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.
Twist 2:
Suppose the logicians did get to know when someone was killed by his screaming. However, once the game is completed once, the interrogator goes back to the back of the line, and repeats the process. He keeps doing so, until everyone alive has answered correctly, on a certain repetition of the game.
