Hilbert's Hotel's plates, apeirotypography, and diminishing returns

Question

Hilbert's Hotel's Plates, apeirotypography, and diminishing returns

1. What?

So I was browsing for videos about some mathematics (as one does), and I stumbled across a comment wondering how the Hilbert's Hotel's staff fit the room number plates on the door. Of course, this is silly, as Hilbert's Hotel is a thought experiment, not an actual hotel, but that got me thinking.

2. Nameplate widths

Hilbert, the manager of Hilbert's Hotel has a problem. He was wasting quite a lot of money on room sizes. Of course, one might wonder how, with the \$$\aleph_0$ his hotel makes every moment, could he ever run out of money, but remember, rent is already 30% (\$$\aleph_0$) of the hotel's expenses, not to mention wage, tax, water, electricity, etc. So Hilbert sent out an investigation (mathmatician) team to investigate...

Givens:

1. All numbers are in 72pt Arial mono
2. The rooms are as wide as the door, which is $(max(36, w_{plate} ))$ inches wide

Q: What is the range of room which is n inches wide (n>36)? Please write a general formula.

3. The attack of the PR department

Having found the issue, Hilbert decides to cut down his plates' width, and thereby that of the rooms. However, just as he was about to make the call, the head of public relations, we'll call him... uhh... Zermelo. He explained that shrinking the room will infuriate $\aleph_0$ people, and also that they are complaining about long travel time between rooms. Being an intelligent man, Hilbert rethinks his decision, and sends another team to investigate again.

Givens:

1. Every given of *Nameplate widths
2. With every mile customers walk, they get p% unhappier
3. A customer starts out 100% happy
4. If a customer is 0% happy, they leave through the emergency exits, which is terrible for the reputation of the hotel
5. You may replace a room with a snack room, which restores the customer's happiness to 100%
6. A snack room is single use
7. A snack room also has a number plate
8. Customers cannot advance any further while using a snack room

Q:

a) How many miles can a customer walk without leaving?

b) When will the width of the rooms be so big that it is impossible to fit another normal room without customers leaving?

All answers in general form.

3. Advertising, Legacy, and a lack of Game Theory

After further consideration, Hilbert, Zermelo, and the rest of the team decided there was simply no way that they can change their hotel such that they are within budget and satisfies everyone. So, they've resorted to advertising. "Hilbert's Hotel: Any width you'd like". It was a terrible slogan, but it somehow worked.

Years later, Hilbert has died. The hotel is now ran by Richard K. Guy. It had stayed unchanged over the years.

John Conway, having heard of this hotel, decides to book a few nights, but, being a game theorist, he doesn't just want a room. He wants the best room. For that, he'd need some normal(-ish) algebra.

Givens:

1. All givens until given 3 of The attack of the PR department
2. A customer may enter a room at any point of its door's span (e. g. if the door is 32 inches, the customer may enter from any points between the start and end of the door)
3. Any rooms, when entered, gives the customer a w% happiness boost, but the customer cannot change rooms.

Q: What is the maximum possible happiness? General form please.

Epilogue

Well, that was a lot more than I expected. These questions are honestly more for fun than for answers. But I do wonder what the answers are.