Suppose n students are in a classroom. Some of the students have dirt on their foreheads; all of the students can see any other students forehead, but cannot see, feel, or otherwise detect, whether their own foreheads themselves have dirt. The professor enters the room, announces that some of the students have dirt on their foreheads, and asks if anyone knows for sure that they have dirt on their forehead. All of the students say NO. The professor asks the same question repeatedly, but the students continue to say NO, until, suddenly, on the n-nth round of questioning, all of the students with dirt on their foreheads answer YES. How many students had dirt on their foreheads?

1. [5 pts] Solve the problem for n = 2, 3 (Must give proof).

2. [5 pts, GR] Solve the problem for any n 1 (Must give proof).