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It seems like PSE's policy on mathematical questions is kinda complicated and has changed over time. What exactly is the policy, and what's the history that led to it?

(There are already lots of Puzzling Meta questions on this topic. The purpose of this one, or more precisely of the answer to it that I'm submitting along with the question, is to be a one-stop shop for the policy and its history.)

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Policy

First of all, if all you want is our actual policy then you can find it in this question:

The policy described there is, briefly:

  • A question is off-topic on Puzzling if it is routine.
    • This means that the question can be answered by following a standard procedure.
    • Other related considerations: is the question itself striking and surprising (e.g., because it asks you to prove something that seems like it can't possibly be true)? Or the answer (e.g., because it resolves in one line something that seems like it should involve a terribly complex calculation)?
  • A question is off-topic on Puzzling if it is incomprehensible to the large majority of PSE readers, even once the answer is given.
    • This doesn't mean that there's anything wrong with [enigmatic-puzzle]s, which are (or should be) entirely comprehensible once the answer is known.
    • Nor does it mean that questions involving higher mathematics as such are off-topic; many such questions are perfectly comprehensible to laypersons. But if the answer to "so what is this question about?" begins with "well, first of all you need to have in your brain the contents of a university-level course on ring theory" then the question doesn't belong here.

For more details and nuances, see the question itself; see also xnor's excellent question and answer from 2015, which was where that "routine" criterion was first stated in something like its modern form.


The issues

Before describing the history that led to our present policy, it's maybe worth saying a few words about why we have any policy on the matter at all and why it's been contentious.

First, an obvious point: many many puzzles are somewhat mathematical in nature, and always have been. (I believe the oldest recorded puzzles are mathematical.) So why would anyone have any sort of problem with mathematical puzzles?

  • Puzzling might not be the right place for them. There are also Math Overflow and Mathematics SE.
    • Usually these are not the right place for anything you would consider calling a "puzzle". Math Overflow is intended for actual research mathematicians and most of the questions there are not at all recreational. Mathematics SE does accept puzzly questions but I think folks there generally prefer them to go here.
  • They might be inaccessible to most readers. If a question requires mathematical expertise, only people with mathematical expertise can make any attempt at solving it.
    • Of course we also have puzzles that are only accessible to people who know about UK-style cryptic crosswords, or video games, or whatever. But those things are easier to learn about, if you want to, than high-level mathematics.
  • They might be boring. We sometimes get a lot of people trying to post their mathematics homework questions here, and while those questions may be good exercises for learning (say) how to solve simultaneous equations, they are usually not much fun as puzzles.
  • They might be bad puzzles in other ways. For instance, those homework questions are homework questions and we generally prefer not to help people cheat on their homework.

The history that follows is the history of PSE's grappling with how to avoid puzzles that are bad in the ways indicated above, while remaining open and welcoming for puzzles that actually add value to the site.


History

Here's a sort of timeline of Meta questions about mathematical puzzles. I have divided it into four periods:

  • Before Xnor, the early days of PSE, before the now-well-established distinction between "mathematics problems" and "mathematical puzzles" was clarified by xnor.
  • The Age of Xnor, starting with that post in 2015-01.
  • A period of debate in 2016 and 2017, about whether we were being too strict and closing perfectly good puzzles just because they had mathematics in, and/or whether we were being too lax and allowing puzzles that were aimed too much at mathematical specialists.
  • The Modern Era, once that debate was resolved.

The prehistoric era before xnor's question and answer that provided the first clear guidance:

(This was also also a time when there was some doubt in the air as to whether Puzzling was meant to be a place for questions that are puzzles (as opposed to questions about puzzles) at all. There was a lot of argument about that, but as any present-day PSE user knows the way it eventually shook out was that now Puzzling is mostly a place for questions that are puzzles. Relevant meta question.)

2014-05 How math-oriented can questions on this site be? [arshajii]

  • Earliest question about this that I've seen.
  • Accepted answer suggests that using advanced mathematics should not be a problem

2014-05 Are probability exercises on topic here? [WendiKidd]

  • Despite the title, this is really about mathematical questions. Interesting answer from user20 suggesting that questions that are purely mathematical should be off topic, a plausible enough idea but not one that later discussion agreed with.

2014-10 Should mathematics questions really be on-topic here? [Gilles]

  • Asks whether any question that "involves non-elementary mathematics" is no longer a puzzle and should be declared off-topic. (Motivated by a number of mathematical questions with highly-upvoted bad answers.)
  • This idea didn't particularly find favour with the People of Puzzling, but it comes back every now and then.

