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For background, see:

We have been less than perfectly consistent in our treatment of mathematics questions here. There's pretty good agreement that we should close questions that just ask for routine calculations, and indeed they generally get closed promptly with helpful explanations for the questioners; but at the other end of the spectrum, when a question turns up that looks as if it requires heavy-duty mathematical machinery it sometimes gets closed and sometimes not, and the justifications aren't perfectly clear either way around.

The purpose of this question is to propose a specific, explicit policy. I'd like to give it a couple of days for refinement and then see whether it meets with general approval. So:

Until 2017-07-15: Please comment, edit, etc., in order to help turn this into a policy that will serve us well.

From 2017-07-16 on: Please upvote if you think the proposal here, or something almost identical, is good; downvote if you think we should have a substantially different policy or no policy; and comment if your opinion is not perfectly and completely expressed by your vote if any.

Proposed policy on mathematical questions

There are at least two ways in which a mathematical question can be problematic.

  • It can be routine, simply asking the solver to turn the handle on some standard procedure.
  • It can be inaccessible, not realistically approachable by any but a small fraction of PSE participants.

Questions that are routine are simply not on-topic here. The only thing this really has to do with mathematics is that a large fraction of the routine questions asked here are mathematical. If your question can be answered by following a standard procedure, it almost certainly doesn't belong here.

(What about questions that can in principle be solved by brute force, but where solving them less stupidly is a good puzzle? Best to make it explicit in the question that a non-brute-force solution is required.)

Questions that are inaccessible are harder to judge, but:

  • The mere fact that a question involves high-level mathematics doesn't disqualify it.
  • A question that is incomprehensible to the large majority of PSE readers is (I hereby declare) off-topic and almost certainly a bad question in any case. Such questions are likely to get closed, and even if they escape they are not likely to be well received.
    • Some perfectly good questions (typically tagged ) are incomprehensible at first, but usually you can see how one might come to understand them, and when the solution is seen everything makes (at least some) sense. I'm talking here about questions that are incomprehensible because you can't understand them without knowledge of something that takes days to years of hard intellectual work to get into a human brain.
  • A question whose solution consists of standard manoeuvres is really a routine question, even if those manoeuvres are difficult for anyone other than a professional mathematician. These questions, too, are liable to be closed as off-topic and/or downvoted.
  • If a question requires heavy mathematical machinery, there had better be a Puzzling-friendly payoff for it: perhaps the solution, despite the machinery, is surprisingly short and elegant, with some beautiful central idea that simplifies it; or perhaps the question itself is obviously attractive even though the solution is tough. Preferably both.

One difficulty facing abstruse questions is that it may be hard to tell whether they have such a Puzzling-friendly payoff, and readers cannot reasonably be expected to tell the difference between a question redeemed by a startlingly elegant answer and a question that's just painful. If you ask an intimidating-looking question, you might consider offering explicit reassurance that despite appearances it has Puzzha-nature.

Puzzle-setters, please feel free to post mathematical puzzles -- but make sure they really are puzzles and not mere problems: they should be fun to solve, and preferably fun to read the solutions of. Please make sure your question makes at least some sense to readers with only amateur mathematical knowledge. If your question is liable to look to readers as if it's no fun, consider providing some sort of evidence that it is. And if the fun consists only of an elegant solution, please consider posting your elegant solution if after a couple of weeks have elapsed no one else has found it.

Others, please vote to close routine questions and questions that are downright incomprehensible. When in doubt, consider seeking advice in The Sphinx's Lair and/or discussing things with the questioner to understand whether there's more to the question than handle-cranking. Please don't try to close questions merely because there is mathematics in them. And, whether closing or querying or whatever, please be polite and welcoming.

Edited 2017-07-14 to clarify what I said about incomprehensibility a bit.

