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Today we revist one of the riddles in the last post and give its solution. (The post can be found here: 5 Math Puzzles)

We are going to be going through the solution of Problem 1: The Hat Problem. If we recall the problem: There are 100 prisoners, and they are going to be executed. However the King is feeling particularly kind today, and decides to give the prisoners a chance to save themselves. He puts a red or green hat on top of each prisoners head (the color is chosen completely randomly) and lines them up from shortest to tallest so that every prisoner can see the hat color of all the prisoners in front of them. Now going from the tallest to shortest person in order, each prisoner has to guess the color of their hat. (They can hear the responses of all the people who guess before them). If more than 50 prisoners guess the color of their hat correctly, they will all go free. If not, then they all die. The prisoners are allowed to meet before the guessing process. Now the question is: Is there any strategy they can come up with that will guarentee the freedom of all the prisoners every time? What is the maximum number of indiviudals that they can ensure will guess their hat color correctly?

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