1) Given the following sequence of words:
John where Pete had had had had had had had had had had had the examiners approval.
insert punctuation marks in such a way that the result is a perfectly sensible English sentence.
2) A slightly contrived, but interesting problem: 24 convicts are being transported together to a special prison, in which they will spend the rest of their life, each in solitary confinement. However they know that they have one last chance of securing freedom if they solve the following puzzle: At random intervals (on average much shorter than the prisoner's lifespan!), a randomly selected prisoner will be led to a room which is completely bare and only contains two switches with two positions each. On every visit exactly one of the switches must be switched exactly once but the visiting prisoner is free to choose which one he wants to switch. When a prisoner can tell that all prisoners have been in this room at least once, all are set free. What strategy should they agree on before being separated upon arrival at the prison?
3) On a remote island there live 30 inhabitants. Ten of them have a pair of red eyes each, the remaining twenty have a pair of green eyes each. All inhabitants obey two rules at all times:
1. They do not communicate with one another (but every inhabitant sees every other inhabitant every day).
2. An inhabitant who finds out that he or she has red eyes has to commit suicide the same night.
One day a missionary arrives. He respects their rules and tries to convert them to his religion, but fails. On the day of his departure he is so angry that he breaks the rules in saying: "At least one of you has red eyes.", whereupon he leaves the island.
What is the subsequent history of the island population?
4) And here a really hard one:
Two logicians meet, let us call them A and B.
A is given a number which is the sum of two natural numbers c and d, i.e. the number (c+d). B is given the sum of the squares of the same two natural numbers c and d, i.e. the number (c^2+d^2). Both know this, but don't know the number given to the other person. The conversation (referring to the two natural numbers c and d) between them runs as follows:
B: I do not know the numbers.
A: I do not know the numbers.
B: I do not know the numbers.
A: I do not know the numbers.
B: I do not know the numbers.
A: I do not know the numbers.
B: I know the numbers.
What are the two natural numbers c and d?
5) Three people take part in a game, as a team. Each of them has a hat placed on his or her head, but cannot see the colour of the hat, which can be blue or red in all three cases. Each person has to guess or pass. If any one guesses correctly, all win. If any one guesses incorrectly, all lose. If everyone passes, all lose. What is the best strategy, which they can agree on beforehand, to maximize their chances of winning? Consider the cases:
a) They may choose the sequence in which they answer.
b) The sequence is fixed beforehand by the rules of the game.
(Thanks to Jeremy for this puzzle!)
6) A gameshow host presents three closed gates - A, B and C - to a candidate in the show, saying that there is a big prize behind one of the gates and nothing behind the other two. The candidate is asked to choose one gate. The candidate chooses gate A. The host responds to this by opening gate B, behind which there turns out to be nothing. The candidate is offered another chance to choose between gate A and C. What choice should the candidate make, or is neither choice better than the other?
This puzzle is called the Monty Hall problem, and is rather famous (follow the link only if you don't mind seeing the solution!).
7) You are given 12 marbles which look identical and are told that one differs from all others in weight. However, you do not know whether it is heavier or lighter. By using a scale three times, find the odd marble and whether it weighs more or less than the others. (Thanks to Jonatan for this one!)
8) Compare the following:
a) You know that Mr and Mrs Smith have two children. They tell you "At least one of our children is a girl." What is the probability of them having two girls?
b) You know that Mr and Mrs Smith have two children. You meet them on the street with a little girl. What is the probability of them having two girls?
(Thanks to James B. for this one!)
9) An evil dictator imprisons 100 logicians, and gives them one last chance to escape death. He says:
"Tomorrow you will all be put into solitary confinement. Then each of you will be called from your cell, one by one, in random order. You will be asked to choose a number between 1 and 100, and led to a room with 100 boxes. In each box there is a number between 1 and 100 (each number occurring exactly once), and you are allowed to open 50 boxes. If you find your number, you go back to your cell. If you don't find it, everyone dies. If everyone finds their number, you all go free."
Assuming the room the logicians are led to is in exactly the same state every time a logician enters, what strategy should the logicians agree on, and what is the probability that they all go free? (Note: It's larger than one quarter!)
(Thanks to Jonatan for this one!)
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