Common knowledge means I know, and I know that you know, and I know that you know that he knows, and she knows that I know that you know that he knows, and so on—any number of iterations.
Each child sees 99 muddy foreheads and therefore knows n >= 99. Each child can tell that each other child knows n >= 98. But, e.g., it isn’t true that A knows B knows C knows that n >= 98; only that A knows B knows C knows that n>=97: each link in the chain reduces the number by 1. So for no k>0 is it common knowledge that n>=k.
Common knowledge means I know, and I know that you know, and I know that you know that he knows, and she knows that I know that you know that he knows, and so on—any number of iterations.
Each child sees 99 muddy foreheads and therefore knows n >= 99. Each child can tell that each other child knows n >= 98. But, e.g., it isn’t true that A knows B knows C knows that n >= 98; only that A knows B knows C knows that n>=97: each link in the chain reduces the number by 1. So for no k>0 is it common knowledge that n>=k.
Thanks, I did end up figuring out my error.