A collection of perfect logicians live on an island. Each of the logicians has an eye color. They can each see the eye colors of everyone else on the island but do not have any information about their own colors. The islanders are not allowed to speak to one another or communicate in any way.
Every night at midnight, a ferry stops on the island. Any islanders who have conclusively determined the color of their own eyes is free to leave the island at this time. The islanders all know of this rule. As it happens, on this island, there are 100 blue-eyed people, 100 brown-eyed people, and one green-eyed person, although the islanders, of course, do not know this. The green-eyed person is called the Guru. One day, the Guru stands on a tall pedestal and makes a statement that all of the islanders can hear: "I can see someone who has blue eyes." This is the only time any of the islanders communicate in any way.
Which of the islanders will be allowed on the ferry, and on which night?
One hundred pirates have come upon a treasure chest consisting of fifty gold coins. The gold coins are each of equal worth to one another and are indivisible. The pirates have a total ordering which they use to determine who is in charge, so Pirate 1 in the leader, Pirate 2 is second in command, and so on. The pirates are also perfect logicians and will always make perfectly logical decisions.
The pirates have a well-defined procedure for determining how to distribute the gold. First, the highest-ranking pirate who is still alive (initially, this will be the leader) proposes a way to distribute the treasure among the surviving pirates. Then, all of the currently living pirates (including the one who proposed this idea) vote on the proposal. If at least half of the voters approve of the proposal, then it is accepted and the gold is distributed that way. Otherwise, the pirate who made the proposal is killed and the process is repeated with the next in command. A pirate's priorities are, in order: survival, wealth, and bloodthirst. That is, a pirate will always favor a decision which results in his survival over one which results in his death. Between two scenarios which both result in his survival, a pirate will always choose the one which results in him getting more money. Finally, if two scenarios both result in survival with the same amount of gold, the pirate will choose the situation that involves the most other pirates dying.
Assuming the pirates make perfectly logical decisions that are in line with their priorities, how will the gold be distributed?
Sam and Polly are perfect logicians. One day, a student walks in and says "I'm thinking of two numbers \(x\) and \(y\) with \(3 \le x \le y \le 97\). I'm going to tell their sum to Sam and their product to Polly. The student does so and then leaves. The following conversation takes places.
(A personal favorite of mine. It's an interesting exercise to write a computer program to solve this. However, I have yet to determine a solution which doesn't require the assistance of a computer.)
A dragon and a knight live on a small island. On this island, there is a freshwater lake that contains ordinary drinking water. There are also six wells, numbered 1 to 6. Each well contains a liquid that is indistinguishable from water in appearance but that is actually a deadly poison. This poison shows no symptoms but will instantly kill the drinker an hour after having been consumed. The poisons, however, are somewhat unique. If someone drinks poison from the same well multiple times, it has the same effect as drinking it only once. However, if someone has already been poisoned and drinks the poison of a higher-numbered well, all poisons from lower wells are cured instantly. This only works of the higher-numbered well is drunk after the lower well(s). As such, Well 6 can cure any of the other wells' poisons but, if a healthy individual were to drink it, would incurably poison them.
Both the knight and the dragon understand the rules of these wells. They also each want the other dead, so they arrange a special contest. Each participant secretly fills a vial with a liquid, either from the lake or from one of the numbered wells. They will then meet, exchange vials, and drink the liquid prepared by their opponent. Each player is free to drink whatever they wish before and after exchanging vials, in preparation for the contest. Further, while the freshwater lake and the first five wells are accessible to both players, Well 6 is located on a steep mountaintop that the knight cannot access.
Is there a strategy that will ensure the survival of either player?
There are two prisoners and a warden. The warden decides to play a game with the prisoners. If the prisoners win, they are free to go, but if they lose then they remain imprisoned forever. The warden has a standard, 8x8 chessboard in his office. He proposes the following challenge to the prisoners. He will position a single quarter on each square of the chessboard. The quarters will be flipped arbitrarily, so that each coin could be either heads or tails. He will then call the first prisoner into his office. The warden will point to a single square on the chessboard, which he calls the magic square. The first prisoner is then required to flip exactly one coin on the chessboard over to its other side. He then leaves, without communicating to the second prisoner. The second prisoner will then be called in, having never seen the board before the first prisoner's change. He is allowed to guess at the location of the magic square. If he chooses correctly, the prisoners win.
The prisoners are not allowed to communicate after the game starts. However, before the game begins they are free to discuss a strategy among themselves. There exists a strategy that will allow the two prisoners to escape with absolute certainty, assuming that they are both perfect logicians. What is this strategy?