What tricky mathematical questions are on topic here?

Question

[EDITED to add:] See followup at Proposed policy on mathematical questions proposing a more specific policy in the light of comments and voting here.

There's strong consensus that straightforward turn-the-handle mathematics questions don't belong here.

But there is another category of mathematics question that's been contentious from time to time: the highly nontrivial advanced-mathematics question. The sort of thing that might be a question in the International Mathematical Olympiad, or a tricky "for enthusiasts" question set to university mathematics students, or even the subject of a short paper in a mathematics journal.

There is (so far as I can tell) no official policy on such questions. On the one hand, they can be a lot of fun for those with the skills to attack them. On the other hand, they're likely completely inaccessible to a large majority of people here, and arguably they would belong better at math.stackexchange.com than here.

The discussion in the meta question I linked above isn't concerned with that sort of mathematics question. Here are some other meta questions that touch on it:

• Should mathematics questions really be on-topic here? -- the questioner's issue in this case wasn't so much that the questions are unsuitable here, as that they attract bad answers. Question has two answers; one (+3-0) doesn't really answer the question; the other (+9-0) says that substantial mathematical content as such isn't a reason to declare something off-topic.
• Should recreational mathematics be on topic? -- closed as duplicate (of the question above), no answers.
• Should we introduce a new off-topic close reason for math problems? -- followup to the question I linked at the start, suggesting a new close reason (which was in fact introduced).

In so far as the community's opinion can be read out of those questions, it seems to be that highly mathematical questions are just fine here. But observe e.g. these questions:

• Polygon wrapping a cube -- closed for being a mathematics question, comment saying "feels more like a math problem" upvoted 10 times. Seems to be pretty tough; it was about 2 weeks between posting and closing, and the only answer submitted is sketchy and doesn't seem to come close to being the actual proof requested.
• https://puzzling.stackexchange.com/questions/52027/the-powerful-groups -- closed for being a mathematics problem. About group theory. No attempts, no objections made to closing it.
• https://puzzling.stackexchange.com/questions/52026/the-impossible-calculation -- closed and even deleted (so, sorry, some readers will be unable to see this one) for being a mathematics problem. About finite fields, kinda. No attempts, no objections made to closing it.

which suggest that at least some of the time, at least some of the community finds these questions inappropriate. Even though all of them are short, seem like they might be fun to attempt for those who enjoy such things, and may for all I know have clever answers that don't require a lot of "grinding".

(One difficulty about questions of this sort is that telling whether they're outrageously too difficult may be a job requiring professional expertise. How many times can any number appear in Pascal's triangle? Open question. Put n runners on a circular track of length n, starting from the same place all running at different constant speeds; must each of them at some time be at least 1 away from all the others? Open question. Is 33 the sum of three integer cubes? Open question.)

So it doesn't seem that we have a clear consensus on the following question:

Are some questions off-topic on Puzzling for being too heavily mathematical, even though they are not routine textbook exercises, and if so what distinguishes them from questions that are OK?

I'll propose some plausible positions in answers. Others should feel free to add more or to improve mine if I haven't made the best possible case for them.

Answer 1

Being heavily mathematical should be no objection to a question

Some questions here are only accessible if you know a lot about rock music. Some, only if you are able to program a computer. Some, only if you know group theory. Why should one of these be treated differently from the others?

A heavily mathematical question may be a bad question, just as (say) a question with a cipher in it may be a bad question. Those questions can be downvoted or, in extreme cases, closed on their (de)merits. But the presence of a specific kind of content, as such, isn't what makes them bad.

Answer 2

Questions should be comprehensible even if their answers require advanced techniques

We should avoid conflating two issues. (1) Questions that make no sense to anyone other than skilled mathematicians. (2) Questions probably soluble only by skilled mathematicians.

A question in category 1 is just no fun for anyone outside that small minority. A question in category 2 can at least be appreciated by everyone.

Look, for instance, at the three questions linked in the OP here. Pretty much anyone can understand the first one and see why it might be interesting. Solving the problem may be beyond the abilities of all but the cognoscenti (or even theirs) but there's at least something there for everyone.

The other two are a different matter. If you don't know what a group is (in the mathematical sense) then question 2 will simply mean nothing to you. If you haven't done some work with groups then even if given the definition you will have no idea why anyone would bother asking the question, why the answer might be yes, or why the answer might be no. Similarly, unless you have worked with polynomials modulo other polynomials then question 3 will be so much gibberish, and even if you figure out what it means you will not care about the answer.

Answer 3

From our policy on maths puzzles and maths problems:

So, what makes something a math puzzle rather than math problem? I think there's a few features.

• Clever or elegant solution, often an "aha" moment
• Unexpected problem statement.
• Unexpected or counterintuitive result.

In contrast, math problems tend to be "textbook". And by that I don't mean that they have to come from textbooks (or that textbooks can't contain math puzzles), but that they use standard, staightforward methods than anyone familiar with the subject is expected to know. They can be difficult, but their goal is to test comprehension of the material, not ingenuity.

I think we can use the same rule of thumb on the questions you're asking about here. Questions such as this one on group theory and this one on field theory (two of the examples linked from your question) are closed not because they require advanced maths, but because they're not particularly interesting - they could be exercises from an algebra textbook, not puzzles as such.

As I said in chat:

• If a question needs a page of boring advanced maths to solve, then it's not really a puzzle, and can be closed under the argument of the existing meta post.
• If a question needs a page of elegant and counterintuitive advanced maths to prove an unexpected result, then I'd say it's interesting enough to count as a puzzle.

I also agree with this answer of yours. Yes, some of our maths questions will be inaccessible to people without advanced maths knowledge. Well, we also have puzzles that are inaccessible to people (such as myself) without programming knowledge or video game knowledge. I don't like, can't solve, and generally avoid such puzzles, but I don't VTC them, and see no reason to prevent others from having their fun with them.

Answer 4

Heavily mathematical questions do not belong here

If someone posts a question that is downright incomprehensible to anyone who hasn't studied algebraic geometry, it's a lousy question because 99% of readers will be unable to appreciate either the question or its answer -- no matter how excellent they may be. Perhaps even worse if the question looks elementary but turns out to be soluble only by combining ergodic theory with differential geometry, or something.

No doubt many professions have puzzles approachable only by the professionals. (Here is an aircraft cockpit: can you see what's wrong? This amplifier circuit will be unstable if fed the wrong input: figure out why and suggest a single component replacement that will fix the problem. Here is an inscription in an unknown language thought to be related to ancient Greek and Etruscan. Figure out the meanings of as many words as you can.) They generally don't belong here; most of their audience isn't here and most of the people here aren't in their audience.