With recent discussions and active attempts to handle the declining riddle quality on the site, I thought I'd also resurrect this question too, as I think it's related (since broad low quality riddles => lots of guesses => HNQ)...
As GentlePurpleRain points out, the algorithm to select posts for HNQ is as follows:
$$
\frac{(min(\text{AnswerCount}, 10) \times \text{QuestionScore}) \div 5 + sum(\text{AnswerScores})}{max(\text{QuestionAgeInHours} + 1, 6) ^ {1.4}}
$$
Issue: As noted, this tends to promote exactly the wrong puzzles from PSE, as it's usually broad, low quality questions that gather many responses in a very short space of time.
Assumption: The SE devs won't be willing/able to change this algorithm on a per-site basis.
Challenge: How could the algorithm be modified across the entire network such that it doesn't make it worse for other sites, whilst still helping to somewhat mitigate the issues we see here on PSE?
Suggestion: Ask the powers-that-be to modify the algorithm slightly, such that it has little to no impact on other existing sites*, but gives us a little more control over keeping low quality puzzling content out of the HNQ.
Specifically, I suggest that we modify $\text{QuestionScore}$, in the algorithm above, such that it is calculated, not as a simple score, but as:
$\text{QuestionScore}$ = $\text{QuestionUpVotes} - 3 \times \text{QuestionDownVotes} - 5 \times \text{QuestionVTC} $
Reasoning: All this change does is to allow downvotes to have a higher impact in preventing a question from qualifying for the HNQ list, and to allow our experienced, 3000+ rep users to have their close votes help even further. As it currently stands, there's a feedback loop where a broad/low quality puzzle hits the HNQ quickly, gaining it more drive-by upvotes than can be balanced out by downvotes from the community, which in turn keeps it "hot". This proposal, I think, would help to slow/reverse that feedback loop on poor quality stuff, whilst neither blocking "good" content nor impacting what qualifies as "hot" on other sites in the network.
* In fact, it would arguably have a slightly positive effect as it would help filter out more controversial posts in favour of universally praised ones.