In Mathematics Stack Exchange (where I came from originally) one is told to ask a question if it needs to be answered and if they do not know how to do it and if they have already tried. It seems different on Puzzling Stack Exchange, and it is unclear from the tour of the site. Can I post a puzzle not because I need to solve it but because I believe that it will be entertaining for those who encounter it on the website? I ask this because I have seen a few questions that seem like this and because this seems a little more informal than Math Stack Exchange.
There's a long and tortuous history behind this, but nowadays most of the 'questions' on this site are puzzles posted just for fun. On Puzzling, unlike nearly all other SE sites, questions are usually intended as challenges to the community: problems for other people to solve for which the OP already knows the answer, rather than questions about problems the OP is facing.
In some ways, yes, this makes Puzzling SE more informal than Maths SE. But of course we have quality standards too: a puzzle which is unsolvable, too easy, or has too many possible solutions is likely to be downvoted or closed. We even have a special close reason (although sometimes "too broad" is used instead) for puzzles which don't have a single verifiable solution.
As Emrakul mentions, it's generally a good idea to make clear whether or not you know the answer, and you should definitely make clear whether or not the puzzle is your own work, otherwise it may be deleted for plagiarism. But the long and the short of it is, yes, fun puzzles are more than welcome, even if (especially if) you don't need help with solving them.
Questions asking for help solving puzzles are also on-topic, but they're a minority here these days.
Yes, just make a note of where it comes from and that it isn't your own creation. You might also want to mention that you already know the answer if it's not implied. It's totally fine, though, for sure.
And, welcome to Puzzling! :]