Math puzzles for dinner

Question

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.

Answer 1

I really like the following puzzle, called the blue-eyed islanders problem, taken from Professor Tao's blog :

"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).

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?"

For those of you interested, there is a huge discussion of the problem at http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/

Malik

Answer 2

You and infinitely many other people are wearing hats. Each hat is either red or blue. Every person can see every other person's hat color, but cannot see his/her own hat color; aside from that, you cannot share any information (but you are allowed to agree on a strategy before any of the hats appear on your heads). Everybody simultaneously guesses the color of his/her hat. You win if all but finitely many of you are right. Find a strategy so that you always win.

Answer 3

You are blindfolded, then given a deck of cards in which 23 of the cards have been flipped up, then inserted into the deck randomly (you know this). Without taking the blindfold off, rearrange the deck into two stacks such that both stacks have the same number of up-flipped cards. (You are allowed to flip as many cards as you please.)

Answer 4

1000 prisoners are in jail. There's a room with 1000 lockers, one for each prisoner. A jailer writes the name of each prisoner on a piece of paper and puts one in each locker (randomly, and not necessary in the locker corresponding to the name written on the paper!).

The game is the following. The prisoners are called one by one in the room with the lockers. Each of them can open 500 lockers. If a prisoner finds the locker which contains is name, the game continues meaning that he leaves the room (and leaves it is the exact same state as when it entered it, meaning that he cannot leave any hint), and the following prisoner is called. If anyone of the prisoners fails to recover his name, they all lose and get killed.

Of course they can agree before the beginning of the game on a common strategy, but after that, they cannot communicate anymore, and they cannot leave any hint to the following prisoners.

A trivial strategy where each prisoner opens 500 random lockers would lead to a winning probability of 1/2^1000. But there exists a strategy that offers a winning probability of roughly 30%.

Answer 5

There are $n$ balls rolling along a line in one direction and $k$ balls rolling along the same line in the opposite direction. The speeds of the balls in the first group and in the second group are equal. Initially the two groups of balls are separated from one another and at some point the balls start colliding. The collisions are assumed to be elastic. How many collisions will there be?

Answer 6

(I learned this puzzle from Ravi Vakil.) Suppose you have an infinite grid of squares, and in each square there is an arrow, pointing in one of the 8 cardinal directions, with the condition that any two orthogonally adjacent arrows can differ by at most 45 degrees.

Can there be a closed cycle? (i.e. start at some arrow, move to the square that arrow points to, follow where the arrow there points and so on, and come back to the square you started at).

Answer 7

Adam Hesterberg told me this one ages ago. It apparently used to circulate around MOP.

Three spiders and a fly are placed on the edges of a regular tetrahedron, and travel only on those edges. The fly travels at the rate of $1$ edge/s, whereas the spiders travel at the rate of $1 + \epsilon$ edge/s for some $\epsilon > 0$. The spiders want to agree beforehand on a deterministic strategy for capturing the fly, whose location they do not know (but they do know each others' locations). The fly is invisible and omniscient; in particular, it is aware of the locations of the spiders and of their strategy at all times. (It also cannot fly.)

Can the spiders guarantee that they will catch the fly in finite time, regardless of the initial positions of the spiders and the fly? Does the answer depend on the value of $\epsilon$?

Answer 8

Simplify (x-a)(x-b)...(x-z).

Answer 9

Here is another of my favorites: Player 1 thinks of a polynomial P with coefficients that are natural numbers. Player 2 has to guess this polynomial by asking only evaluations at natural numbers (so one can not ask for $P(\pi)$). How many questions does the second player need to ask to determine P?

Answer 10

Most of us know that, being deterministic, computers cannot generate true random numbers.

However, let's say you have a box which generates truly random binary numbers, but is biased: it's more likely to generate either a 1 or a 0, but you don't know the exact probabilities, or even which is more likely (both probabilities are > 0 and sum to 1, obviously)

Can you use this box to create a unbiased random generator of binary numbers?

Answer 11

A princess inhabits a flight of 17 rooms in a row. Each room has a door to the outside, and there is a door between adjacent rooms. The princess spends each day in a room that is adjacent to the room she was in the day before. One day a prince arrives from far away to woo for the princess. The guardian explains the habits of the princess and also the rules to him: Each day he may knock at an outside door of his choice. If the princess is behind it she will open and in the end marry him. If not, nothing happens, and he gets another chance the next day. Unfortunately his return ticket expires after 30 days. Does he have enough time to conquer the princess? (Adapted from "simpler-solutions.net")

Answer 12

It is very important that you tell these two puzzles in the correct order, i.e., first the first puzzle and then the second one. The first puzzle is very easy but messes with people's minds in just the right way. In my experience some mathematicians are driven crazy by the second puzzle.

