Can I ask questions that serve as puzzles? I.e. I already know the answers but post the questions anyway to see if other people can solve the puzzles?
I don't see any problem in principle with posting puzzles as questions, assuming that they're being asked because the poster is genuinely interested in the answers, and that the post makes it clear that it is posted as a puzzle (or at least makes it clear that the poster already knows an answer and would like to see other solutions).
There have been a number of questions along the lines of "I know a proof of X, but I would like to see some others." That isn't quite the same, but is closely related, and these have often gone over well.
If someone doesn't want to answer a question that the poster already has an answer to, then as long as the question is up front about the fact that it is posted as a puzzle, it will be easy to ignore.
I have changed my mind on this issue. In light of the greater number of off-topic and otherwise poorly-asked questions on the site these days, I would like to advocate a stricter policy with regard to questions that do not fit the SE model: in particular, questions where the OP already knows the answer. Posting a puzzle you don't know the solution to is fine, but if you are posting a puzzle you know the solution to, you are more or less just sharing the puzzle, and math.SE is not for sharing puzzles - there are plenty of other ways to do this (blogs, Facebook, etc.).
There may be good ways to ask a puzzle question. For example, if the actual question is "what kind of math is this? How does it generalize?" or "what other solutions are possible?" or something like that, that would be fine. The key is that the OP does not know the answer to these questions.
Here is one reason I don't think puzzles are a good fit for math.SE: how do you tag a puzzle? If your tags are too specific, you may give away a hint for the puzzle. The asking of a puzzle hinges on the answerers not having all of the information that the OP has, which I think fundamentally goes against the spirit of the SE enterprise. (Again, a modified question like the ones I suggest above would be more than fine.)