A basilisk came out of one of the rooms last night. The prefects were standing at their posts and saw nothing, which means the snake never crossed a room they were watching. There are seven rooms and only six prefects, so one room was left without a post. Put every prefect back where they stood and the empty room gives itself away.
Rules
Place all six prefects. Exactly one arrangement fits, and the puzzle solves itself the moment you find it.
Every line in Filch's notes is true. The work is in what they rule out.