Cat's LogicLogic Level 3
Albert the cat is living in one of the \(3\times 3\times 3\) cubic cat condominium rooms, where every room has conjoined cat doors on all sides including the top and bottom (but no diagonal connection), and the cat can pass through any door in any direction if he so chooses.
All condominium rooms are labeled with 3-digit numbers from 1-3. The first digit is the floor number while the other 2-digit combination can be laid out as shown in the table above. For example, in this 3D model, the room numbers are 111 for the blue, 211 for the pink, and 311 for the yellow, etc.
You know that Albert is living somewhere on the third floor, and after interviewing his neighbor cats, whose rooms he needs to pass in respective order, this is the information you've got:
Cat #1: I'm living in one of the corner rooms on the first floor. Albert always comes in my room first.
Cat #2: My room number is even.
Cat #3: The sum of my room number digits is \(5\).
Cat #4: The sum of the 3 digits can divide the room number itself.
Cat #5: My room number is prime with no repeated \(1\) digits. Albert's room is just next door.
Assuming Albert never retraces his steps en route to his room, what is his room number?