Tuesday, February 12, 2013

Problem of Hats logic

Here it is a quiz of logic.
A hundred men, all of them are genius, are sitting around a huge table.
They wear a hat on their head. They can see hats anyone wears other than the hat of their own.
Now the facilitator announced, “Let’s play a game. I gave you hats whose color is either red or white. And at least one man wears a white hat. Who can know the color of your own hat? If so, raise your hand. The first man who can get the color is the winner. Let’s begin!”
Actually, the hats they wear are all white.
Then what will be happened?

It is a famous problem. To solve this some techniques similar to mathematical induction are needed. Additionally, you have to imagine the thought of other persons.

You got it, or confusing?
The answer is below;

First, if the first man (called A) saw 99 red hats, A would know that he wears a white hat, because the only man who wears a white hat must be himself.
But A did not raise his hand. So, at least one man in the last 99 men other than A wears a white hat. It is objectively true information.

Second, if the second man (called B) saw 98 red hats besides A’s hat, B would know that he wears a white hat, because at least one man in the 99 men other than A wears a white hat, and it must be B.
But B did not raise his hand. So, at least one man in the last 98 men other than A and B wears a white hat.

Same as before, C is not able to raise his hand. D, E, and F…

At last, the 100th man noticed that he wears a white hat.

He raises his hand.
In the assumption of this problem, the men are all genius. So everyone will notice the final fact and raise their hand at the same time.
This is the answer.

It is very interesting. One can know the fact according to the fact that other people cannot know. It’s a little tricky.
However, this is not realistic situation. In real world, we are not genius. We are not equal. And unfortunately we are not trustful each other. I am sorry I would not be able to see that everyone raises their hands synchronized.

1 comment:

  1. That is an interesting story. In real world, sadly this is true. Sometimes we know the truth from someone by his reaction, not what he said. Politicans behave much worse about it. This is the reason why I hate politicans.