Dr Bees likes bees. Dr Bees is not his real name, but that is what everybody calls him.
I should actually say that he is obsessed with bees. Bees are his whole life. He knows everything about them. Anything anyone has published about bees, he knows.
He never learned the alphabet past the second letter.
Anyway. He made a fortune locating the best honey you could find anywhere in the world and selling it to people who can afford to pay ridiculous prices for a minuscule jar of honey. He wants to invest his money.
His latest project is a hotel for bee lovers. He noted that there is no bee-themed hotel, which he sees as a huge and untapped market opportunity. I mean, who doesn't like bees? Who wouldn't like to spend a night surrounded by the soft humming of bees? Or better, who wouldn't like to live like a bee for a night? So he decided that he would build the first hotel for bee lovers.
He pays a visit to an architect friend and brings along the concept he drafted for his hotel. A large hotel with 127 rooms in the shape of a beehive.
The architect looks at the plan and tries not to laugh.
- Is it a hotel for ... bees?
- No, no! This is a hotel for people. But I got my inspiration from an actual beehive. The rooms are hexagonal to minimize the length of the walls. Inside I wanted to show people. But I never managed to draw people.
- Don't worry, I know you all too well. But I see a problem. Real people don't fly. How are they going to go in or out of the hotel rooms?
- I would say "with the bees, do like the bees". But I see your point. I didn't think of that. Can you fix that?
- I think so. First you need an entrance somewhere on the outer wall. Then you will have to replace some rooms with hallways. Every room must be adjacent to a hallway and the hallways must provide a path from the room to the hotel entrance.
- I see. But that is not good. People pay for the rooms, not for the hallways. These hallways are just maintenance and no revenue. How many hallways do I need? I mean, what is the absolute minimum number of rooms I need to convert to hallways to make my hotel usable for real people?
- Hmm. That is a good question. I will have to consult my wife. She is good at these things.
So, dear reader, how would you solve Dr. Bees' problem?
- You can see a preliminary plan of the hotel. Bees represent hotel rooms.
- You have to replace a number of rooms with hallways in order to provide a path from each room to a single main entrance. This means that the hallways must form a connected region and at least one of them must lie on the perimeter of the hexagon.
- Every remaining room must be adjacent to at least one hallway.
- The number of rooms converted to hallways must be minimal.
So, what is the minimum of hallways the hotel needs? Show how it can be done.
Some solutions assume the outer rooms don't need an access to a hallway since they can have a door to the outside. While this is arguably what Dr. Bees would prefer, it is not what the puzzle asks you to solve. It explicitly says that all rooms must be adjacent to a hallway and the hallways must form a connected region.
I will add to the specifications that the outside of the hotel must not be considered a giant hallway.
This is my own creation. I have a solution; I am curious to see if you can find a better one.