Let's say I asked "How do I do X". Then I found it's impossible. I thought such questions should be closed but there's not "impossible" option on list.
Should I answer myself "X is impossible" and accept own answer or what?
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That's exactly what I'd do. "This can't be done" is a legitimate answer, and it's fine to answer your own questions, particularly if nobody else seems to know the answer. You don't want to close the question, because someone else might be wondering how to do X and search SO, and your answer that it's impossible will help them too
There are degrees of impossibility. I like the idea of posting that as an answer if you really believe it, but hold off a little while on accepting your own answer. Someone might take it as a challenge, and they might even recognize a solvable special case that, for your purposes, "achieves the impossible".
Someone may have a technique or workaround to achieve the same result.
You can't say "impossible": it's may only be "impossible" for you to achieve without inspiration from an answer.
There are around 500 SO questions with "impossible" with many accepted answers...
Of course, there are always impossible requests but I'm trying to answer the "impossible. give up" mentality...