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Often when people post their solutions to a particular problem on a question of theirs and they tag it , the solution that they give is flawless, or can be improved a small amount eg by cutting down on unnecessary assumptions.

In these cases, there isn't much that you can do to answer the actual question: you can just post a comment saying "Yes, your solution is correct. Perhaps it could be improved slightly by...".

However, what I often see in these sorts of scenarios is that people then post answers to the question offering alternative solutions, although the OP doesn't ask for one. This often is very useful to the OP, as can be seen in the numerous instances when the OP thanks the answerer profusely for giving them a new method in their toolkit. I've done this myself sometimes.

So, here's my question:

Should we be answering questions tagged with alternative solutions when the OP doesn't don't ask for one and the argument that they give is fine? What is MSE's policy about this?

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    $\begingroup$ Saying "yes, that's correct." is...pretty terse. And it's just proof by Authority. Saying "Yes, that's correct, and here's another way of thinking about it...." at least provides insight into the underlying question. $\endgroup$
    – lulu
    Commented Jul 5, 2021 at 20:23
  • $\begingroup$ Judgiing from the myriad of cases where a user wants to know if their solution is correct, and another user asks "Does this post answer your question ..................", the most frequent response is an OP staying, "I just want to know if my own solution is correct!" An alternative method should be used only when the OP presents their solution, asks for verification, and also asks: "Is there another more efficient method"? etc. Please do not read an OP's mind thinking "I'm so clever, I'll wow her with *my solution!" when it is not explicitly asked. $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 20:35
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    $\begingroup$ That said, solution/proof verification questions have a shaky status on math.se. E.g., we refuse to read prepublication manuscripts, because "it is not the purpose of this site to review others work." It's not a far stretch to suggest that, because the only way to successfully actually answer such a question is telling them something they did not ask for. When there are profound mistakes, a decent answer can arise, but we are not graders, reviewers, etc. $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 20:39
  • $\begingroup$ @lulu (1) Answerers aren't privy to what a user has in their tool box, and to use a tool they have no knowledge of, say until it's covered a month later in their text/class, is downright problematic. Say a user posts everything you suggest? And the OP ended up with a correct answer, via erroneous moves/assumptions, but the user answering doesn't see that immediately, so approves of faulty work, then goes on to show how they would do it (at a level beyond the OP). What do we do about students who submit their "approved work" only to fail an exam? $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 20:48
  • $\begingroup$ Exams and homework often stipulates: *Use induction to prove ............ *. They post their induction proof, and you are going to suggest, "induction isn't needed!" Dah, but the OP has specific instructions. That happens all the time on math.se! $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 20:51
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    $\begingroup$ @amWhy That seems like a rather extreme scenario, no? Surely the more likely case (at least the case we all hope for) is that the OP is introduced to a new technique or approach. If the OP can't follow the new solution they can ask about it, or they can simply discard it and accept the "Yes, that's correct." which precedes it. $\endgroup$
    – lulu
    Commented Jul 5, 2021 at 20:52
  • $\begingroup$ That might be what you think is more likely, but I don't think so, So we are both conjecturing, but the example I spoke of has happened, and too many times. Potentially such questions/answerers work. But it is not our place to replace students instructors, nor publication reviews by professional people in the precise field pertaining to the asker. I'm afraid there are too many users reviewing posts who may not realize their own limits, or those of the OP. $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 20:58
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    $\begingroup$ I think situations where "the solution that they give is flawless" are rare, and we should acknowledge this and not have a policy for it. Proofs can usually be shortened or insight can be shared, whilst keeping with the flavour of their original proof (e.g. "you could have shortened that step by doing this"). So try and find some grumble or even something to complement. (I dislike both other options: people saying "yes your proof is correct, now here's mine!" makes the original solution/question inconsequential, while answers saying "yes, this is correct" are low effort & frustrating.) $\endgroup$
    – user1729
    Commented Jul 5, 2021 at 21:31
  • $\begingroup$ Related: Is this correct? Questions. $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 21:50
  • $\begingroup$ See also: proof-verification types of questions $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 21:51
  • $\begingroup$ @lulu Not at all extreme! solution to my question exceeds my knowledge. $\endgroup$
    – amWhy
    Commented Jul 5, 2021 at 21:53
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    $\begingroup$ A point no doubt already covered in one of the above links: If the proof is suffiiciently complicated, then even the ‘optimal’ proof can be helped by giving a birds eye view/proof sketch that tries to clarify the important parts, like where certain assumptions are needed and how the proof fails when those are relaxed. $\endgroup$
    – Calvin Khor
    Commented Jul 6, 2021 at 1:45
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    $\begingroup$ A problem with an answer consisting of an alternative solution is that the underlying question is often a duplicate. At least those alternative solutions often are. I elaborated on my views here. Of course, opinions differ on how much duplication is tolerable. I am leaning towards allowing very little by veterans of the site, but to be more forgiving to a relative newcomer, who has had less time to grasp how well this site actually has covered most of the basic questions. $\endgroup$
    – Jyrki Lahtonen
    Commented Jul 6, 2021 at 5:10
  • $\begingroup$ Thanks, @Jyrki, I was looking for your post. $\endgroup$
    – amWhy
    Commented Jul 6, 2021 at 13:18

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It's a good question, and one I have no universal response to.

I dislike the flat "Yes, that's correct" response though I have used it.

It's effectively meaningless. At best it is Proof by Authority. At worst? Well, many users here are heavily credentialed but it's all anonymous. I have seen many instances in which people confirmed entirely senseless and incorrect arguments. Sometimes cases in which the underlying claim was false. So, what value is there in simply consenting to the argument?

Providing alternate proofs at least removes doubt as to whether or not the underlying claim is correct. And, possibly, it introduces the OP to new and valuable ideas.

There are, of course, cases in which one can streamline or otherwise improve the proof and in those cases one should do so. But it sometimes happens that the proof the OP proposes is optimal or nearly so. In those cases, in which mere confirmation is called for, supply that. But, generally, there is more to be said about the underlying problem or the provided solution and in those cases one should write more.

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