It may be easy to argue by induction, but what is actually going on?
Let's take a group of three children for a larger example: Abe, Ben and Cindy. Let's denote the situation "Abe has a clear forehead and Ben and Cindy have muddy foreheads" as [OXX], and so on.
Before the teacher tells them anything, Abe knows that either [XXX] or [OXX] (because he sees the mud on Ben's and Cindy's foreheads). If [OXX] were the case, Ben would know that either [OXX] or [OOX]. If [OOX] were the case, Cindy would know that either [OOX] or [OOO].
This picture demonstrates the situation.
The teacher's announcement makes it common knowledge that [OOO] is not possible. Then if Abe supposed [OXX] and provided that Ben supposed [OOX], Cindy would be sure that [OOX] (meaning she has mud on her forehead). After the first call, she doesn't stand up so we know that if Abe supposed [OXX], in that idea Ben could no longer suppose [OOX] and would be instead sure that [OXX] (meaning he has mud on his forehead).
This example suggests that the children don't need the general notion of common knowledge to efficiently reason whether they have muddy foreheads. It suffices to use any chain "Abe considers possible that Ben considers possible that Cindy considers possible etc.". that visits every child exactly once in an arbitrary order.