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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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closed as no longer relevant by Yemon Choi, Mark Meckes, Andrés E. Caicedo, Scott Morrison Feb 2 '11 at 3:46

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Since I see this has accumulated a couple of votes to close, I've started a meta thread: – Anton Geraschenko Jun 24 '10 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 '10 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 '10 at 1:53
I find it a bit odd that there is a bounty on a CW. Can we discuss this on meta? – Willie Wong Jul 25 '10 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 '11 at 3:02

67 Answers 67

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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Could be anything from epsilon to 100. I suspect the actual problem is just to prove that it's non-zero. – Gerry Myerson Jan 27 '11 at 3:40
Hmm...showing it's non-zero seems too trivial, but indeed the percentage can be arbitrary as long as it's greater than zero. Perhaps the birds are distributed according to some known distribution on the wire, and we wish to find the expected value of this proportion? – Daniel Litt Jan 27 '11 at 3:44
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 '11 at 4:37
Using Ocam's razor I would assume that $n$ crows are independently uniformly distributed on $[0,1]$. – Christian Blatter Jan 27 '11 at 13:30
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 '11 at 17:07

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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BX, V fbyirq gur svefg bar (Nfx bar: "Ner lbh sebz gur fnzr ivyyntr?", vs ur fnlf "Lrf" gura gur bgure bar unf gb or gur gehgu-gryyre, vs ur fnlf "Ab" gur bgure bar vf gur yvne - urapr jr whfg nfx gur bgure bar jurer gb tb.) Ohg guvf bar V pna'g penpx - nal uvagf? – Harun Šiljak Jun 26 '10 at 6:15
@H. M. Gung vf abg gur fbyhgvba gb gur svefg bar. V znqr vg irel pyrne gung gurl !pbhyq! obgu or sebz gur fnzr ivyyntr, naq obgu ivyyntrf !pbhyq! or ivyyntrf bs yvnef/gehgugryyref, ohg arvgure ner arprffnevyl gehr. – BlueRaja Jun 26 '10 at 16:13
Bu, gur svefg irefvba bs gur evqqyr (orsber gur rqvgvat) gevpxrq zr - V nffhzrq bar ivyyntr vf gur ivyyntr bs gehgu-gryyref, naq gur bgure vf gur ivyyntr bs yvnef. – Harun Šiljak Jun 26 '10 at 16:26
Since every riddle of mine but this one has been solved, I'll post the answer here: Gur gevpx vf gb znxr fher gur -frpbaq- bar jr fcrnx gb vfa'g gur enaqbz nafjrere; gura jr pna hfr gur fnzr gevpx jr hfrq va 'Sbex va gur Ebnq 1' gb svaq gur pbeerpg cngu. Sbetrg nobhg gur 'cvfu/cbfu' sbe n frpbaq. Jr nfx gur svefg crefba "Jung jbhyq lbh fnl vs V nfxrq lbh vs crefba O vf gur enaqbz nafjrere?" Vs ur vf gur yvne be gehgu-gryyre, jr'yy trg gur gehgu bhg bs uvz naq or noyr gb nfx gur frpbaq dhrfgvba gb n aba-enaqbz nafjrere. (pbag..) – BlueRaja Jul 8 '10 at 21:37
(pbag..) Vs gur svefg crefba vf gur enaqbz nafjrere, vg qbrfa'g znggre jung ur fnlf fvapr jr pna svantyr gur gehgu bhg bs rvgure bs gur bgure gjb. Guvf tvirf hf bhe nafjre, rkprcg sbe gur 'cvfu/cbfu' ceboyrz. Gb trg nebhaq gung, jr nygre bhe dhrfgvba gb or "Vs V jrer gb nfx lbh vs crefba O vf gur enaqbz nafjrere, jbhyq lbh fnl 'cbfu?'" Jurgure cbfu zrnaf lrf be ab, obgu gur yvne naq gehgu-gryyre jvyy nafjre 'cbfu' jura O vf gur enaqbz nafjrere, naq 'cvfu' jura ur vfa'g. Jr nfx fvzvyneyl sbe gur pbeerpg ebnq, naq jr ner qbar. – BlueRaja Jul 8 '10 at 21:39

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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Already posted. – Qiaochu Yuan Jun 26 '10 at 2:31
"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 '10 at 12:30
@MichaelBurge It may not be inside, but also is not outside - it's the boundary of a locked (and so closed) box. So much for thinking outside of the box ... – Hagen von Eitzen May 1 at 21:10

I understand your feeling , I myself know lots of them . Among original ones

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

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Yes, I suggested the Winkler books in my answer of 24 June. – Gerry Myerson Oct 1 '10 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 '10 at 12:15
wtf?........... – Boyarsky Jun 24 '10 at 14:15
6 – Qiaochu Yuan Jun 24 '10 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 '10 at 14:33
Actually you can solve it with $2^2$ bulbs, and only once entering the room. – Pietro Majer Jun 24 '10 at 18:06

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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The point of having a list is that people should read the answers before adding their two cents... – Yemon Choi Feb 2 '11 at 2:52
yes, I agree... and actually what I mean is that I didn't solve all the puzzles, and I don't know if some or them are equivalent to this one here. Can someone point me at equivalent puzzles? I don't think that the infinite cases are, because for the solution of this one you need finiteness (at least, for the solution I found!) – klaraspina Feb 2 '11 at 11:26

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