### 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 wade-athon 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 as a good puzzle—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 disembowelled.**Be mathematical (as much as possible):**It's true that logic*is*mathematics, but puzzles beginning '*There is a street where everyone has a different coloured house …*' are tedious as hell. Note: there is a happy medium between this and trig substitutions.**Not be too hard:**Any level of difficulty is cool, but if coming up with an 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.**Have that indefinable spark that makes a puzzle awesome:**Like a situation that seems familiar, requiring unfamiliar thought …

Ideally include where you found the puzzle so we can find more cool stuff like it. For ease of voting, one puzzle per post is best.

# 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

The collection of $n \times n$ 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