Riddle of the Week: The Puzzle of 100 Hats
Difficulty level: Hard
Problem
One
hundred prisoners are lined up single file. A blue or red hat is placed
on each of their heads randomly. The prisoners cannot see the color of
the hat on their own head, but they can see the colors of all the hats
in front of them. The prisoner in the back can clearly see all 99 hats
in front of him. The 50th prisoner in line can see the 49 hats in front
of him, and the prisoner in the front of the line cannot see anything
but the forest before him. Also, the prisoners don't know the proportion
of red hats to blue hats in advance—it could be 50/50, but it could
also be any combination that adds to 100.
A
guard is going to walk down the line, starting in the back, and ask
each prisoner what color hat they have on. They can only answer "blue"
or "red." If they answer incorrectly, or say anything else, they will be
shot dead on the spot. If they answer correctly, they will be set free.
Each prisoner can hear all of the other prisoners' responses, as well
as any gunshots that indicate an incorrect response. They can remember
all of this information.
Before
the executions begin, the prisoners get to huddle up and make a plan.
How can the prisoners ensure that the most people possible survive?
Hint
There is a way to ensure that the vast majority of the prisoners survive.
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