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.
3. The only allowed response is a short, unemotional "white" or "black".
4. Having said that, I will add that you can put any value you like into each of these words.

Ok the last person would say the colour which is equal to the majority of the hats he sees. Now if the 9th person sees a tie in the number of hats he will know what colour his hat is (since the previous person mentioned a majority) If there isn't a tie then that person will say the majority. of the hat colour he sees. At which point the next person would go. There is one exception. If a person can figure out what colour his hat is (for sure) then he should say that colour. This will cause every single person to say the same colour and save the number of people equal to the number of hats that are the majority. An even simpler way to express this startegy is that 10th person says the colour of hats which is the majority and every single person says that same colour. Because hats a re distributed randomly it can cause any hat from combination from 5-5 to 10-0 which means the survival rate equally ranges from 50% to 100% averaging out to 75%. I realy can't think up any other way.

Posted by Alan on 2003-03-21 11:35:47