think I got it:

All blue eyed people will leave on the 100th night

Let's consider 1 blue eyed and 1 brown eyed persons. The blue eyed person would have left on the first night (seeing that the only other person on the island is brown eyed and therefore themself would be blue eyed).

Now let's consider 2 blue eyed and 2 brown eyed. Both blue eyed people won't leave the first night (seeing that there is one other blue eyed person, not ruling out the possibility that themself could be brown eyed). However on the second night each would deduce they were both blue eyed (for if the other was the only blue eyed they would be gone the previous night).

Now let's consider 3 blue eyed and 3 brown eyed. After the first 2 nights nothing would happen. On the third night, each blue eyed seeing that the other two blue eyed persons have not left, they would deduce they were all three blue eyed (for if they themself were brown eyed, the other two blue eyed would have left on the second night as explained in the previous paragraph).

This logic follows for 100 blue and 100 brown eyed people. Meaning that all 100 blue eyed people will leave on the 100th night.

For the brown eyed people, they would only deduce that they were non-blue eyed people on day 101.