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 $$√2+√3\sqrt{2+\sqrt{3}}$$

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 $$R\mathbb{R}$$ 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!

The Blue-Eyed Islander problem is one of my favorites. You can read about it here on Terry Tao’s website, along with some discussion. I’ll copy the problem here as well.

There is an island upon which a tribe resides. The tribe consists of
1000 people, with various eye colours. Yet, their religion forbids
them to know their own eye color, or even to discuss the topic; thus,
each resident can (and does) see the eye colors of all other
residents, but has no way of discovering his or her own (there are no
reflective surfaces). If a tribesperson does discover his or her own
eye color, then their religion compels them to commit ritual suicide
at noon the following day in the village square for all to witness.
All the tribespeople are highly logical and devout, and they all know
that each other is also highly logical and devout (and they all know
that they all know that each other is highly logical and devout, and
so forth).

[For the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.]

Of the 1000 islanders, it turns out that 100 of them have blue eyes
and 900 of them have brown eyes, although the islanders are not
initially aware of these statistics (each of them can of course only
see 999 of the 1000 tribespeople).

One day, a blue-eyed foreigner visits to the island and wins the
complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their
hospitality.

However, not knowing the customs, the foreigner makes the mistake of
mentioning eye color in his address, remarking “how unusual it is to
see another blue-eyed person like myself in this region of the world”.

What effect, if anything, does this faux pas have on the tribe?