In Hogwarts Legacy, the Faculty Tower is in the south wing of Hogwarts School of Witchcraft and Wizardry and can only be unlocked during The Caretaker’s Moonwail quest. Sneaking around the faculty tower at night, you might notice one of the school’s arithmancy gates, but there’s a prefect patrolling nearby, so it’s probably best to finish the quest and then come back during the day.
Related: How to solve the Great Hall door puzzle in Hogwarts Legacy
How to open the faculty tower puzzle door
Gamepur screenshot
From the Floo Flame faculty tower, head south, then turn right and go through the door next to which Mr. Moon, the caretaker, usually stands. Go up the first staircase and the puzzle door is on the other side of the first landing.
Gamepur screenshot
To unlock the Faculty Tower puzzle door, you need to find the “?” And “??” triangles and rotate them until they show the creature icon that represents the number that completes the equations written on the door. The equations are presented as triangles, so the numbers at each point of the triangle must add to the number in the middle of the triangle. Some of the dots have creature icons instead of numbers, and the creature icons are numbered, left to right, from 0 to 9. So the top triangle reads spider (8) + 5 + ? = 13. This means that the “?” should be 0, so it throws the “?” triangle (to the left of the door) until it shows half way.
Gamepur screenshot
The bottom triangle indicates 3 + dragon (2) + ?? = 14, so the “??” must be 9, which is represented by the snake creature icon. So, find the “??” triangle to the right of the door and rotate it until it shows the snakes icon. You can now open the door and loot the two chests in the attic beyond, one of which contains the natural terrain spell.
Source