Common Knowledge and the Red Hats Puzzle Here is a puzzle that is supposed to help illuminate common knowledge in game theory. Three girls are sitting in a circle, each wearing a red or white hat. Each can see the color of all hats except their own. Now suppose they are all wearing red hats. 

It is said that if the teacher announces that at least one of the hats is red, and then sequentially asks each girl if she knows the color of her hat, the third girl questioned will know her hat is red. I understand the reasoning there. The first must have seen at least one red hat on the other two to say I don't know. And the second girl must have seen a red hat on the third, or else she would deduce that the first girl saw a red hat on her. 

What I don't understand is the necessity of the teacher. Everyone knows there is at least one red hat. And, if we start with common knowlege, they should figure out that everyone else knows that. So is the teacher only introduced if common knowledge is not an assumption?

Source: http://cowles.econ.yale.edu/~gean/art/p0882.pdf
Without the teacher, everyone knows that there is at least a red hat, but nobody knows that everyone knows - the fact is not common knowledge. 

With the introduction of the teacher, 


Girl 1 doesn't answer. Due to common knowledge, 2 and 3 can reason: "1 knows there is at least one red hat, and since she doesn't know her hat color, 2 and/or 3 must have a red hat.


Without the introduction of the teacher,


Girl 1 doesn't answer. Without common knowledge, there is nothing 2 and 3 can reason on top of their prior knowledge: 2 will keep knowing that 3 has a red hat, and 3 will keep knowing that 2 has a red hat. Nothing more.


In other words: Without the teacher, the set of knowledge is:


1: 2+3 have red hats
2: 1+3 have red hats
3: 1+2 have red hats


The teacher works as an injector of additional knowledge:


1: 2+3 both know that there is at least one red hat
2: 1+3 both know that there is at least one red hat
3: 1+2 both know that there is at least one red hat


And, common knowledge means that in the next level, everyone knows that everyone knows


1: 2+3 both know that I know that there is at least one red hat


etc, ad infinitum. This additional information is required to solve the puzzle.