Solving the Infinity Hotel Problem: 161 Guests

[Yankel]

Hello all,

You are probably familiar with the problem of the infinity hotel, a hotel in which there is an infinite number of rooms. Each room is filled with a guest, i.e. there are infinite number of people too. It is well known that two infinite sets have the same number of elements if there is a bijection from one to another. For example, if an infinite group of new people arrive at the hotel, then if the person in the n's room goes to the room in the 2n's room, then this function f(n)=2n is the bijection and they all fit in.

I am trying to find a bijection to solve the problem of a finite number of people arriving at the hotel. For example, 161 people arrive to the "full" hotel. How do they fit in ?

Thank you in advance.

 

[I like Serena]

Hello all,

You are probably familiar with the problem of the infinity hotel, a hotel in which there is an infinite number of rooms. Each room is filled with a guest, i.e. there are infinite number of people too. It is well known that two infinite sets have the same number of elements if there is a bijection from one to another. For example, if an infinite group of new people arrive at the hotel, then if the person in the n's room goes to the room in the 2n's room, then this function f(n)=2n is the bijection and they all fit in.

I am trying to find a bijection to solve the problem of a finite number of people arriving at the hotel. For example, 161 people arrive to the "full" hotel. How do they fit in ?

Thank you in advance.

Actually, f(n)=2n is not a bijection, since for instance 1 does not have an original.
But it does free up all the odd rooms, so that a new arrival with number i can occupy room 2i-1.

To fit in 161 people, we can move person n to n+161, which frees up the first 161 rooms for the new arrivals.