Puzzles/Logic puzzles/3 Hats in a Circle/Solution
Person A (who figured it out) looked at Person B and Person C and thought to himself, "Both these men are wearing red hats, and each of them sees the other man and I both have our hands up. If I were wearing a green hat, then person B would look at person C and see that Person C's hand was up, but not for me -- since my hat would be green. Therefore, he would immediately figure out his own hat was red and leave the room. But ten minutes have gone by and neither Person B nor Person C have figured out what would be an obvious conclusion; therefore, they're stuck because my hat is red, not green."
To solve this problem, we must answer the following question: what happens when the number of red hats is between 0 and 3?
Case 1. If there are no red hats, then no man raises his hand, each man sees that the others do not raise their hands and thus concludes that his hat must be green, so he immediately leaves the room. This did not happen, so there is at least one red hat.
Case 2. If there is exactly one red hat, then the man wearing the red hat does not raise his hand but the other two men do. The man wearing the red hat sees no red hats but does see the raised hands, so he concludes that his hat must be red and he leaves the room.
Case 3. If there are exactly two red hats, then all hands are raised. Each red hat-wearing man sees a red and a green hat and deduces that his hat must be red because the other red hat-wearing man's hand is raised. Thus the two red hat-wearing men leave the room.
Case 4. If there are three red hats, then all hands are raised, so cases 1 and 2 are eliminated. Each man sees that the other two men do not leave the room simultaneously. Since this must be the logical conclusion of case 3, there must be at least three red hats. Thus all men leave the room knowing their hat is red.
In order to arrive at the conclusion that only one man leaves the room, we conclude that the men reason logically at different rates and that the man to leave the room first has reasoned faster than the others.