Solving variants of the blue-eyes puzzle
Python 2, 60 48 bytes



f=lambda b,r,c:all(map(eval,c))and-~f(b-1,r+1,c)


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How it works

By induction:


If the island composition is (b, r) and it is known that (b − 1, r + 1) is not possible, then on day 1, the blue-eyed islanders, seeing (b − 1, r), will conclude that they are blue-eyed and leave.
If the island composition is (b, r) and it is known that the blue-eyed islanders would have left on day n if the composition were (b − 1, r + 1), then on day n + 1, the blue-eyed islanders, seeing (b − 1, r), will conclude that they are blue-eyed and leave.


This second inference fails if b = 1, but in that case, the blue-eyed islander would never leave if they hadn’t already, and the test case would be invalid.

Note that non-blue-eyed islanders will never leave, in this version of the puzzle, because even if they use similar logic to conclude that they are not blue-eyed, they still don’t know which non-blue color their eyes are.