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?

You are given an irreflexive symmetric (but not necessarily transitive) “enemies” relation on a set of people.  In other words, if person A is an enemy of a person B, then B is also an enemy of A.  How can you divide up the people into two houses in such a way that every person has at least as many enemies in the other house as in their own house?

Hint:  Planar configuration of straight connecting lines puzzle may provide a hint to solving this puzzle.

Given an even number of points in general positions on the plane (that is, no three points co-linear), can you partition the points into pairs and connect the two points of each pair with a single straight line such that the straight lines do not overlap?

You are given one 44-meter piece of fence and 48 one-meter pieces of fence.  Assume each piece is a straight and unbendable.  What is the large area of (flat) land that you can enclose using these fence pieces?

Someone picks, at their will, two cards from a deck of cards. The cards have different numbers, so one is higher than the other. (In other words, the person picks two distinct numbers in the inclusive range 1 through 13.)  The cards are placed face down on a table in front of you.  You get to choose one of the cards and turn it face up.  Now, you will select one of the two cards (one of whose face you can see, the other one you can’t).  If you select the highest card, you win.  Design a card-selection strategy for which your chance of winning is strictly greater than 50%.