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One of my questions has been closed as being a math-textbook problem. However, this problem is not meant to (and in fact cannot) be answered through standard methods, and instead requires other knowledge to find the true answer. The question I have is how do I demonstrate this? The only solutions I can think of would be to provide the answer in the question text, which seems a little poor-form. Is there something else I should do?

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    $\begingroup$ My close-vote on that question is not for math textbook-ness but instead for Needs Details/Clarity - unfortunately they only display one reason. $\endgroup$
    – bobble
    May 3 at 14:34
  • $\begingroup$ I did vtc as math textbook, but realize that as an error. Should have gone with details/clarity. $\endgroup$ May 4 at 13:32

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The only solutions I can think of would be to provide the answer in the question text

Nah, don't do that. Even if you spoilertag it, it kind of ruins the fun because any answerer could have checked the solution before posting.

How do we prove that a math-based question has a clever solution?

Something that could help a lot is simply to show that you're aware of the policy on maths puzzles.

We see a lot of questions, mostly but not only from low-rep first-time posters, where the OP is clearly unaware that we have a policy against simple textbook-style maths problems. Closing a question isn't a punishment as such; hopefully it would make the OP aware that such questions are off-topic, so that they can post more on-topic ones in future.

If you make clear that you're not one of those people, even if just by posting a footnote in the question to say something like "this isn't just a simple maths problem; there's more to it than meets the eye", then people will think twice before using that close reason. We do have to operate a lot on trust here at PSE: it would be possible, for example, for someone to create a perfectly plausible-looking puzzle that actually has no solution, just to troll the community into wasting their time; and on the other hand, many brilliant puzzles can look like complete nonsense at first glance. If you say it's not a maths textbook-style problem, then you've at least passed the first hurdle of knowing the policy, and you're more likely to get the benefit of the doubt from voters.

One of my questions has been closed as being a math-textbook problem.

Let me give some feedback on your specific question. Although I didn't downvote or vote to close, I did comment on it, and felt strongly doubtful about it even before the votes started rolling in. Here's why: it's not clear what sort of puzzle this is. And not in the good sort of way, more in the sense that you could mean a number of different things but none of them seems to make a good puzzle. If I had voted to close, I would've used the "unclear what you're asking" close reason.

  • Are we meant to just give the formula $\frac{4}{3}\pi R^3$ for the volume of a sphere? That's trivial, and would justify the existing close reason.
  • Are we meant to find the decimal expansion of $\frac{4}{3}\pi R^3$ without knowing the value of $R$? That's impossible.
  • Is there some lateral thinking involved, like "its volume" doesn't mean what we think it does (as in the existing answer)? But there's no tag.

The problem with this puzzle is not only that it's not clear what we're meant to do, but that there don't seem to be any clues pointing us towards what to do. The puzzle is very short and doesn't contain much context to indicate which of the above three options, or something else, is intended. That's why a lot of people add stories to their puzzles: not only for flavour text, but because it's a great place to hide clues.

(An example I always love is this puzzle from 2015. Just the image with no context? Pretty poor puzzle. A little bit of flavour text packed with clues? Everything fits together beautifully and naturally, no need to spend hours futilely trying approaches that were never intended to succeed. The only criticism I'd make, now after years of PSE development, is that there might be too many clues, making it too obvious how to proceed. Some of them still pretty subtle though, like the apparent typo in "oh my god", and I love how the title does double duty to clue two different facets of the solution.)

So my advice is to extend the question a bit. Try to add some clues, even very subtle ones, indicating what sort of solution method you're thinking about. If people can understand that there is something non-trivial to be solved, even if they can't see how to solve it, they're less likely to assume it's impossible or a simple maths homework question.

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My thoughts:

Questions with math-problem type solutions are not appropriate for this site and can be asked on other mathematics forums.

A question can be closed if it is solvable with standard math. It can also be closed if it appears solvable with standard math, or if others mistakenly assume that this is so.

So, the answer to your question is to update the problem so that it is clear and apparent that the question is not solvable with standard math.

This can be done with tags, text, or a combination of these.

Example (will be closed):

2+2 = ?
tags: Mathematics

Fixed: (will not be closed, although it's still a pretty awful puzzle):

2+2 = ?
The answer is 4 letters long and ends in 'U'
tags: enigmatic-puzzle, word

Here, the second line prevents someone from answering "4" or "FOUR".
The more information you add, the less likely it will appear to be a math problem to others.

You can be as creative as you like, adding information like:

  • The answer is not mathematical
  • Find a solution that involves clothing
  • The answer is a four letter reduplicative word

Note that your example question will still be closed if another issue is not addressed. A question cannot have multiple equally valid answers, or 'invite' guessing. As it stands, tobu's answer seems perfectly valid and I can think of plenty of other random guesses. The fix is again the same; add better tags and information.
For example:

  • the answer will be obviously correct and intentional
  • the answer is thirteen numbers starting with '7'
  • add a tag like 'anagram' or 'braille' to show the correct approach etc.
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