a bit more (which then shifts it into the former category), or at least to say how specifically it applies. I cannot help but feel that some "hints" are mainly popular with those that don't need them actually, but find them nice and are happy that they understood them.

  • "Hint" because it give some relevant piece of information that when interpreted correctly is helpful. Again, that's a hint in my book. It's risky though since it might just not help at all, too. Personally, I'd avoid to write answer posts only comprised of this. But, alright, that's an actual hint. I'd still recommend to give at least some complementary information.

  • "Hint" because somebody was a bit too lazy or too busy to write an answer proper and want some fallback. Just don't.

  • "Hint" because somebody answers a question they know they should not answer and wants some fallback. Just don't.

  • "Hint" because somebody was actually not sure about the answer. But, hey, why not throw it out there and let it validate or correct by commentators. Again, just don't.

  • I agree with the first two points. For the third, are you sure there's nothing missing in the sentence that starts "It is annoying for everybody ..."? I have trouble understanding what you're talking about there. Is it something about moderator flags? – Henning Makholm Aug 22 at 18:54
  • I am glad we agree on the first two points. The second part of 3. is about "Not an answer" flags that are raised (sometimes in error yet still) with some frequency on rhetorical questions intended as hints/answers. In review there is then an option to recommend deletion with auto-comment "If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context." Which is intended for new Q posted as answers in error. Then confusion ensues. Thus, avoid rhetorical Q (as A). – quid Aug 22 at 19:10
  • (Aside: I've seen those canned comments, but I've never figured out how to get to a point where the system would offer to post one of them for me. Which review queue do they come from?) – Henning Makholm Aug 22 at 19:16
  • As for rhetorical questions, what would you think about an answer to this (assuming for the same of argument the OP shows some effort) that went roughly: "Hint: If the graph has exactly one node with degree $1$, it also has exactly one node with degree $n-2$. Can the isomorphism between $G$ and $G^\complement$ map two nodes to each other?" – Henning Makholm Aug 22 at 19:19
  • @HenningMakholm I think that this would be unnecessarily indirect. But it is unlikely to create the problem that I mentioned. I'd just leave the word "hint" away and state that this cannot happen. (Maybe indicating that it would be good to justify this in detail.) [I did not think the question through fully, maybe the answer is off. But that is not really the point.] – quid Aug 22 at 19:27
  • More broadly an issue with this type of formulation is that not few faced with it actually rather than thinking about the math will try "to read your mind" and infer what the answer is that you likely expected based on the formulation. – quid Aug 22 at 19:39
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    x @quid: The reason for that formulation, rather than something declarative-but-terse, is to make it obvious that it won't work just to copy it verbatim and hand it in as a homework solution -- the OP will need to have some minimal understanding of what's going on before they can transform the hint into something that looks like an argument. – Henning Makholm Aug 22 at 19:47
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    I understand that but I believe that there are other ways to convey this. Even something like 'Investigate whether it is possible that ...' is better. To ask a question, especially in writing, just adds a layer of confusion and indirection. – quid Aug 22 at 20:05
  • To answer the aside that I had not seen before, sorry: that's the "very low quality queue" On the site there is a "From review" link, when you follow that you get to the review summary. – quid Aug 22 at 23:31

Sometimes it is obvious that a question-asker is a student struggling with a homework question. In this case, I argue that a hint not only can, but should be posted as an answer:

  1. It is pedagogically beneficial for the student to work out the problem as much as possible on their own; a hint that clears up an error or misconception on the part of the student, or points a key idea the student needs to apply to solve the problem, is exactly what the student needs. The student can always comment and ask for more help if needed.

  2. Posting a full answer, that the student copy-pastes into their solution without understanding, is pedagogically damaging. Posting a hint as an answer discourages others from spoiling a full solution.

  3. Posting the hint as an answer removes the question from the Unanswered queue, which is important for keeping the queue clutter-free.

tl;dr

If it answers the question or adds to the answer, yes. If it skips to answer the question and just gives a hint: no

argument

This is from the official, overruling Stack Exchange help center (emphasis mine):

Answer the question

Read the question carefully. What, specifically, is the question asking for? Make sure your answer provides thator a viable alternative. The answer can be “don’t do that”, but it should also include “try this instead”. Any answer that gets the asker going in the right direction is helpful, but do try to mention any limitations, assumptions or simplifications in your answer. Brevity is acceptable, but fuller explanations are better.

