Is the distinction too subtle between downvoting and voting to close?

Question

Q1 (rhetorical). What distinguishes these three things?

• A downvote

• A vote to close

• A vote to remain closed

I’m beginning to wonder if we really know, and am beginning to suspect that we sometimes mistakenly use the ability to vote for closure as an amped up ability to downvote and to usurp the community's evaluation function.

Q2 (dismayed). Why did this fresh example get at least 2 votes to remain closed and apparently 0 votes of support for reopening it so far? It was originally closed for being unclear and has been revised to be as clear as almost any puzzle here.

• It is related to a type of mathematics game

Q3 (optimistic). Could we follow this simple guideline?

• An unimpressive valid puzzle merits a downvote, at worst, and/or a constructive comment, at best, but not closure.

Stack Exchange’s a-vote-is-a-vote-is-a-vote system has masses-are-asses weaknessess, no doubt, but my-close-vote-counts-more-than-your-vote is susceptible to groupthink-like tunnel vision, alienates newcomers, and seems to me downright unfair.

Further reading:
Downvote vs vote to close question
Is it appropriate to downvote a question without closevoting it
Close reason: No permanent value?
Proper rules for closing

Answer 1

A CLOSE VOTE IS NOT A SUPER-DOWNVOTE.

^{Sorry for shouting.} But this is a really important point which far too many people apparently fail to grasp.

• Downvotes are used at the discretion of the individual voter, and mean "this is a bad question". They can reflect the voter's opinion of the quality of the question, rather than any objective criteria. You can downvote for any reason you want and those votes will still be counted as valid¹.

• Close votes are used by the community as a whole, and mean "this question doesn't belong here". They reflect whether or not the question is on-topic according to the objective standards decided by the community on meta. On-topic questions which are closed should be reopened.

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You can see that this is the way the system is designed just by looking at how it works in practice.

• Any number of people can cast downvotes on a post, and these aren't subject to any kind of peer review. But as soon as someone votes to close, the question enters the review queue where the rest of the community can decide whether or not the close-vote was valid.
• There's also an option for 'undoing' close votes by reopening the question. But one can't cancel out downvotes, only counter them by upvoting (which isn't a good reason to upvote anyway).
• Question closures can be challenged on meta: someone might make a meta post calling for the reopening (or the closure) of a particular question, and after discussion as to whether that question meets the community-created scope guidelines, a consensus will be reached on whether it should be closed or open. But if anyone posted "hey, this question shouldn't be downvoted", they'd be laughed out of meta.

Note also that questions which are closed can't be answered. One reason for closing a question is that it shouldn't get answers here, but there's nothing wrong with answering questions which are simply bad without being off-topic. They should be downvoted, sure, but not closed, since that blocks people from answering them.

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Further reading:

• Why do you cast downvotes on answers? from main meta. The essential takeaway from this is that people downvote for whatever the hell reason they want, and the system doesn't stop them from doing so.

• Proper rules for closing from Puzzling meta. This lays out some rules for when a question should be closed here. Note that there are rules for when a question should be closed - close-votes aren't just for "this is a bad question", as downvotes are.

• Downvote vs vote to close question and Is it appropriate to downvote a question without closevoting it from SO meta (although some of the talk about downvoting there applies to answers more than questions)

• A Close Vote is not a Super-Downvote. Please don't use it as one from Arqade meta.

• Is there an explanation of duties for each milestone? from SFF meta. Both my answer there² and the above-linked question from Arqade give a good run-down of the criteria for when close-votes should be cast. Of course, the above-linked PSE meta post from Avigrail is more relevant to our site, but it's not fully filled out yet (and is written by someone with less broad experience of SE).

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¹ _{Unless you vote for a person rather than a post and trip the serial voting detector, but that's not really the point here.}

² _{Self-promotion alert! :-P}

Answer 2

1. I don't have VTC privilege on this site yet, but, if I did, I would be inclined to leave It is related to a type of mathematics game closed because it is a mathematics problem and not a mathematics puzzle.  A question shouldn't be reopened just because the original close reason has been resolved; it should be reopened only if it is worthy to be open.
2. Maybe this is an incorrect way of thinking, but: The question appears to have a unique solution (answer), and it has gotten that answer (twice).  It's not clear what purpose is served by opening the question to new answers.