I think it's much better because the teacher doesn't have to match guesses to students. For example, for each student the odds are 30/100, roughly one in three. And any duplicates can be matched by a single guess.
E.g. teacher picking 1-30 and then each student has 0.3 odds of picking 1-30 or 31-100.
The issue is I think all it would take to beat the teacher is one unusual student.
(Incidentally, even with N=99 the probability is 0.37 ≈ 1/e, and the probability is lowest at N=37 ≈ 100/e. This is not a coincidence.)
Then it would be 1 to the power of 100.
I guess I should've clarified that the 0.3 refers to being able to choose 30 out of 100 numbers?
The teacher picks 30 numbers out of 100. Then each student (independently) picks one number. If random, that is 0.3 ^ 30. Obviously, the students are not picking random.
If I had to pick for the teacher:
multiples of 10: 10,20,30,40,50,60,70,80,90
double digits: 11,22,33,44,55,66,77,88,99
Not sure where to go next.
i would pick a 3-7, 30-37, 70-77, then some other fews from there like 1, 100, 50 etc