Puzzletober 2025: Skeletal

The newcomer asks the innkeeper about the doors upstairs. He says, "The numbering system is straightforward enough. The two deluxe suites are opposite each other, and the clerk's room is opposite the maid's. The patrons tend to stay in the same rooms each year, leading every room to be mildly personalized. For example, all nine suites upstairs are painted a different color. Was there something else you wanted to know?"

"I assume you use warded locks since you don't have electrical systems here." "That is correct, yes. I am aware of the security risks. I am at least confident that neither the patrons nor the staff know how to pick locks. Customers are required to turn in their keys when checking out, so I'd know if anyone tried to replicate my skeleton key. I don't believe the clerk or maid have lost their keys either, so I know I have the only one." "But these keys are short. You ought to ensure that your lock system isn't as skeletal as your passkey."

Rules: Use the following information to determine the notch sequence for every room key. Every key has five relevant tooth slots which may either contain or not contain teeth.

• The key to room 9 has no teeth and the key to room 8 has three teeth; all the other keys have two teeth.
• Exactly three of the keys have at least two teeth in consecutive slots.
• If the keys are interpreted as binary such that the presence of a tooth is a 1 and the slot closest to the inserted end is the ones' place, the value of the key to room 1 is double that of room 5, which itself is double that of room 6. Furthermore, the key to room 8 encodes a prime number which is less than the value of the key to room 4.
• The key to room 2 can become the key to room 1 or to room 3 by moving the tooth that is not on an end of the warded section to an adjacent slot.
• The tooth arrangement for the key to room 7 is somewhere in the picture below. It does not open the door to room 8. A key can open another key's lock if it is missing some of the teeth.