Pow Hats

Problem Statement

In a classroom there are 3 students named Arturo, Belicia, and Carletta who complain about all the POWS and homework they get. The teacher decides to make a deal with them. The teacher has 3 blue hats and 2 red hats he makes the students close their eyes and he puts a hat on each of their heads and then hides the remaining 2 hats. Then when he finishes putting the hats on each of their heads he allows Arturo to guess which hat he has on. Arturo can look at Belicia and Carletta to see which hat they have on, but he can't look at his own. If he guesses the hat correctly he doesn't have to do any POW's for the rest of the semester, if he guesses wrong he has to do the POW's and he has to grade his classmates. If he passes then his workload stays the same. Arturo decides to pass because he can't know for sure what hat he has on. Next Belicia opens her eyes, she looks at Arturo and Carletta and decides to pass as well because she can't know for sure which hat she has on. Then without even opening her eyes Carletta guesses which hat she has on her head correctly. Each of the students are incredibly smart and if they could figure it out they would've guessed it correctly. Our job is to find out which hat Carletta had on her head and how she knew.

Process:

First I drew out the problem and tried to find combinations when the students couldn't guess it correctly with the information given to them. Then I thought about how Carletta could've guessed without looking and what each of the other 2 students passing told her about which hat she had on her head. I soon figured out that Arturo and Belecia couldn't have possibly seen two red hats because if they did then they would've known that they had a blue hat on their head. I soon figured out that the only way that both of the students could've passed is if the combination was blue, red then blue. The reason for this is that if Arturo sees red and blue or blue and blue then he can't guess for sure and he has to pass. Belecia knows that those are the only two things that Arturo can see so when she looks up and sees that both him and Carletta are wearing a blue hat then she still has know way of knowing because the only way she could know what he saw is if Carletta was wearing a red hat. If Carletta had been wearing a red hat she would've seen this and ruled out that he had saw blue and red and so she could guess that she was wearing a blue hat. When she saw that Carletta was wearing a blue hat this means that her hat could either be blue as well or red because she still doesn't know what Arturo saw.

Soultion:

Carletta is wearing a blue hat and she knows this simply by listening to what their classmates answers were and knowing what their answers meant in terms of what they saw. Arturo looks up first and sees that Belecia is wearing a red hat and Carletta was wearing a blue hat. When he sees this he has no way of determining what color hat he has on because the only way he could do that would be if he saw two red hats. Seeing two red hats allows you