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Old 2005-10-15, 12:53   #1
fetofs
 
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Aug 2005
Brazil

2·181 Posts
Default Classic: Hats puzzle

Many people already know this puzzle, but others might find it interesting, at least.

There is an island with 10 inhabitants. One day a monster comes and says that he intends to eat every one of them but will give them a chance to survive in the following way:
In the morning, the monster will line up all the people - single file so that the last person sees the remaining 9, the next person sees the remaining 8, and so on until the first person that obviously sees no one in front of himself. The monster will then place black or white hats on their heads randomly (they can be all white, all black or any combination thereof).
The monster will offer each person starting with the last one (who sees everyone else's hats) to guess the color of his/her own hat. The answer can only be one word: "white" or "black". The monster will eat him on the spot if he guessed wrong, and will leave him alive if he guessed right. All the remaining people will hear both the guess and the outcome of the guess. The monster will then go on to the next to last person (who only sees 8 people), and so on until the end.
The monster gives them the whole night to think.

The Task:
Devise the optimal strategy that these poor natives could use to maximize their survival rate.

Assumptions:
1) All the 10 people can easily understand your strategy, and will execute it with perfect precision.
2) If the monster suspects that any of the people are giving away information to any of the remaining team members by intonation of words when answering, or any other signs, or by touch, he will eat everyone.

Most people know this puzzle by the "prisioner" version:

10 straight-jacketed prisoners are on death row. Tomorrow they will be arranged in single file, all facing one direction. The guy in the front of the line (he can't see anything in front of him) will be called the 1st guy, and the guy in the back of the line (he can see the heads of the other nine people) will be called the 10th guy. An executioner will then put a hat on everyone's head; the hat will either be black or white, totally random. Prisoners cannot see the color of their own hat. The executioner then goes to the 10th guy and asks him what color hat he is wearing; the prisoner can respond with either "black" or "white". If what he says matches the color of the hat he's wearing, he will live. Else, he dies. The executioner then proceeds to the 9th guy, and asks the same question, then asks the 8th guy ... this continues until all of the prisoners have been queried.

This is the night before the execution. The prisoners are allowed to get together to discuss a plan for maximizing the number of lives saved tomorrow. What is the optimal plan?

P.S: After you try to solve it by the "black and white" case, assume the 7 rainbow colors are used. After that, generalize it to N people and K colors of hats. Good luck
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