Let's look at some examples:

  • Question A is "I have trouble finding which theorem to use in this case. I want to solve X, I have this [insert of where OP is]. Where to go now?"
  • Question B is "I want to solve X. I have this, but the answer does seem wrong. Where did I go wrong?"
  • Question C is "I have this problem. I have no idea how to solve it. How does one solve it?"

let's look at what the questions ask for:

  • Question A asks for a hint. Only a hint to the right direction. Providing a Hint-answer is acceptable, as it is providing what the question is asking for.
  • Question B asks to judge the answer, pick it apart and point out where the solution went wrong. As long as the answer tells where the error is (and best what to do instead) anything additionally might be acceptable.
  • Question C does not ask for a hint but a solution. So it is not an answer to just give a hint, providing a hint should not be acceptable. In case of a Textbox Question, though, a full answer detailing all the steps and giving all the intermediate steps should be neither. It's better to outline the steps with enough information to allow someone working themselves through the problem to solve it.

As amWhy pointed out, a lot of type C questions seem to be what physics.SE would ban as "homework questions" or, as they phrased it bare PSQ's (problem statement "questions" that read like an exercise copied (often literally copied) from a textbook or problem set, or a test). A hint to such is (after the guidelines) not acceptable. Yet a full answer does not help the student in the same way as a collection of hints as they won't learn from getting their problems solved for them. (roughly quoted form amWhy).


And before someone says "we are different from the communities at large", please check https://math.stackexchange.com/help/how-to-answer which has the identical text.

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    On question C, in quite a good number of such questions of form C which sometimes are called bare PSQ's (problem statement "questions" that read like an exercise copied (often literally copied) from a textbook or problem set or a test) I would agree that a a hint should not be acceptable, but ADD nor should a full answer be considered acceptable. Generally, there is no *One size fits All guideline from SE that applies to each and every site in the SE network. Physics has successfully prohibited "homework-style questions", after all; why shouldn't math.se do so similarly? – amWhy Aug 22 at 21:24
  • That actually IS a good suggestion: banning homework style questions would encourage to take apart the problem and thus critical thinking. However, it would be a correct (and imho acceptable) answer to type C to layout the process of how to solve it without actually solving it. Like "To solve this integration you first substitute with n=this, then solve it with integration variable n, sesubstitute with n=that – Trish Aug 22 at 21:28
  • It all depends on how much the asker brings to the table, initially, or in an edit. If asker X asks question C: "I have this problem [question; explanation of where they are stuck, or inclusion of definitions they figure are needed], I have no idea how to solve it. [I thought, by the definitions I list here, that given $\alpha$ implies $\beta$, procedure A might work". But I'm completely stuck after that.]. How does one solve it?" Then I think it is perfectly appropriate to point out another definition they list, reminding them of a Theorem to apply, ... – amWhy Aug 22 at 21:33
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    and outlining the solution, being encouraging, without doing it for them, but inviting them to run their solution by the answerer, in a comment below the answer. – amWhy Aug 22 at 21:37
  • Well, then many users on this site have a problem with the PSQ form, "do it for me" insistence of Question C. I tried to offer what kind of question is better than what you insist is Question C. If we must stick to "question C" as you frame it, then I don't think any "hint" or "full solution" is appropriate to post in an answer field. – amWhy Aug 22 at 21:39
  • To your first comment, note, I wrote out a sketch of an example answer that doesn't actually do all the work for the asker, hence, is not a full solution but that provides an outline of how the asker can follow, and move forward and find a solution, in my sixth(?) comment below Henning's answer. – amWhy Aug 22 at 21:44
  • @amWhy Worked on getting that in, I hope it is now. – Trish Aug 22 at 21:51
  • Very much appreciate it, @Trish +1 – amWhy Aug 22 at 23:13

You can post your hint, someone else might post an answer, and you'll see which is accepted, or which one gets more upvotes. It's not really up to us to decide whether a hint is acceptable, it's up to the people looking at and asking the question.

I guess it's also up to the mods; you pointed out there's been a discussion about whether there should be a policy against it.