Puzzle 1: Grandma made a cake whose base was a square of size 30 by 30 cm and the height was 10 cm. She wanted to divide the cake fairly among her 9 grandchildren. How should she cut the cake?

Puzzle 2: Grandma made a cake whose base was a square of size 30 by 30 cm and the height was 10 cm. She put chocolate icing on top of the cake and on the sides, but not on the bottom. She wanted to divide the cake fairly among her 9 grandchildren so that each child would get an equal amount of the cake and the icing. How should she cut the cake?

Answer 13

You have a glass of red wine and a glass of white wine (of equal volume). You take a teaspoon of the red wine and put it in the glass of white wine and stir. You then take a teaspoon of the white wine (which now has a teaspoon of the red wine in it) and put it in the glass of red wine and stir.

Which glass has a higher ratio of (original wine)/(introduced wine)?

Answer 14

What is the resistance between 2 adjacent vertices of an infinite checkerboard if every edge is a 1 ohm resistor?

Answer 15

A certain rectangle can be covered by 25 coins of diameter 2. Can it always be covered with 100 coins of diameter 1?

Answer 16

When you watch yourself in a mirror, left and right are exchanged. But why aren't top and bottom?

Answer 17

1. Alice shuffles an ordinary deck of cards and turns the cards face up one at a time while Bob watches. At any point in this process before the last card is turned up, Bob can guess that the next card is red. Does Bob have a strategy that gives him a probability of success greater that .5?

2. Let $x_1, x_2, \dots, x_n$ be $n$ points (in that order) on the circumference of a circle. Dana starts at the point $x_1$ and walks to one of the two neighboring points with probability $1/2$ for each. Dana continues to walk in this way, always moving from the present point to one of the two neighboring points with probability $1/2$ for each. Find the probability $p_i$ that the point $x_i$ is the last of the $n$ points to be visited for the first time. In other words, find the probability that when $x_i$ is visited for the first time, all the other points will have already been visited. For instance, $p_1=0$ (when $n>1$), since $x_1$ is the first of the $n$ points to be visited.

3. Let $\pi$ be a random permutation of $1,2,\dots,n$ (from the uniform distribution). What is the probability that 1 and 2 are in the same cycle of $\pi$?

4. Choose $n$ points at random (uniformly and independently) on the circumference of a circle. Find the probability $p_n$ that all the points lie on a semicircle. (For instance, $p_1 = p_2 = 1$.) More generally, fix $\theta<2\pi$ and find the probability that the $n$ points lie on an arc subtending an angle $\theta$ .

Answer 18

This is a hat problem I heard only two days ago: you have a hundred people and each one has a (natural) number between 1 and 100 written on his hat. (Numbers may repeat.) As ususal, everybody can see only the numbers on other people's hats. Give these guys a strategy for guessing so that at least one will surely make the right guess. (They do not hear each other guesses.)

Answer 19

Not a very difficult one but I like it since it is even suitable for non-mathematicians:

A small boat carrying a heavy stone is floating in a swimming pool. What happens to the level of water (up, down or remains equal) in the swimming pool if one removes the stone from the boat and throws it in the swimming pool?

The very easy solution suggests the following joke (illustrating the well-known ignorance of mathematicians of reality): Instead of sending scores of ships for saving passengers from the Titanic, one should have sunk all possible rescue-ships in order to lower the sea-level.

Answer 20

Instead of recommending some puzzles, I'll recommend some books containing many puzzles. Peter Winkler, Mathematical Puzzles; Peter Winkler, Mathematical Mind-Benders; Miodrag Petkovic, Famous Puzzles of Great Mathematicians.

Answer 21

Via the great Martin Gardner: A cylindrical hole is drilled straight through the center of a solid sphere. The length of hole in the sphere (i.e. of the remaining empty cylinder) is 6 units. What is the volume of the remaining solid object (i.e. sphere less hole)? Yes, there is enough information to solve this problem!

Answer 22

(I learned this problem from Persi Diaconis.) A deck of $n$ different cards is shuffled and laid on the table by your left hand, face down. An identical deck of cards, independently shuffled, is laid at your right hand, also face down. You start turning up cards at the same rate with both hands, first the top card from both decks, then the next-to-top cards from both decks, and so on. What is the probability that you will simultaneously turn up identical cards from the two decks? What happens as $n \to \infty$? And does the answer for small $n$ (say, $n=7$) differ greatly from $n=52$?

