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To solve this one, I had to read carefully the text. Let’s find who is the lady.
She is not the one with the black hair, because this one answers her. Her hair is not brown, so she can not be Professor White because White’s hair is not white.
She is not Professor Black, because he bit into his sandwitch.
Therefore, she is Professor Brown and her hair is white.
Professor’s Black hair is brown (neither white nor black) and Professor’s White’s hair is black.