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 thathilariousscene 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: logicismathematics, 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 audienceActually 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 itHave 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

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 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!

**Answer**

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?

**Attribution***Source : Link , Question Author : Community , Answer Author :
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