"We must've been through hundreds of books already and we can't find him anywhere —"
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The Head of Gryffindor House is a Professor at Hogwarts School of Witchcraft and Wizardry with the specific role of looking after the students in their House. It is unknown how a professor would be elected for the post, but it's most likely that the headmaster chooses the professor whom embodies all the qualities Godric Gryffindor himself had (bravery, daring, conscientiousness, etc.) and was probably in Gryffindor House when he/she was at Hogwarts.