1.6 Some Brain Teasers

Here are three problems to experiment with. The important point here is not to get the correct answer but to think about the forms of reasoning you are using to resolve the difficulty.

1. You are a police officer on a highway patrol. You come across an accident in which two cars have collided in an off-highway rest area. Each driver claims that he has been at the rest area for over two hours eating lunch and sleeping and that the other driver drove in from the highway and ran into his car a few minutes ago. You cannot tell from the position of the vehicles which one is telling the truth. There are no witnesses. Can you think of how you might sort out the claims on the spot? What form of reasoning have you used?

2. Two friends of yours are having a bitter argument over the question of whether or not two women could have exactly the same number of hairs on their heads. They want you to determine the question. Can you think of some deductive way to resolve their problem? What would an inductive resolution of the issue require?

3. A man is walking to the town of Ipswich. He comes to a fork in the road, with the two branches leading in two different directions. He knows that one of them goes to Ipswich, but he doesn't know which one. He also knows that in the house right beside the fork in the road there are two brothers, identical twins, both of whom know the road to Ipswich. He knows that one brother always lies and the other always tells the truth, but he cannot tell them apart. What single question can he ask to whoever answers his knock on the door which will indicate to him the correct road to Ipswich?

4. Three men are placed directly in line facing a wall. The man at the back can see the two in front of him, the man in the middle can see the man immediately in front, and the man at the front can see only the wall. Each man has a hat on his head, taken from a supply of three black hats and two white hats (the men know this). They are told to remain in line silently until one of them can guess the colour of the hat on his head. That man gets a large cash prize. After five minutes of standing in line, the man facing the wall (at the front of the line) correctly identifies the colour of the hat on his head. What colour must it be? How did he arrive at the correct conclusion? Note that he did not guess.