There are five holes arranged in a line.  A hermit hides in one of them.  Each night, the hermit moves to a different hole, either the neighboring hole on the left or the neighboring hole on the right.  Once a day, you get to inspect one hole of your choice.  How do you make sure you eventually find the hermit?

You have 240 barrels of wine, one of which has been poisoned.  After drinking the poisoned wine, one dies within 24 hours.  You have 5 slaves whom you are willing to sacrifice in order to determine which barrel contains the poisoned wine.  How do you achieve this in 48 hours?

You are given a balance (that is, a weighing machine with two trays) and a positive integer N.  You are then to request a number of weights.  You pick which denominations of weights you want and how many of each you want.  After you receive the weights you requested, you are given a thing whose weight is an integer between 1 and N, inclusive.  Using the balance and the weights you requested, you must now determine the exact weight of the thing.

As a function of N, how few weights can you get by requesting?

You are given four points (on a Euclidian plane) that make up the corners of a square.  You may change the positions of the points by a sequence of moves.  Each move changes the position of one point, say p, to a new location, say p’, by “skipping over” one of the other 3 points.  More precisely, p skips over a point q if it is moved to the diametrically opposed side of q.  In other words, a move from p to p’ is allowed if there exists a point q such that q = (p + p’) / 2.

Find a sequence of moves that results in a larger square.  Or, if no such sequence is possible, give a proof of why it isn’t possible.  (The new square need not be oriented the same way as the original square.  For example, the larger square may be turned 45 degrees from the original, and the larger square may have the points in a different order from in the original square.)

N people team up and decide on a strategy for playing this game.  Then they walk into a room.  On entry to the room, each person is given a hat on which one of the first N natural numbers is written.  There may be duplicate hat numbers.  For example, for N=3, the 3 team members may get hats labeled 2, 0, 2.  Each person can see the numbers written on the others’ hats, but does not know the number written on his own hat.  Every person then simultaneously guesses the number of his own hat.  What strategy can the team follow to make sure that at least one person on the team guesses his hat number correctly?

A game is played as follows. N people are sitting around a table, each with one penny.  One person begins the game, takes one of his pennies (at this time, he happens to have exactly one penny) and passes it to the person to his left.  That second person then takes two pennies and passes them to the next person on the left.  The third person passes one penny, the fourth passes two, and so on, alternating passing one and two pennies to the next person. Whenever a person runs out of pennies, he is out of the game and has to leave the table. The game then continues with the remaining people.

A game is terminating if and only if it ends with just one person sitting at the table (holding all N pennies).  Show that there exists an infinite set of numbers for which the game is terminating.

Find two positive integers that together with 23 are the lengths of a right triangle.

Hint: There’s a simple technique that, given any odd positive integer, allows you to figure out the other two integer sides of a right triangle in your head (or with pen and paper if the numbers get too large).  Find this technique.

Think of a positive integer, call it X.  Shuffle the decimal digits of X, call the resulting number Y.  Subtract the smaller of X,Y from the larger, call the difference D.  D has the following property:  Any non-zero decimal digit of D can be determined from the remaining digits.  That is, if you ask someone to hide any one of the non-zero digits in the decimal representation of D, then you can try to impress the other person by figuring out the hidden digit from the remaining digits.  How is this done?  Why does it work?

You’re given a regular deck of 52 playing cards.  In the pile you’re given, 13 cards face up and the rest face down.  You are to separate the given cards into two piles, such that the number of face-up cards in each pile is the same.  In separating the cards, you’re allowed to flip cards over.  The catch:  you have to do this in a dark room where you cannot determine whether a card is face up or face down.