2014-11 Should recreational mathematics be on topic? [Lembik]

  • Closed as duplicate of the previous one. Questioner feels that recreational mathematics should be considered on-topic. Not much further discussion.

The Age of Xnor, with puzzles and problems distinguished clearly for the first time, and ensuing discussion:

2015-01 Are math-textbook-style problems on topic? [xnor]

  • Draws the distinction between "math puzzles" and "math problems" that's been central ever since.
  • "What makes something a math puzzle rather than math problem? ... Clever or elegant solution ... Unexpected problem statement ... Unexpected or counterintuitive result."

2015-02 Should we introduce a new off-topic close reason for math problems? [user20]

  • First proposal for doing this. We do now have an "off-topic close reason for math problems", with pretty much the same wording proposed by user20 here.

2015-03 Would this potential post be a math puzzle or math problem? [Kevin]

  • Seeks clarification on xnor's criteria for a particular mathematical puzzle.
  • Accepted answer says: yes, seems like a puzzle.
  • Also an answer from xnor saying: having seen the answer, I now think it's more problem than puzzle. (An early example of how it can be hard to classify without seeing the solution.)

2015-05 Is this question I asked on Math.SE better suited for Puzzling.SE? [ghosts_in_the_code]

  • Asks about a single specific question. Answer is probably "it's fine in either place". Answer from Rand al'Thor citing xnor's criteria.

2015-08 I say that a maths puzzle is suitable for this site. The puzzle-setter disagrees. What do you say? [chasly]

  • Asks about a single specific question. Answer is that that specific question seems definitely to be "problem" rather than "puzzle". The question has since been closed for exactly that reason.
  • It's a nice example of how one can't always distinguish "problem" from "puzzle" without seeing the solution; there might have been a nice elegant aha!-y solution, but it seems there isn't.

Doubts about difficult problems explicitly considered:

2016-05 Revisiting the 'math question or math puzzle' policy [manshu]

  • Questioner seems concerned that PSE isn't closing enough questions for being mathematical. "If we are using advanced mathematics in a question then the probability of the solution being elegant is pretty low."
  • Questioner is also concerned that the puzzle/problem distinction proposed by xnor, because it can be hard to evaluate before the answer is known, makes it difficult to tell what to do with a question until it's too late.
  • Accepted answer by humn argues that, contrariwise, 'having "math problem" as a close reason has caused demonstrable harm' by encouraging things to be closed that definitely belong on PSE. Gives a couple of examples of nice puzzles that were closed for being mathematics problems and then reopened.

2016-06 Should we disable the "math problem" close reason? [user20]

  • Motivated by the previous question, and other discussion in chat suggesting that maybe things were being closed on these grounds when they shouldn't be.
  • Answer from humn saying "yes, we're doing it wrong, but the overall intent is OK" and suggesting revised wording for the close reason. (The wording was not changed.)
  • Answer from Deusovi saying: no, too many homework problems being posted and this close-reason is useful for them, but we need to clarify exactly what counts.

2017-07 What tricky mathematical questions are on topic here? [Gareth McCaughan]

  • Motivated by several questions that had recently been closed for being mathematics questions, and that didn't (to me) seem to be obviously textbooky.
  • Sought to get explicit agreement on whether simply involving difficult mathematics is a reason for considering something off-topic.
  • Offered as answers a few possible policies and asked for up/down-votes on those. Strong consensus emerged that mathematics is fine, but some concern that questions should be comprehensible to the non-expert.

2017-07 Proposed policy on mathematical questions [Gareth McCaughan]

  • This is the same post linked up above; it's our current official policy.
  • There's some interesting discussion in comments.

2017-07 Puzzling SE refers me to Mathematics SE refers me back to Puzzling SE [Jonathan Fowler]

  • Questioner was looking for somewhere to post mathematics questions and found that math.se rejected them for being too puzzly while puzzling.se rejected them for being too mathematics-textbook-y.
  • Lengthy answer from Rubio, mostly emphasizing the old puzzle-versus-problem distinction; some links to useful discussion in TSL chat.

The modern era after our current policy was agreed and codified:

2018-10 Area 51 Proposal: Math Challenges [TheSimpliFire]

  • Proposing a new SE site for "math challenges" which would include questions too "routine" for Puzzling and questions too "puzzly" for math.se.
  • The proposal didn't end up going anywhere.

2020-10 Why is this question considered a math problem? [Snack Exchange]

  • Questioner had posted a problem from an IQ test or something of the sort, which was closed for being basically an arithmetic problem, and was asking why.
  • There's quite a nice answer from Rand al'Thor, linking to xnor's question from 2013 and addressing the reasons the questioner gave for why their question shouldn't have been closed.
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