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  • $\begingroup$ I like your parenthetical remark. That type of puzzle is exactly what the no-computers tag is for. $\endgroup$ – Rand al'Thor Jul 14 '17 at 12:16
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    $\begingroup$ "A question that is incomprehensible to the large majority of PSE readers is (I hereby declare) off-topic and almost certainly a bad question in any case." - disagree. Or at least, this needs to be refined. What do you mean by "incomprehensible", and how does it not cover, say, most of our enigmatic-puzzles? What's wrong with being incomprehensible to most users, as long as at least some can understand it? Again, many computer-puzzles are incomprehensible to me, but I don't VTC or even DV them - I simply pass over them. [cont] $\endgroup$ – Rand al'Thor Jul 14 '17 at 12:23
  • $\begingroup$ [cont] Put yourself in the shoes of a non-mathematician looking at an advanced mathematical puzzle - how can they tell whether it's incomprehensible to the majority of users or just to them? For example, some people have never heard of combinatorics; does that give them carte blanche to VTC all combinatorics questions as incomprehensible? $\endgroup$ – Rand al'Thor Jul 14 '17 at 12:23
  • $\begingroup$ Also, in your final bullet point, "there had better be" needs clarifying. Are you saying that puzzles without this should be closed, or only downvoted? (I would say downvote rather than close, simply because of the obvious inherent subjectivity in making a judgement of elegance or attractiveness.) $\endgroup$ – Rand al'Thor Jul 14 '17 at 12:26
  • $\begingroup$ Finally, is "Puzzha-nature" a typo? $\endgroup$ – Rand al'Thor Jul 14 '17 at 12:27
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    $\begingroup$ Incomprehensibility The difference between enigmatic puzzles and what I'm talking about here is that the former become comprehensible once you see what's going on. A puzzle that's incomprehensible to everyone but Alexandre Grothendieck will remain so even once its solution is seen. I'll think about ways to clarify that. $\endgroup$ – Gareth McCaughan♦ Jul 14 '17 at 12:55
  • $\begingroup$ It's curious that you use combinatorics as an example; that seems to me a field of mathematics in which incomprehensibility is more rarely a problem. Take a look at those questions tagged combinatorics that you linked to; can you honestly say that any of them is incomprehensible because of mathematical jargon or notation? $\endgroup$ – Gareth McCaughan♦ Jul 14 '17 at 12:57
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    $\begingroup$ Despite your shouting in the answer linked there, I think it's entirely appropriate for there sometimes to be "confusion", because the boundary between really bad questions and ones that just don't belong here is a fuzzy one. This is especially true with criteria like routine-ness which are subjective and admit of degrees. A puzzle that's merely somewhat routine may be a bad one and get downvoted. One that's totally routine may be closed as not-really-a-puzzle. But you can interpolate almost continuously between the two. $\endgroup$ – Gareth McCaughan♦ Jul 14 '17 at 15:35
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    $\begingroup$ "A question that is incomprehensible to the large majority of PSE readers is (I hereby declare) off-topic." This should be policy in general. Any puzzle whose solution can neither be found nor even properly explained without relying on "knowledge of something that takes days to years of hard intellectual work to get into a human brain" and for which there is no objectively defensible reason to expect any meaningful part of the user base possesses, is off-topic ... (continued) $\endgroup$ – Rubio♦ Jul 16 '17 at 1:44
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    $\begingroup$ ... for the same reason (pretty literally!) that a perfectly valid word-puzzle, say, written entirely in Ulithian and relying on wordplay specific to that language is off-topic. A puzzle mustn't rely on knowledge of a subject or language—be it mathematics, spoken, computer, or otherwise—in which the considerable majority of P.SEers are not conversant ("Familiar or acquainted by use or study; well-informed; versed"); if it does, though it might be a perfectly fine puzzle otherwise, it is off-topic for THIS group of puzzle appreciators. $\endgroup$ – Rubio♦ Jul 16 '17 at 1:48
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    $\begingroup$ @Rubio I disagree. It should be fine to have a puzzle which requires specialised knowledge (of any kind) to be solved, provided it's tagged as such so that people without that knowledge don't waste time on it. Or are we going to close all those riddles about video games, and all the puzzles requiring advanced programming skills? There have even been puzzles requiring knowledge of the Japanese language, and nobody kicked up a stink about them. $\endgroup$ – Rand al'Thor Jul 16 '17 at 10:37
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    $\begingroup$ Those puzzles are all comprehensible after the fact to a layperson willing to put in a few minutes' effort. "Oh, I see, there's a video game with those things in? Fair enough." "Ah, so those are the Japanese words for those things? Yeah, I can see how that works." "Wow, that's a lot of computation, and I've no idea how it worked, but the end result is an arrangement of those rectangles that fits together nicely. Neat!" [continues] $\endgroup$ – Gareth McCaughan♦ Jul 16 '17 at 11:11
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    $\begingroup$ ... whereas for some highly-mathematical questions it could be the case that even with the solution right there someone without extensive mathematical training couldn't make any sense of the question, the answer, or the proof without prolonged study. $\endgroup$ – Gareth McCaughan♦ Jul 16 '17 at 11:14
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    $\begingroup$ I remark that there is "objectively defensible reason to expect a[] meaningful part of the user base [to] possess[]" a knowledge of quite a lot of mathematics, simply because lots of PSE regulars are actually mathematicians. So Rubio's version of the incomprehensibility criterion is quite generous to arguably-incomprehensible questions. $\endgroup$ – Gareth McCaughan♦ Jul 16 '17 at 11:15
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    $\begingroup$ Gareth is correct on both counts as to the intent. The wording "solution can neither be found nor even properly explained without [knowledge]" was completely intentional - many (maybe most) puzzles need a particular bit of knowledge or insight to solve, but understanding the solution doesn't; these are fine. And the criterion is intentionally generous precisely so we don't entirely preclude puzzles about programming, or MathJax, or foreign languages, that actually have an audience here. (But 'a[ny] meaningful part" complements "considerable majority"; it wasn't meant to stand in a vacuum.) $\endgroup$ – Rubio♦ Jul 16 '17 at 15:50

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