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.