That's just a good heuristic, though, it does not guarantee finding a solution. Also note that NP is a class of decision problems, and finding a solution for the n-Queens problem is not.
None of the two decision problems that arise naturally from n-queens, deciding whether there exists solutions for a given board size, and verifying whether a candidate is indeed a solution, are NP-hard (solutions exist for n \not\in {2, 3}; verification can clearly be done in time quadratic in n). Counting solutions is #P-complete, though.
If you read the cites on that link, you'll see that the value for 26 was only added in 2016. They do not have a closed form formula, or they would have shown it.
I mean, they could just be slow revealing. I doubt it, though.
Fair enough. I was obviously reading the above as having a polynomial time as having an efficient time one. Where efficient was shorthand for "quick." Both leaps, I concede were misguided.
None of the two decision problems that arise naturally from n-queens, deciding whether there exists solutions for a given board size, and verifying whether a candidate is indeed a solution, are NP-hard (solutions exist for n \not\in {2, 3}; verification can clearly be done in time quadratic in n). Counting solutions is #P-complete, though.