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Maybe I'm not seeing the concept exactly, but math was never my thing. I know that infinity plus one is still infinity, so they are still a one-to-one ratio even if you add another person...
I guess the problem is that if the VIP customer were to say, try every door until he found the first unoccupied room, he'd be trying doors into infinity, never finding the vacant room. So you'd have to do something like tell all the other guest to move up one room, and the VIP could take the first room, since it's occupant just moved up one. Since there are infinite rooms, every guest would have a room to to move up to.
If that's not the answer, then I'll have to think about it a bit longer.
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Hot Pastrami!
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