Arrangements in Hilbert's Hotel

Calculus Level 2

In Hilbert's Hotel, there are infinitely many rooms labeled $1, 2, 3, \ldots,$ with one room for every natural number.

An infinite number of guests stay in the hotel: $\text{Mr. }P_1,\, \text{Mr. }P_2,\, \text{Mr. }P_3, \ldots,$ with $\text{Mr. }P_n$ for every natural number $n$ .

Hilbert, the manager, realizes that every room is occupied by at least one guest.

Does that mean no two guests are in the same room?

Yes, always No, not necessarily

3 solutions

Joseph Newton
Dec 30, 2017

Consider the scenario where all rooms have exactly one person in them, i.e. $\text{Mr. }P_1$ is in room $1$ , $\text{Mr. }P_2$ is in room $2$ , etc.

Now imagine that everyone except $\text{Mr. }P_1$ moved from their room to the previous room, i.e. $\text{Mr. }P_2$ moves to room $1$ , $\text{Mr. }P_3$ moves to room $2$ , etc.

The Hotel is infinite, and every room has another room after it that a person can come from. This leaves no room empty at the end, simply because there is no end, and so all the rooms in the Hotel still have at least one person in them.

However, after the move there are now $2$ people in room $1$ : $\text{Mr. }P_1$ and $\text{Mr. }P_2$ . This means there is a possible arrangement where two guests are in the same room without violating the rule that all rooms must be filled, and so the answer is "No, not necessarily."

But there was nothing in the question about the guests moving from their rooms, also when they move to their previous room , mr infinity ' s room is empty cuz he's at room no. Infinity-1 so if they move they are violating all the rooms must be filled rule

Joyjyoti Jana - 3 years, 5 months ago

The guests do not necessarily need to move rooms. I am simply making them move rooms to give an example of a situation where two are in the same room.

Remember that Mr. Infinity doesn't exist! The guests are labelled by the natural numbers (1, 2, 3, etc.) and the natural numbers definitely do not contain infinity, despite the fact that there are an infinite number of them. Every single guest in the hotel has another guest after them, so if Room Infinity existed, then surely Mr. Infinity+1 could move into Room Infinity, Mr. Infinity+2 could move into Room Infinity+1, and on and on again.

This is one of the key properties of infinity: there is no last room, and so the issue of the last room being empty is not a problem.

Joseph Newton - 3 years, 5 months ago

Karthikeyan Parasuraman
Feb 4, 2018

Pigeon hole principle

Raven Herd
Jan 17, 2018

Pigeon hole principle is the answer!

@Raven Herd Can you explain HOW!!??

Aaghaz Mahajan - 3 years, 4 months ago

Arrangements in Hilbert's Hotel

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