Okay, so this question was bound to come up sooner or later- the hope was to ask it well before someone asked it badly...
We all love a good puzzle
To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that drives us. Indeed most puzzles (cryptic crosswords aside) are somewhat mathematical (the mathematics of sudoku for example is hidden in latin squares). Mathematicians and puzzles get on, it seems, rather well.
But what is a good puzzle?
Okay, so in order to make this question worthwhile (and not a ten-page wadeathon through 57 varieties of the men with red and blue hats puzzle), we are going to have to impose some limitations. Not every puzzle-based answer that pops into your head will qualify for answerhood- to do so it must
- Not be widely known: If you have a terribly interesting puzzle that motivates something in cryptography; well done you, but chances are we've seen it. If you saw that hilarious scene in the film 21, where kevin spacey explains the monty hall paradox badly and want to share, don't do so here. Anyone found posting the liar/truth teller riddle will be immediately disemvowelled.
- Be mathematical: as much as possible- true: logic is mathematics, but puzzles beginning 'There is a street where everyone has a different coloured house...' are much of a muchness and tedious as hell. Note: there is a happy medium between this and trig substitutions.
- Not be too hard: any level is cool but if the answer requires more than two sublemmas, you are misreading your audience
- Actually have an answer: crank questions will not be appreciated! You can post the answers/hints in Rot-13 underneath as comments as on MO if you fancy.
And should
- Ideally include where you found it: so we can find more cool stuff like it
- Have that indefinable spark that makes a puzzle awesome: a situation that seems familiar, requiring unfamiliar thought...
For ease of voting- one puzzle per post is bestest.
Some examples to set the ball rolling
Simplify $\sqrt{2+\sqrt{3}}$
From: problem solving magazine
Hint:
Try a two term solution
Can one make an equilateral triangle with all vertices at integer coordinates?
From: Durham distance maths challenge 2010
Hint:
This is equivalent to the rational case
nxn Magic squares form a vector space over $\mathbb{R}$ prove this, and by way of a linear transformation, derive the dimension of this vector space.
From: Me, I made this up (you can tell, can't you!)
Hint:
Apply the rank nullity theorem
Happy puzzling!