A room has 100 light switches, numbered by the positive integers 1 through 100.  There are also 100 children, numbered by the positive integers 1 through 100.  Initially, the switches are all off.  Each child k enters the room and changes the position of every light switch n such that n is a multiple of k.  That is, child 1 changes all the switches, child 2 changes switches 2, 4, 6, 8, …, child 3 changes switches 3, 6, 9, 12, …, etc., and child 100 changes only light switch 100.  When all the children have gone through the room, how many of the light switches are on?

Each of two players picks a different sequence of two coin tosses.  That is, each player gets to pick among HH, HT, TH, and TT.  Then, a coin is flipped repeatedly and the first player to see his sequence appear wins.  For example, if one player picks HH, the other picks TT, and the coin produces a sequence that starts H, T, H, T, T, then the player who picked TT wins.  The coin is biased, with H having a 2/3 probability and T having a 1/3 probability.  If you played this game, would you want to pick your sequence first or second?

100 coins are to be distributed among some number of persons, referred to by the labels A, B, C, D, ….  The distribution works as follows.  The person with the alphabetically highest label (for example, among 5 people, E) is called the chief.  The chief gets to propose a distribution of the coins among the persons (for example, chief E may propose that everyone get 20 coins, or he may propose that he get 100 coins and the others get 0 coins).  Everyone (including the chief) gets to vote yes/no on the proposed distribution.  If the majority vote is yes, then that’s the final distribution.  If there’s a tie (which there could be if the number of persons is even), then the chief gets to break the tie.  If the majority vote is no, then the chief gets 0 coins and has to leave the game, the person with the alphabetically next-highest name becomes the new chief, and the process to distribute the 100 coins is repeated among the persons that remain.  Suppose there are 5 persons and that every person wants to maximize the number of coins that are distributed to them.  Then, what distribution should chief E propose?

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?

Consider a game that you play against an opponent.  In front of you are an even number of coins of possibly different denominations.  The coins are arranged in a line.  You and your opponent take turns selecting coins.  Each player takes one coin per turn and must take it from an end of the line, that is, the current leftmost coin or the current rightmost coin.  When all coins have been removed, add the value of the coins collected by each player.  It is possible that you and your opponent end up with the same value (for example, if all coins have the same denomination).  Develop a strategy where you take the first turn and where your final value is at least that of your opponent (that is, don’t let your opponent end up with coins worth more than your coins).

Prove that for any positive K, every K^th number in the Fibonacci sequence is a multiple of the K^th number in the Fibonacci sequence.

More formally, for any natural number n, let F(n) denote Fibonacci number n.  That is, F(0) = 0, F(1) = 1, and F(n+2) = F(n+1) + F(n).  Prove that for any positive K and natural n, F(n*K) is a multiple of F(K).

There are three boxes, one containing two black balls, another containing two white balls, and another containing one black and one white ball.  You select a random ball from a random box and you find you selected a white ball.  What is the probability that the other ball in the same box is also white?

The people in a country are partitioned into clans.  In order to estimate the average size of a clan, a survey is conducted where 1000 randomly selected people are asked to state the size of the clan to which they belong.  How does one compute an estimate average clan size from the data collected?

A spy is located on a one-dimensional line.  At time 0, the spy is at location A.  With each time interval, the spy moves B units to the right (if B is negative, the spy is moving left).  A and B are fixed integers, but they are unknown to you.  You are to catch the spy.  The means by which you can attempt to do that is:  at each time interval (starting at time 0), you can choose a location on the line and ask whether or not the spy is currently at that location.  That is, you will ask a question like “Is the spy currently at location 27?” and you will get a yes/no answer.  Devise an algorithm that will eventually find the spy.

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.)