Hints can come off as hostile to the asker and are often not very useful unless you already have a good idea on how to solve the problem. While we shouldn't be giving the answers to homework questions we should be giving answers that explain how to solve the class of that problem.

One problem with a hint is, when the asker is getting it, but you are not.

Think of an interesting question. Now someone leaves a hint, the asker can solve his problem and then accepts the answer with the hint. This will prevent most people from adding actual answers.

Now you googled the problem and found the question and cannot get the hint, but would like to have the solution.

Leaving hints is nice for forum discussions, where you can help one by leading him to the solution, but it is help for one person and not creating an Q&A item.
On Stackexchange, we try to create a knowledge base, not a list of exercises with hints how to solve them.

And even when you think leading people to understand the problem better is better than just providing an answer, sometimes people (often not the asker himself, but anonymous readers) just want the answer to use it without understanding all details.

  • The same problem may appear (and has appeared on our site many times) with an "answer". A future visitor may miss a step in an argument quite irrespective of whether the argument is titled a hint or an answer. I somewhat prefer the process of the said future answer posting another question, asking for more details in a step they don't find clear. Unfortunately the "eager beaver answerers" use such questions as an excuse to reproduce an earlier argument and/or complain when they are accused of answering a dupe. – Jyrki Lahtonen Sep 4 at 12:04
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    @JyrkiLahtonen there is still a range and certain types of hints are especially hit-or-miss in that they can be very useful but they can also be utterly useless if one does not get them. Honestly, I am not that keen on this new category of questions you propose the HINT-explanation questions. Will those at least be answered by a proper answer, or also a hint? Then the next can ask what that HINT-on-the-HINT is trying to say. You and I agree on many things on the site, but IMO your (and some others) positive attitude towards hints is part of the problem, a pretty big one actually. – quid Sep 4 at 12:27
  • @quid Fair enough. I have seen here many questions motivated by earlier threads, when the asker didn't understand some step in the argument. More often than not that older argument was posted as an answer (rather than as a hint). We already have these "request for clarification" questions. I'm not proposing anything new. In fact I'm not proposing anything at all. – Jyrki Lahtonen Sep 4 at 14:42
  • The last sentence in my first comment is about my frustration with such "request for clarification"-threads. Namely, certain users have a tendency to post answers to such requests that are simple reproductions of answers to the original question, and pay no special attention to the points needing clarification. Effectively turning the request for clarification into a dupe. Much the same way you outlined here @quid. – Jyrki Lahtonen Sep 4 at 14:47
  • But, yeah, I failed to communicate my thinking clearly in my first comment. I really miss the ability to edit my comments long after :-/ – Jyrki Lahtonen Sep 4 at 14:51
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    @Jyrki sorry for having jumped onto that. What I just find surprising is that a main preoccupation seems to explain how hints are actually not that bad. To me this answer is spot-on and highlights a quality-problem of the site. Many answers are mediocre at best and others stashed out quickly and a bit incompletely (that includes not few of my own answer). This practice is then glorified to some pedagogical principle, the most visible symbol of which HINT. I mean if users would at least say in words what they mean by HINT but no it must a blurted out HINT. – quid Sep 5 at 0:26
  • @quid My thinking about this is still evolving. Last week I searched for the buzzword "Hint" within my own answers. 406 hits IS more than I expected. While I was rather proud of the few highest scoring ones, certain other (mostly older posts) did make me facepalm. When/if I have the time I will edit some of them. Particularly those answers that I would now describe with one of the following: Sketch / Roadmap / Plan of attack / Extended hint. – Jyrki Lahtonen Sep 5 at 5:09
  • But, yes, I stilil approve of the following: 1) breaking the problem into tiny steps, each an exercise, 2) math golf (condensing the argument to 2-liner). Nowadays I will definitely post as a comment a single idea/suggestion that I want the asker to use to produce partial effort/context. And I strongly disprove of the practice of some of posting as "a hint" either an essentially complete answer, or a thought they haven't followed thru. – Jyrki Lahtonen Sep 5 at 5:16
  • I do expect that a criticial perusal of my old "hints" will make me see the beam in my own eye, adding a healthy dose of humility to my thinking. – Jyrki Lahtonen Sep 5 at 5:18

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