To get to the arithmancy door, fast travel to the Divination Classroom. Once you do that, walk towards the wooden platform and make a right on the first turn. There'll; you'll find the door. To solve ...

To solve the arithmancy door near the door in the hidden room, make sure you solve the puzzle mentioned above. After that, switch the ? block to 4 (the owl) and the ?? block to 5 (the Quintaped, ...