My opinion is similar to Mau's and Rory's, that puzzles should be allowed, but that multiple specific puzzles from the same category (e.g. Sudoku or River-Crossing) should not be allowed.
Currently, if I were to flag a question as being in the same family as another question, the only realistic option I have in the list of flags is
This question has been asked before and already has an answer. I recognize that an answer can be tips on how to solve the puzzle, as opposed to a solution to the puzzle, but other users may not.
I recommend that we either clarify that text, or add a new option. For example, we could try creating a new flag type:
The puzzle in this question belongs to a family of questions that already has a general answer. Then the flagger could select the question. This would allow users whose questions are flagged to know that we recognize their question, but we will not solve their specific sudoku/river-crossing-puzzle/etc. At the same time, we are trying to give them tips in the right direction.
However, this leads to another interesting problem. Some puzzles (i.e. End of the world on Sunday, or my bulb question) have one specific answer, and cannot really be generalized. On the other hand, other questions (i.e. dealing with things like sudoku and balls and scale) can be generalized. So, which ones should be generalized?
My opinion is that if a question can be generalized, it should be. Then others of the same type can be marked as duplicate. However, if the type of problem is too narrow to be generalized (such as the bulb puzzle), then one specific example should do.
Also, as others have said, let's make sure not to rush to conclusions/policies here. Yes, this question (should specific brain-teasers be allowed) is valid, but the policy we set will have a long-term effect on the site.
Also, in the comments to the question, a valid question was raised: "Should an answer be hidden in spoiler tags?" There is a good discussion about that here. (Thanks to durron597 for pointing that out.)