Answer 23

Okay, so it's somewhat more numeric than the others, but I quite enjoy the simplicity of:

Simplify:

$$\sqrt{2+\sqrt3}$$

Answer 24

Countably many little dwarfs are going to their everyday work to the mine. They are marching and singing in a well-ordered line (by natural numbers), so that number 1 watches the backs of all the other ones, and, in general, number n watches the backs of all the others from n+1 on. Suddenly, an evil wizard appears on the top of a small hill, and magically puts a name on the back of each dwarf. Any name may be used, even more than once: existing ones, old-fashioned ones, or just weird sounds sprang out of his sick imagination, included grunts, sneezes and any snort-like name (you may enjoy providing your listerners with examples if they ask for). Then, he claims that, at his signal, everybody has to guess his own name, and say it loudly, all together. Whoever fails, will disappear immediately. Poor dwarfs are not new to these bully spells, and do have a strategy, that allows all but finitely many of them to survive. How do they do? To formalize, we may think the evil wizard has attached a real number to each dwarf's back.

Answer 25

You have a large pile of ropes and some matches. All you know about the ropes:

• Each rope has a different length
• Each rope burns completely (starting from one end) in exactly 64 minutes
• Each rope has non-uniform density, meaning it is thicker at some points than others. Consequently, burning half a rope cannot be guaranteed to take 32 minutes.

The goal is to identify when exactly 63 minutes have passed.

Answer 26

You have 1000 bottles of wine. Exactly one of the bottles contains a deadly poison, but you don't know which one. The killing time of the poison varies from person to person, but death is imminent in at most $t$ hours after ingestion. You are allowed to use 10 notorious criminals as poison fodder (they are on death row). How much time do you need to correctly determine the poisoned bottle?

Answer 27

There is a square with seven monkeys on the floor and seven bananas on the top. Seven ladders go up the square, from one monkey to the banana over it, and the monkeys can climb them. Moreover there are some ropes which connect the ladders.

A monkey will go up towards the bananas, but whenever it meets a rope it cannot resist the temptation to stray and hang on it. Prove that every monkey will reach a banana, no matter the configuration of ropes.

There are at least two different solutions to this.

Answer 28

When they came to diner some shook hands. Ask them to prove that that two of them shook hands the same number of times.

Answer 29

Oldie but a goodie (Monty-hall problem):

You are on a game show with three doors, behind one of which is a car and behind the other two are goats. You pick door #1. Monty, who knows what’s behind all three doors, reveals that behind door #2 is a goat. Before showing you what you won, Monty asks if you want to switch doors. Should you switch?

Answer 30

There is a plane with 100 seats and we have 100 passengers entering the plane one after the other. The first one cannot find his ticket, so chooses a random (uniformly) seat. All the other passengers do the following when entering the plane (they have their tickets). If the seat written on the ticket is free, one sits on this seat, if not he chooses a other (free) seat at random (uniformly). What is the probability the last passenger entering the plane gets the correct seat?

Answer 31

Here's one of my favorites. There are 99 bags, each of which contains some number of apples and some number of oranges. Prove that there's a way to select 50 out of the 99 bags, such that these 50 simultaneously contain at least half the total number of apples and at least half the total number of oranges.

One fun aspect of this problem is that there are a number of distinct solutions, inspired by different areas of math. I know of at least three...

Answer 32

Given three equal sticks, and some thread, is it possible to make a rigid object in such a way that the three sticks do not touch each other? (all objects are 1 dimensional; sticks are straight and rigid, and the thread is inestensible).

Answer 33

I had a good outcome with this one once. It probably helped that the other two mathematicians had a drink or two before dinner. (Otherwise they would have solved it in 5 seconds...) When salad was served, somebody had oil and vinegar in separate little pitchers...

Suppose you have two containers, one with oil, one with vinegar, equal volume. Take one teaspoon of the oil, put it into the vinegar, stir. Than take one teaspoon of the mixture, put it into the oil, stir. Now: is there more vinegar in the oil or more oil in the vinegar?

Answer 34

Suppose 100 ants are placed randomly (with random orientations) along a yard stick. Each ant walks at a pace of an inch a minute. Each time two ants meet, they instantaneously reverse direction, and if an ant meets the end of the yardstick, it instantaneously reverses direction.
Do the ants ever return to their starting positions? At what time? (A yardstick is 36 inches long.)

