## Monday, March 23, 2009

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### What colour is my hat?

In his TierneyLab blog in the New York Times, John Tierney recently posted an old puzzle, of a kind of which I’m not fond. It goes like this:

Three wise men are told to stand in a straight line, one in front of the other. A hat is put on each of their heads. They are told that each of these hats was selected from a group of five hats: two black hats and three white hats. The first man, standing at the front of the line, can’t see either of the men behind him or their hats. The second man, in the middle, can see only the first man and his hat. The last man, at the rear, can see both other men and their hats.

None of the men can see the hat on his own head. They are asked to deduce its color. Some time goes by as the wise men ponder the puzzle in silence. Finally the first one, at the front of the line, makes an announcement: “My hat is white.”

He is correct. How did he come to this conclusion?

The answer depends, as is conveyed in this version of the story by calling them “wise men”, on the assumption that the men are all ideal subjects for the puzzle — they they all use perfect reasoning:

1. If the third man, who can see both other hats, say two black hats, he would know that his own hat was white. Otherwise, he could not be sure — he’d have a 2/3 chance of having a white hat if he saw one black and one white in front of him, and a 2/3 chance of having a black hat if he saw two whites.
2. Because the third man said nothing after being given a moment to reason it, the others can assume that they do not both have black hats.
3. Now if the second man sees that the first has a black hat, he knows that his own is white, and should say so, since his reasoning is also perfect.
4. Because the second man also said nothing, the first man knows that he has a white hat.

I don’t like these kinds of puzzles because they require assumptions. You have to assume that the men are all perfect, that the second man, for instance, would understand the information the third man’s silence conveys. You have to assume that they have the right sense of how long to wait before making an inference from the others’ silence. What if the first man announced that his hat was white just as it dawned on the second man that the first man’s black hat and the third’s silence told him that his was white?

People’s opinions of these sorts of puzzles differ. Some feel that they show an ability to “think outside the box”. Some, like me, just consider them trick questions. Sometimes I get them, and sometimes I don’t... but I always dislike them, either way.

Like this one, which I remember from an old episode of All in the Family:

Archie: Edith, did I tell you about the guy who bowled three hundred and one?

Edith: Whaaaat?

Archie: Yeah, I know a guy who bowled three hundred and one.

Edith: But Archie, the most you can get is three hundred.

Archie: Well, this fella bowled three hundred and one.

Edith: Oh, no, Archie, that can’t be.

Archie: You don’t think so. Edith?

Edith: No, I don’t. It’s impossible.

Archie: Well... did you ever see anyone who bowled three hundred... and lost?