When I was at school, I thought 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 it 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 dimF means. The challenge is 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. 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.
What colour is the bear?
Bolleke puzzle on beermat
Simultaneous equations
Four colliding ships
Manhole covers
The Monty Hall puzzle
Six points on a disk
Numbers in a car park
Ants and triangle
Two points on the equator
A sliding block puzzle
Doppler effect question
Square question
Square square pyramidal number
Tornado question
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