Math puzzles for dinner [closed]

You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the problem together.

I love puzzles like that. But there's a problem -- I running into the same puzzles over and over. But there must be lots of great problems I've never run into. So I'd like to hear problems that other people have enjoyed, and hopefully everyone will learn some new ones.

So: What are your favorite dinner conversation math puzzles?

I don't want to provide hard guidelines. But I'm generally interested in problems that are mathematical and not just logic puzzles. They shouldn't require written calculations or a convoluted answer. And they should be fun - with some sort of cute step, aha moment, or other satisfying twist. I'd prefer to keep things pretty elementary, but a cool problem requiring a little background is a-okay.

One problem per answer.

If you post the answer, please obfuscate it with something like rot13. Don't spoil the fun for everyone else.

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Since I see this has accumulated a couple of votes to close, I've started a meta thread: meta.mathoverflow.net/discussion/471/… – Anton Geraschenko Jun 24 2010 at 16:00
It's easy to forget the question, read one of the problems below, then write down an answer... – Gerald Edgar Jun 25 2010 at 0:28
Out of curiosity, is there a way of posting hidden text in answers that can be revealed by clicking on "Hidden Text?" (kind of like on Art of Problem Solving forums). – Alex R. Jun 26 2010 at 1:53
I find it a bit odd that there is a bounty on a CW. Can we discuss this [on meta](meta.mathoverflow.net/discussion/550/…)? – Willie Wong Jul 25 2010 at 14:46
Am I the only person who goes to social events with mathematicians and drinks, banters and has pointless debates about politics or films? – Yemon Choi Feb 2 2011 at 3:02
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closed as no longer relevant by Yemon Choi, Mark Meckes, Andres Caicedo, Scott Morrison♦Feb 2 2011 at 3:46

Saw this recently:

There is a board with a grid drawn on it. You hammer a few nails into the board at intersection points and then stretch a rubber band around the nails. Then you observe that

(1) you can't take away any of the nails without changing the shape,

(2) the rubber band does not enclose (or pass over) any grid point without a nail in it, and

(3) you can't add another nail and extend the rubber band around it without (1) or (2) becoming untrue.

How many nails are there?

(The answer is obvious and easy to show in a few lines, but I like it because it's quicker to figure out the corresponding problem in d dimensions and then see what pops out for d=2 than to make a specific two-dimensional argument that isn't the generic argument with d=2.)

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There was a puzzle in the Journal of Recreational Mathematics. I apologize if I am telling it incorrectly. A wire is stretched between two telephone poles. A flock of crows lands simultaneously on the wire. When they land, each crow looks at his nearest neighbor. What percentage of the flock are looking at a crow that is looking back at it?

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This problem appeared in the Australian Mathematical Society Gazette 35 (July 2008) page 151: "An odd number of wombats are standing in a field so that their pairwise distances are distinct. If each wombat is watching the closest other wombat to them, show that there is at least one wombat who is not being watched." Maybe this is the problem OP is trying to remember. – Gerry Myerson Jan 27 2011 at 4:37
It can't be independent of the number of crows: For n=2 it is 100% and for n=3 it is 2/3. – Sune Jakobsen Feb 1 2011 at 17:07
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Fork in the road 2

You're once again at a fork in the road, and again, one path leads to safety, the other to doom.

There are three natives at the fork. One is from a village of truth-tellers, one from a village of liars, one from a village of random answerers. Of course you don't know which is which.

Moreover, the natives answer "pish" and "posh" for yes and no, but you don't know which means "yes" and which means "no."

You're allowed to ask only two yes-or-no questions, each question being directed at one native.

What do you ask?

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Bob and Alice want to marry each other, so Bob decides to send Alice a ring. The problem is that they both live in different countries, and any valuables they send through the mail are sure to be stolen, unless they are sent in a locked box. The box can be locked by a padlock which can only be opened by the right key. Both Alice and Bob have an infinite supply of boxes and padlocks with corresponding keys. However, neither Alice nor Bob have keys to each other's padlocks, only for their own. Suppose you can put boxes inside each other. How can Bob send Alice the ring? Of course, the solution to the problem must end with Alice putting the ring on her finger. To reiterate, anything outside of a padlocked box is guaranteed to be stolen.

This problem has numerous solutions as well as interpretations which makes for a fun discussion. It can also be solved in your head without pencil or paper.

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"anything outside of a padlocked box is guaranteed to be stolen." - So if I send a locked box, it'll be stolen? It is, after all, not inside a locked box. – Michael Burge Jul 26 2010 at 12:30
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I understand your feeling , I myself know lots of them . Among original ones

http://www.amazon.com/Mathematical-Puzzles-Connoisseurs-Peter-Winkler/dp/1568812019

This Peter Winkler does something that is rarely done and is a must not only for a mathematician but for a connoisseur: He produces declination of a problem.

In fact there are two books of his.

Another source of problems that you may like is "IBM ponder this" AT

http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/pages/index.html

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Yes, I suggested the Winkler books in my answer of 24 June. – Gerry Myerson Oct 1 2010 at 12:34

A room contains 3 bulbs and 3 switches outside controlling the bulbs. Is it possible to determine which switch controls which bulb by entering the room only once and observing bulbs?

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guvf vf abg zngu! – Yaakov Baruch Jun 24 2010 at 12:15
wtf?........... – Boyarsky Jun 24 2010 at 14:15
en.wikipedia.org/wiki/ROT13 – Qiaochu Yuan Jun 24 2010 at 14:29
It should be noted that this has several possible answers depending on what kind of hardware is used in the question. – Ketil Tveiten Jun 24 2010 at 14:33
Actually you can solve it with $2^2$ bulbs, and only once entering the room. – Pietro Majer Jun 24 2010 at 18:06
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There are some dwarves approaching to a bridge. They have to cross it to come back home from the cave where they work. Unfortunately, a dragon has just decided to reside under that bridge, and it's hungry. But it's also bored, so it doesn't want just to eat the dwarves, but proposes them a game: it will put on each of them one hat, either black or white, in no specific proportion (for example, it can happen all hats to be white). Of course they can't see their own hat, but they can see the others'. They will be then queueing at the beginning of the bridge, and each of them can just say one word. If this word matches with the colour of the hat that dwarf is wearing, then he's allowed to pass and to come back home. Otherwise, he'll be eaten by the dragon. Of course, they can decide for a strategy before the game begins.

What's the best strategy, and how many dwarves die on average?

EDIT: (deleted the previous PS, modified into this one) PS: Since I didn't solve all previous puzzles posted, I would be glad if someone could point me at equivalent puzzles, if any!

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