Tend to think Marcuss is right.
It's easy to extend n=2 to n=3. If there are 3 people with blue eyes, each one of them expects the other 2 to commit suicide on 2nd day if there are exactly 2 blue-eyed men (as Marcuss explained). When no one commits suicide on 2nd day, each one of them knows there are more than 2 such men and thus each one commit suicide.
In case of n=4, each of them expects other 3 to commit suicide on 3rd day and so on...
Perfectly logical but I am intrigued by what did the outsider really do to trigger this chain of thought.