Answer 35

How many times a day is it impossible to tell the time by a clock with identical hour and minute hands, provided you can always distinguish between a.m and p.m? P.S. Ask them for a fast answer.

Answer 36

Start with four beads placed at the corners of a square. You are allowed to move a bead from position x to position y if one of the other three beads is at position (x+y)/2. In other words, you may `reflect a bead with respect to another bead.' Find a sequence of such moves that places the beads at the corners of a bigger square, or show that the task is impossible.

Answer 37

Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope. You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.

Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?

Answer 38

Here is a classic:

Plant 10 trees in five rows, with 4 trees in each row.

I like this because there are two basic approaches to the problem: the one almost everyone thinks of and uses to grind slowly towards a solution, and the one they should think of instead, which leads quickly to many solutions.

Gerhard "Ask Me About System Design" Paseman, 2010.07.28

Answer 39

A puzzle(rather, a tale to lure the reader into the domain of complex numbers) lifted from George Gamow's "One, Two, Three, Infinity":

There was a young and adventurous man who found among his great-grandfather’s papers a piece of torn parchment that revealed the precise location of a hidden treasure. The instruction reads:

Sail to North latitude __ and West longitude __ where thou wilt find a deserted island. There lieth a large meadow, not pent, on the north shore of the island where standeth a lonely oak and a lonely pine tree. There thou wilt see also an old gallows on which we once were wont to hang traitors. Start thou from the gallows and walk to the oak counting thy steps. At the oak thou must turn right by a right angle and take the same number of steps. Put here a spike in the ground. Now must thou return to the gallows and walk to the pine counting thy steps. At the pine thou must turn left by a right angle and see that thou takest the same number of steps, and put another spike into the ground. Dig halfway between the spikes; the treasure is there.

The instructions being quite clear and explicit, our young man chartered a ship and sailed to the South Seas. He found the island, the field, the oak and the pine, but to his great sorrow, the gallows was gone. Too long a time had passed: rain and sun and wind had disintegrated the wood and returned it to the soil, leaving no trace of the place where once it had stood. Our adventurous man fell into despair. Digging all over the field at random, he found nothing and sailed back empty-handed.

A sad story for sure, but sadder to think that he might have easily located the treasure had he known a little about the arithmetic of complex numbers!!

Question: How???

Answer: Read on from Here.

Answer 40

You and your adversary have a sufficiently large bag of identical coins, and are seated on opposite sides of a rectangular table. You take turns placing coins on the table. The first one that cannot put a coin on the table without overlapping any other coin loses. What is your strategy to always win if you're allowed to start?

Answer 41

Can one partition the plane $\mathbb{R}^2$ by closed intervals of equal length? How? The answer to the first question is "yes". In other words, can one cover the plane with translates and rotations of a given closed line segment such that every point lies on exactly one segment?

Answer 42

Here's an easy but fun one (probably suitable for a class or for mathematicians who have had some drinks). You have ten bags of coins, one of which contains fake coins. We may of course assume that each bag contains infinitely many coins. The real coins weigh 1 gram each, while the fake coins weigh .9 grams each. You have a scale, which is capable of only one accurate reading before breaking. Determine which bag contains the fake coins.

Answer 43

Cryptography riddle - that's a branch of mathematics, right? :)

Jan and Maria have fallen in love (via the internet) and Jan wishes to mail her a ring. Unfortunately, they live in the country of Kleptopia, where anything sent through the mail will be stolen unless it is enclosed in a padlocked box. Jan and Maria each have plenty of padlocks, but none to which the other has a key. How can Jan get the ring safely into Maria’s hands?

Answer 44

An evil sorceress is holding 100 princes captive. Right now they are all in the same prison cell and can discuss strategy. However, in a moment, they will each be taken to individual prison cells, where no communication is possible. After that, the sorceress will start randomly calling princes to her bedroom (one at a time). This continues indefinitely, so a prince can visit the bedroom many times. The bedroom has two light switches, whose state can be observed only from inside the bedroom. When a prince is called to the bedroom, he can observe the state of the switches, and then must change the state of exactly one of the light switches. The initial state of the light switches is not known.

The princes will be set free if any one of them can determine if all of them have been called to the room.

Puzzle: Determine a strategy for the princes so that they are guaranteed to be set free eventually. The strategy should never output a false positive. For example, if a prince has been called one million times he can reason that on average, everyone else has been called one million times. Thus it is very likely that all the princes have been called to the bedroom, but it is possible that one prince still hasn't been called.

