The house next door has some new neighbors.  They have two children, but you don’t know what mix of boys and girls they are.  One day, your wife tells you “At least one of the children is a girl”.  What is the probability that both are girls?

Your wife then tells you “The way I found out that at least one of the children is a girl is that I saw one of the children playing outside, and it was a girl”.  Now, what is the probability that both are girls?

A particular basketball shootout game consists of a number of duels.  In each duel, one player is the challenger.  The challenger chooses another player to challenge, and then gets one chance to shoot the hoop.  If the player makes the shot, the playing being challenged is out.  If the player does not make the shot, or if the player chooses to skip his turn, then the game continues with the next duel.  A player wins when only that player remains.

One day, this game is played by three players: A, B, and C.  Their skill levels vary considerably:  player A makes every shot, player B has a 50% chance of making a shot, and player C has a 30% chance of making a shot.  Because of the difference in skill levels, they decide to let C begin, then B, then A, and so on (skipping any player who is out of the game) until there is a winner.  If everyone plays to win, what strategy should each player follow?

[For this follow-up question, it will be helpful to have a paper and pen–not because the calculations are hard, but because it helps in remembering the numbers.]

If A, B, and C follow their winning strategies (as determined above), which player has the highest chance of winning the game?

You’re on a government ship, looking for a pirate ship.  You know that the pirate ship travels at a constant speed, and you know what that speed is.  Your ship can travel twice as fast as the pirate ship.  Moreover, you know that the pirate ship travels along a straight line, but you don’t know what that line is.  It’s very foggy, so foggy that you see nothing.  But then!  All of a sudden, and for just an instant, the fog clears enough to let you determine the exact position of the pirate ship.  Then, the fog closes in again and you remain (forever) in the thick fog.  Although you were able to determine the position of the pirate ship during that fog-free moment, you were not able to determine its direction.  How will you navigate your government ship so that you will capture the pirate ship?

A (presumed smart) insurance agent knocks on a door and a (presumed smart) woman opens.  He introduces himself and asks if she has any children.  She answers: 3.  When he then asks their ages (which for this problem we abstract to integers), she hesitates.  Then she decides to give him some information about their ages, saying “the product of their ages is 36”.  He asks for more information and she gives in, saying “the sum of their ages is equal to our neighbors’ house number”.  The man jumps over the fence, inspects the house number, and the returns.  “You need to give me another hint”, he begs.  “Alright”, she says, “my oldest child plays the piano”.  What are the ages of the children?

There’s a certain kind of egg about which you wonder:  What is the highest floor of a 36-story building from which you can drop an egg without it breaking?  All eggs of this kind are identical, so you can conduct experiments.  Unfortunately, you only have 2 eggs.  Fortunately, if an egg survives a drop without breaking, it is as good as new–that is, you can then conduct another dropping experiment with it.  What is the smallest number of drops that is sure to determine the answer to your wonderings?

A duck is in circular pond.  The duck wants to swim ashore, because it wants to fly off and this particular duck is not able to start flying from the water.  There is also a fox, on the shore.  The fox wants to eat the duck, but this particular fox cannot swim, so it can only hope to catch the duck when the duck reaches the shore.  The fox can run 4 times faster than the duck can swim.  Is there always a way for the duck to escape?

You and a friend each has a fair coin.  You can decide on a strategy and then play the following game, without any further communication with each other.  You flip your coin and then write down a guess as to what your friend’s coin will say.  Meanwhile, your friend flips her coin and writes down a guess as to what your coin says.  There’s a third person involved: The third person collects your guesses and inspects your coins.  If both you and your friend correctly guessed each other’s coins, then your team (you and your friend) receive 2 Euros from the third person.  But if either you or your friend (or both) gets the guess wrong, then your team has to pay 1 Euro to the third person.  This procedure is repeated all day.  Assuming your object is to win money, are you happy to be on your team or would you rather trade places with the third person?

