POW 11 A Hat of a Different Color, what is the process? -steps in finding out the problem-?

Question

A teacher made a deal with 3 of his students. He said that if you can guess what color hat you have on your head without looking, I will pass you. There were 2 red hats and 3 blue hats. The deal worked like this: The three students would close there eyes, and the teacher would put a hat on each of their heads and then hide the other 2. Then one at a time, the students would open their eyes and try to figure out what color hat was on their head. The student could guess or pass. This is what happened when they put their deal to the test: A boy named Arturo was first and opened his eyes but wasn't sure so he passed so he didn't get it wrong. Belicia was next and she passed too because she wasn't sure. Also she thought about the fact that Arturo didn't know. Carletta was last and without even opening her eyes, she knew for sure what color hat she was wearing and her answer was right. So i have to figure out what color hat she's wearing for sure. and how she knows. the answer is blue

Answer 1

Since there are only 2 red hats, if Arturo had opened his eyes and seen 2 red hats he would have known immediately that his was blue and won. Since he wasn't sure, he must have seen either 2 blue hats or 1 blue/1 red. (Arturo's could therefore be either red or blue.)

Belicia knows that Arturo didn't see 2 red hats, so she knows that between herself and Carletta at least 1 must have a blue hat. Thus, if she had opened her eyes and seen a red hat on Carletta, she would have known that her own hat was blue and would have won. Since Belicia didn't know what color she had after looking at Carletta, Carletta must have been wearing a blue hat. (Belicia's could therefore be either red or blue.)

Carletta followed the above logic and figured out her hat was blue without even having to look. Smart cookie!