Answer 45

Here's a balance scale problem that I decided to post because a little bit of googling around for it came up negative. It differs from most balance scale puzzles I've seen because it doesn't involve "bad weights". I learned of it from a friend of mine who is an engineer.

There are 10 balls which come in two possible weights. Using a balance scale at most 3 times, determine whether all the balls are the same weight or not.

Answer 46

You are the captain of a team of N players, in charge of choosing a strategy that your adversary will overhear (and therefore rig the game for you to lose unless the strategy is perfect). To play the game, the adversary writes a distinct name on each player's forehead and you are brought into a situation where each of you can learn the name given to every other player, but not your own. Naturally you cannot communicate once the game has started. Each of you is blindfolded and given a single invertible glove. On a signal, each of you silently places your glove on one hand or the other. You are then lined up in alphabetical order by the names on your foreheads, all facing the same direction, and you join hands in one long chain. If any of you touches another player's glove with your bare hand the team loses, but if it is always hand-to-hand and glove-to-glove, you are victorious.

For what values of N can you give your team a winning strategy, and what is it?

Answer 47

Here's one I saw a while ago:

A prisoner is presented with the following challenge by one of the guards of the jail. The prisoner is to be blindfolded and then the guard will place $n$ coins on a circular turntable with any combination of heads and tails facing up (with at least one tails showing initially). The prisoners goal is to flip over coins until all heads are showing.

This would be easy enough if the guard did not interfere. The prisoner could just try all $2^n$ combinations, and one of them would be guaranteed to result in all heads. However, to complicate matters, the guard may turn the table during this process. More specifically, the following process is repeated. First, the prisoner chooses a set of positions of coins to flip over. Then, before the coins are flipped, the guard turns the turntable so as to try to prevent the prisoner from flipping all of the coins to heads. Finally, the prisoner flips over the coins that are at the positions chosen in the first step. If all heads are showing, the game stops and the prisoner is set free.

The question is, for what values of $n$ does the prisoner have a winning strategy and how many moves does it take?

What if the guard uses 6-sided dice instead of coins with the goal of showing all ones (assuming the orientations of the dice are preserved relative to their positions on the turntable between rounds)?

In general, what values of $n$ allow a solution with $k$-sided dice?

Answer 48

Take a convex polyhedron and any point inside of it. To every face, drop a normal line from the point. Note that it is both possible to land inside the face or outside. Construct a polyhedron where every such normal line drops outside of the corresponding face, or prove such a polyhedron cannot exist.

Answer 49

Simple puzzles; unfortunately, I do not know how to formulate them in a whimsical fashion suitable for a dinner, they very much sound like math puzzles.

1. Take n labeled points $x_1, \dots, x_n$ in the plane. How do you construct a n-gon $a1, \dots, an$ such that for all i, $x_i$ is the midpoint of $[a_i, a_{i+1}]$ (with the convention $a_{n+1}=a_1$ of course). I was surprised to come across this problem in the puzzle pages of Le Monde. I think non-mathematicians would have a hard time with it.

2. For mathematicians who don't already know it, the Sylvester Gallai Theorem can offer stimulating after dinner discussions (or during those long proctoring sessions).

3. A napkin should be enough for this one (if even needed!). Consider a map f from the plane to the reals such that the sum of the values of f on the vertices of any square is zero. Find all such maps.

Answer 50

Here's one that I like that I just heard a few days ago. Alice and Bob play the following game. Alice is randomly dealt 5 cards from an ordinary deck of cards. She is allowed to show Bob 4 of the 5 cards (in order). Bob must then guess what the 5th card is.

Prove that Alice and Bob have a strategy where Bob can always guess correctly.

Edit. Actually, there is a strategy that works for 124 cards, but it is probably not human implementable (unlike the 52 card problem).

Answer 51

What is four thousand and ninety-nine plus one?

Answer 52

Okay, I've got one, and as far as I know it hasn't been analyzed before.

I was watching a travel show the other night -- they were in Korea, and a group of people were playing a drinking game. It works like this:

One person is "it". This person says something like, "ready, set..." then points at one other player and calls out a number between 2 and n (where n is the number of people playing). At the same time, everyone else also points at one other player. Then, for whatever number got call out, you jump that many steps from the "it" person, and that person has to drink. So if I call out "two" and point at Joe, and Joe points at Bob, then Bob has to drink.

I think the game is pretty interesting, mathematically, especially when you allow numbers greater than n to be called. One interesting thing I found: with n=3, if you call 7 (or 7+6x, where x is a non-negative integer), you are guaranteed to stick the player you initially point at, no matter who points to whom.

