Pre-game: There are 1000 logicians in a line, each wearing a black or a red hat, completely at random. No one knows the color of their own hat. Each can see the hats on the next 10 people. The logicians are allowed to communicate before the game, but not once the hats have been placed.
Game: Starting with the back, each logician must call out loudly, a color, red, black, or white. White is obviously always wrong. This calling is, however, the only communication in the game. Every person who fails to say his own hat color is silently killed.
What is the best strategy for the logicians?
Twist 1:
Suppose there is exactly one blind man in the line. His location is not known to anyone, but he can hear and speak.
Answer
The person in the back looks at the next $10$ people and adds $1$ for each black hat and $0$ for every red hat. If the sum is odd then he says "black". Otherwise he says "red". This gives him a 50% chance of survival.
The next 10 people can sum the 9 people they can see or have heard the color of from their block of 10 people. If their sum is the same parity (even or odd) as the one the back person had, they yell "red". Otherwise they yell "black".
The result of the limitation that each peson can only see 10 people ahead of them is that no information from the back person will continue to person 12. This means he will behave as a new back person. This means that the people in positions $1,12,23,etc$ labeled from the back are forced to act as "back" people and therefore have a 50% chance of survival. On average 45.5 people will die each game.
Twist 1:
The effect of a blind person will depend on his position $\mod 11$. If he is in positions $1,12,23, etc$, ($1 \mod 11$) guessing means there is a 50% chance he will die and there is an independent 50% chance everyone in the next 10 will die. He should say "white" which will kill him, result in the person in front of him having a 50% survival rate and the next 10 people after the person in front of him a 100% survival chance. If this occurs, on average 46.5 will die.
If he is in position 1000, he should just guess; everyone else has already gone.
If he is in position $11,22,etc$ ($0 \mod 11$) he has a 100% chance of survival because he could hear what everyone said.
If he is in position $10,21,32,43,54,65,76,87,etc$, ($10 \mod 11$) when he guesses if he is correct, both him and the person in front of him survive. If he is wrong they both die. If he says white, he dies and the person in front of him acts as a back person. If he guesses, on average 46.5 people will die. If he does not guess, 47 people will die on average. This means he should guess.
If he is in position $9,20.31,42,53,64,75,86,etc$, ($9 \mod 11$) when he guesses if he is correct, both him and the 2 people in front of him survive. If he is wrong they all die. If he says white, he dies and the person in front of him acts as a back person. If he guesses, on average 47 people will die. If he does not guess, 47 people will die on average. It doesn't matter if he guesses.
If he is in any other position, he should say "white". This means the person in front of him will act as a back person and, on average, 47 people die.
This means if the blind man is moral and randomly placed on average 46.8 people will die.
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