100 prisoners are lined up and assigned a random hat, either red or blue. There can be any number of red hats. Each prisoner can see the hats in front of him but not behind. They also cannot see their own hat. Starting with the prisoner at the back of the line and moving forward, each prisoner must say only one word, either "red" or "blue". If the word matches their own hat colour, they are released. If it doesn't match their hat colour, they are killed on the spot. They can hear each others answers, no matter how far they are on the line. A friendly guard warns them of this test one hour beforehand and tells them that they can formulate a plan. The prisoners have to come up with a plan to save as many lives as possible, with certainty. What is the plan to achieve the goal?

Number the prisoners from 1 to 100, with 100 being the last person in line, being able to see 99 other hats. Prisoners agree that if the number of red hats that the 100th prisoner can see is even, then he will say “red”. If they add up to an odd number, he will say “blue”. This way, number 99 can look ahead and count the red hats. If they add up to an even number and number 100 said “red”, then 99 must be wearing a blue hat. if they add up to an even number and number 100 said “blue”, signalling an odd number of red hats, number 99 must also be wearing a red hat. Prisoner 98 knows that prisoner 99 said the correct hat color and uses that information along with the 97 hats in front in order to figure out what color hat is on his head. This way, the chance that the 100th prisoner survives is 50%, but for all other 99 prisoners it's 100%.

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Prisoner Problem #1

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