I think an interesting question is, given n players, what is the smallest number the 'it' person can call that guarantees he will not stick himself? (I have an answer, but I want to see if you all come up with the same thing. :-) And what's the best strategy for the caller if you enforce the rule that you must call out a number between 2 and n? What's the best strategy for the other players, if they're allowed to collude on who they're going to point to? Etc.

Answer 53

A magician places $N = 64$ coins in a row on a table, then leaves the room. A person from the audience is then asked to flip each coin however he likes [so there are $2^{64}$ possible states]. He is also asked to mention a number between 1 and $N$. After this, the magician's assistant flips exactly one coin. The magician reenters the room, looks at the coins and "guesses" the number chosen by the audience.

What is their strategy? For which values of $N$ can the trick be performed?

Answer 54

Let $I$ be the set of irrational numbers, with the usual topology. Is $I$ homeomorphic to $I\times I$?

Edited to add: In fact, it's an even better puzzle if you replace $I$ with $Q$, the rational numbers.

Answer 55

A table with three legs does not wobble.

How about a quadratic table with four legs? Can it be rotated to fix the wobbling?

(Please assume reasonable conditions on life the universe and everything.)

Answer 56

Assuming you have unlimited time and cash, is there a strategy that's guaranteed to win at roulette?

Answer 57

Fork in the road 1

You're on a path on an island, come to a fork in the road. Both paths lead to villages of natives; the entire village either always tells the truth or always lies (both villages could be truth-telling or lying villages, or one of each). There are two natives at the fork - they could both be from the same village, or from different villages (so both could be truth-tellers, both liars, or one of each).

One path leads to safety, the other to doom. You're allowed to ask only one question to each native to figure out which path is which.

What do you ask?

Answer 58

First, I apologize that this puzzle is not "clean". It's more suitable for the bar rather than dinner (in fact I heard it at a party with mathematicians). I understand if it should be taken down or rephrased. I scanned the previous puzzles and I don't think this puzzle is a duplicate.

Suppose you are male(female) and stranded on an island with three females(males). You wish to have protected sex with all three females(males), but you only have two condoms and no other forms of protection. Clearly you can have protected sex with two of them by using one condom, throwing it away and using the other. Can you have protected sex with all three of them?

By protected I mean there is no exchange of fluids from one person to another, i.e. you can't use a condom with one person and then use it "as is" with another person. Also, despite being surrounded by the ocean you cannot just rinse them off - that's dirty.

Answer 59

I think this has not been published yet. Apologies otherwise. I learnt it from Antonio Sánchez Calle in my first year of undergraduate and I had 3 non-mathematicians thinking about it for about 4 hours, so there is a guaranteed success if you tell around :)

5 people are shipwrecked in a deserted island. They find a monkey and lots of coconuts. They spend the whole day collecting coconuts that they keep together and since they are tired they go to sleep. The first person wakes up, attempts to divide the amount of coconuts in five parts, but one of them is spared, so he gives to the monkey. Then he eats one fifth of the coconuts and goes back to sleep.

The second person wakes up and follows the same procedure. He divides the coconuts in five, one is spared and he gives it to the monkey, eats his share and goes back to sleep.

The third, forth and fifth people do the same thing. How many coconuts were there at the beginning? (modulo something, of course).

Answer 60

Quite simply, a monkey's mother is twice as old as the monkey will be when the monkey's father is twice as old as the monkey will be when the monkey's mother is less by the difference in ages between the monkey's mother and the monkey's father than three times as old as the monkey will be when the monkey's father is one year less than twelve times as old as the monkey is when the monkey's mother is eight times the age of the monkey, notwithstanding the fact that when the monkey is as old as the monkey's mother will be when the difference in ages between the monkey and the monkey's father is less than the age of the monkey's mother by twice the difference in ages between the monkey's mother and the monkey's father, the monkey's mother will be five times as old as the monkey will be when the monkey's father is one year more than ten times as old as the monkey is when the monkey is less by four years than one seventh of the combined ages of the monkey's mother
and the monkey's father.

If, in a number of years equal to the number of times a monkey's mother is as old as the monkey, the monkey's father will be as many times as old as the monkey as the monkey is now, find their respective ages.

(Answer at http://7c6j.sl.pt)

Answer 61

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.)

Answer 62

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?

Answer 63

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?

Answer 64

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.

Answer 65

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

Answer 66

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?

Answer 67

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!