Old questions with hint-only answers. Can we provide whole answers?

Question

Summary

I'm encouraging users to provide whole answers to sufficiently old questions that have only received hints thus far.

• I argue that the reasons to give hint-only answers are no longer (sufficiently) relevant after a period of time (e.g. 3 months).

• I encourage:

  • Users who have given hint-only answers in the past, to replace them with whole solutions. Alternatively, add whole solutions in addition to the hint, if the hint itself is valuable (possibly using spoiler space).

  • If we see a question with only hints as answers, give a whole solution.

  • Up-voting new whole solutions (or upgraded solutions) to these questions (even if they have an "accepted" hint-only answer).

  • Those who have received hint-only answers to their questions to provide worked solutions to their own questions.

---

Reasons for hints

The main reasons for hint-only answers seem to be:

1. Help educate the OP.

2. Avoid participating in cheating.

3. At the OP's request.

(I won't provide citations for these reasons; they seem well-known or self-evident.)

After a certain period of time (say, 3 months), these reasons seem irrelevant: the OP has presumably moved on with their life, and the vast majority of homework deadlines would have passed.

Reasons against hints

• Hints have been argued to come at the expense of the site due to their inherent "incomplete" nature:

One purpose of stack exchange sites is to have long lasting questions and answers, this is why we close as duplicates, because the original is meant to be found on search engines, and be a reference for future question askers. Having an inordinate number of questions with only incomplete hint-answers, and where posting a complete answer is not allowed, nearly defeats this purpose. I am not saying hints are bad, they have their place, but having a policy which only allows hints on a whole class of questions is ludicrous. -- Eric Naslund, 2012. (Emphasis mine.)

Hints are good for the OP, but not so good for others. An interesting solution to this is to later go back and edit your hints into a full solution. I haven't done this, nor have I seen it done, but it makes sense to me. (I'm not discouraging hints - I sometimes give them too - but I acknowledge that they do not improve the quality of the site. They will never be pearls, as I reference below). -- mixedmath, 2013. (Emphasis mine.)

• Hint-only answers can be plain wrong (unintentionally, of course):

I have seen some cases, where hints were not correct - the poster of hint probably missed some step in the solution. But in such cases we have many other experienced users here, probably some of them will notice this and point out the problem. -- Martin Sleziak, 2012. (Emphasis mine.)

I remember the first time I gave a hint, it was quickly pointed out that it was completely wrong. I have been subsequently much more careful.

In fact, 2-year old (accepted) hint was what motivated this post (I won't name and shame). The hint seems like a sensible approach that would work (if it actually worked). Since it was up-voted and accepted, I presumed someone actually tried it and found that it worked, and consequently I up-voted the answer myself. After some reflection, however, I found it doesn't work (I give a counter-example in my new answer).

---

Examples

Here's a couple of graph theory examples of questions that I think could be improved by having non-hint-only answers:

• Bipartite graph non-isomorphic to a subgraph of any k-cube

• A lower bound on the chromatic number of regular graphs

(I'm not convinced these answers are even correct. They might be, I'm just not sure.)

Of course, I'm not encouraging users to forego thinking: answers should contribute something of value (they don't necessarily need to be "better" than the current answers, just add something.)

Here's a graph theory hint which is a gem:

• Graphic sequence $\langle n-d_1-1, \cdots, n-d_n -1 \rangle$

But even then, a well-written answer to this question would still help teach about mathematical communication.

Of course, most hint-only questions are somewhere in the middle, e.g.

• Find all of the distinct non-planar graphs with 6 vertices.

It has two decent hints, which would lead to a solution with careful work. But a reader like me, who only has a passing interest, would not be interested in doing the work. Nevertheless, I'd like to know the actual answer (ideally with illustrations).

---

Please up-vote/down-vote, comment and answer to express your opinion on the above.

Answer 1

I argue that hints are just as useful for future readers as complete answers are, and we should strive for better questions to be asked, rather than modifying the hints into complete answers.

1. Hints can be useful to others. If the OP posts into the question their attempts, and some of the relevant knowledge they have, future readers can identify whether or not they have a similar position, and the hints are again useful.

2. Complete answers can be less-useful to others. Many theorems appear throughout mathematics with very different proofs. In calculus I you may see a theorem whose proof is long and tedious, but measure theory, or topology, the theorem is a trivial conclusion from one of the fundamental theorems. Without proper context, a complete answer can be either too tedious - as it addresses the freshman, or too abstract - as it addresses the senior.

   Yes, it is true that with complete answers it's easier to judge. So here's one proposed solution for giving hints, one can give them with some perquisite:

   HINT:

   If you already know theorem X, then note that Y and Z hold for your question.

   Yes, it's sometimes tedious, and we can exercise our judgment to decide if such introduction is needed, as sometimes theorem X is so fundamental that the question can't be solved, and shouldn't be given without it.

3. Good hints are equal to good answers. I see this site as an extension to the classroom and the books. I strive to teach, not just answer, and I'm sure that there is a majority of the core answerers which feel the same.

   Hints are excellent for teaching, because they leave the user to think about the problem on their own. I recently commented on someone who asked a question, and self-answered it after realizing it's a mistake, that I learned from similar mistakes in books, much more than I did from the correct questions. Overcoming a typo in a book require you to solve a problem without guidance. (I'm not advocating for intentional typos, though.)

   If we want to be a teaching site, then we need to assume that the future reader comes here to learn, just as the OP did. If they come to learn on the site, then hints can help them just as they helped to the OP.

4. Better questions make a better student. More than half the questions I set to ask on this site never got here because when I write a question, I try to be very complete. Often the solution pops out of the definitions and problems when I finished writing them.

   If we encourage people to write better questions, this benefits not only the site, but themselves. More than that. Being a mathematician (in my eyes) is a lot about understanding the question correctly, and not just answering the question. The former is usually essential for the latter. So if we encourage people to write better questions, then we help them develop as mathematician, perhaps even more than any help answering their questions.

5. Complete answers are still good. Despite all the above, one can think that I'm going to argue that hints are better than complete answers. But this is not true, of course. Complete answers, when well written, are still helpful. Someone can't proceed using hints only, and from time to time a complete solution is needed in order to learn how to write a proof, how to use certain information from the assumptions, and so on.

   What I am not going to answer, because I don't think anyone can, is when one is better than the other. This requires, I feel, some ineffable and unquantified ability to sense the question and the confusion of the OP. If the confusion seems to run deep, I'll prefer writing a complete answer (if possible). But if the OP's misunderstandings seem shallow, then hints can go further along the way.

   I can't say how I'm deciding that. I suppose it come from experience of teaching people in real life, something not everyone using the site actually did. And even those who did, not necessarily did well.

So all in all, I don't think that hints are bad for the future. Although I do agree that seeing a question which has six answers containing just hints can be a bit off putting. I'm not saying that complete answers are not as good as hints. But I'm denying the suggestion that they are better.

We can improve the way we give hints, and we can improve the questions. But I don't think there is a real problem with hints staying here for future readers, they just might be helpful to them too!

Answer 2

Another reason is

$4$. Laziness, or lack of time

Posting a telegram is faster and easier, and does not require working out details.

I do that regularly, for the stated reason, but opt for "condensed answer" rather than "hint" style of presentation, because deliberately withholding information can (quite rightly) be interpreted as condescenscion.