There are two kinds of coins, genuine and counterfeit.  A genuine coin weighs X grams and a counterfeit coin weighs X+delta grams, where X is a positive integer and delta is a non-zero real number strictly between -5 and +5.  You are presented with 13 piles of 4 coins each.  All of the coins are genuine, except for one pile, in which all 4 coins are counterfeit.  You are given a precise scale (say, a digital scale capable of displaying any real number).  You are to determine three things: X, delta, and which pile contains the counterfeit coins.  But you’re only allowed to use the scale twice!

Ten friends walk into a room where each one of them receives a hat.  On each hat is written a real number; no two hats have the same number.  Each person can see the numbers written on his friends’ hats, but cannot see his own.  The friends then go into individual rooms where they are each given the choice between a white T-shirt and a black T-shirt.  Wearing the respective T-shirts they selected, the friends gather again and are lined up in the order of their hat numbers.  The desired property is that the T-shirts colors now alternate.

The friends are allowed to decide on a strategy before walking into the room with the hats, but they are not otherwise allowed to communicate with each other (except that they can see each other’s hat numbers).  Design a strategy that lets the friends always end up with alternating T-shirt colors.

Given is a (possibly enormous) rectangular chocolate bar, divided into small squares in the usual way. The chocolate holds a high quality standard, except for the square in the lower left-hand corner, which is poisonous. Two players take turns eating from the chocolate in the following manner: The player whose turn it is points to any one of the remaining squares, and then eats the selected square and all squares positioned above the selected square, to the right of the selected square, or both above and to the right of the selected square. Note, although the board starts off rectangular, it may take on non-rectangular shapes during game play. The object of the game is to avoid the poisonous square. Assuming the initial chocolate bar is larger than 1×1, prove that the player who starts has a winning strategy.

Hint: To my knowledge, no efficient strategy for winning the game is known. That is, to decide on the best next move, a player may need to consider all possible continuations of the game. Thus, you probably want to find a non-constructive proof. That is, to prove that the player who starts has a winning strategy, prove just the existence of such a strategy; in particular, steer away from proofs that would construct a winning strategy for the initial player.

A man has a medical condition that requires him to take two kinds of pills, call them A and B.  The man must take exactly one A pill and exactly one B pill each day, or he will die.  The pills are taken by first dissolving them in water.

The man has a jar of A pills and a jar of B pills.  One day, as he is about to take his pills, he takes out one A pill from the A jar and puts it in a glass of water.  Then he accidentally takes out twoB pills from the B jar and puts them in the water.  Now, he is in the situation of having a glass of water with three dissolved pills, one A pill and two B pills.  Unfortunately, the pills are very expensive, so the thought of throwing out the water with the 3 pills and starting over is out of the question.  How should the man proceed in order to get the right quantity of A and B while not wasting any pills?

Hilary and Jocelyn are throwing a dinner party at their house and have invited four other couples.  After the guests arrive, people greet each other by shaking hands.  As you would expect, a couple do not shake hands with each other and no two people shake each other’s hands more than once.  At some point during the handshaking process, Jocelyn gets up on a table and tells everyone to stop shaking hands.  She also asks each person how many hands they have shaken and learns that no two people on the floor have shaken the same number of hands.  How many hands has Hilary shaken?

In a room are three boxes that on the outside look identical.  One of the boxes contains a car, one contains a key, and one contains nothing.  You and a partner get to decide amongst yourselves to each point to two boxes.  When you have made your decision, the boxes are opened and their contents revealed.  If one of the boxes your partner is pointing to contains the car and one of the boxes you are pointing to contains the key, then you will both win.  What strategy maximizes the probability of winning, and what is the probability that you will win?

You have 3000 bananas that you want to transport a distance of 1000 km.  The transport will be done by a monkey.  The monkey can carry as many as 1000 bananas at any one time.  With each kilometer traveled (forward or backward), the money consumes 1 banana.  How many bananas can you get across to the other side?