When at school, I beleved that a good puzzle should have a solution. In my first year at college I encountered the idea of a "solution set". This was mind-expanding. A puzzle has a set of solutions: often the set has one member, but it may have none, or many.
When I was at college, I came across puzzles like this:
If dimF v = n prove that dimF (Hom(V,V)) = n2.
I realised that the setter will have been thinking "the pupil will know what all this stuff means. The challenge is to find a proof." In reality, the pupil might not know what, say, dimF means. The challenge is then to come up with a plausible meaning for dimF such that the proof is possible but not trivial.
I regard such puzzles as metapuzzle/puzzle pairs. Often, in mathematics, the challenge is to parse a problem statement "Show that < set of words and symbols specifying some structure > has < specified relationship > to < set of words and symbols specifying some structure >" in such a way that it makes sense and has an interesting solution. In general, the metapuzzle is to come up with a plausible interpretation of the puzzle statement; the puzzle is to solve that interpretation. If the puzzle is too easy or too hard, maybe you've got the metapuzzle wrong.
I once regarded such metapuzzle/puzzle pairs as a defective puzzles, which had accidentally acquired a metapuzzle wrapping. Now, I regard any puzzle statement as a metapuzzle/puzzle pair. Sometimes the metapuzzle is trivial, but I don't know that until I've solved the puzzle, or at least convinced myself that it may well have an interesting solution. This viewpoint doesn't just apply to abstract algebra puzzles like the one above, but also to some old chestnuts like "what colour is the bear", "manhole covers", and "Monty Hall" in the list below.
Accidental metapuzzles resulting from careless wording of the problem statement
Four colliding ships
The Monty Hall puzzle
Six points on a disk
Ants and triangle
Simultaneous equations
Doppler effect question
Square question
Deliberate metapuzzles with problem statements that are intended to misdirect or to be hard to parse
Two points on the equator
A sliding block puzzle
Numbers in a car park
Square square pyramidal number
"Found" metapuzzles where the lack of a clear puzzle statement is not due to the puzzle's creator
Wooden puzzle
Plastic puzzle
Bolleke puzzle on beermat
Unclear which category is most appropriate
What colour is the bear?
Manhole covers
Other
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