How Does A Problem with Infinite Hotel Keys - Numberphile Work?

The Infinite Hotel has rooms that never run out, even when it’s full!

Imagine a hotel where every room is taken, and a new guest arrives. In a normal hotel, you’d have to turn someone away or make them wait. But in the Infinite Hotel, there's always space for one more person.

Here's how it works: When a new guest comes in, the person in Room 1 moves to Room 2, the person in Room 2 moves to Room 3, and so on, like everyone is shifting up by one room. Since there are infinitely many rooms, no one runs out of space! The new guest gets Room 1.

Now imagine two new guests arrive at once. Everyone shifts up by two rooms: the person in Room 1 goes to Room 3, the person in Room 2 goes to Room 4, and so on. Now there are two empty rooms, Room 1 and Room 2, for the two new guests.

It’s like having a never-ending line of boxes, you can always add more things, even if all the boxes are already full! The Infinite Hotel has rooms that never run out, even when it’s full!

Imagine a hotel where every room is taken, and a new guest arrives. In a normal hotel, you’d have to turn someone away or make them wait. But in the Infinite Hotel, there's always space for one more person.

Here's how it works: When a new guest comes in, the person in Room 1 moves to Room 2, the person in Room 2 moves to Room 3, and so on, like everyone is shifting up by one room. Since there are infinitely many rooms, no one runs out of space! The new guest gets Room 1.

Now imagine two new guests arrive at once. Everyone shifts up by two rooms: the person in Room 1 goes to Room 3, the person in Room 2 goes to Room 4, and so on. Now there are two empty rooms, Room 1 and Room 2, for the two new guests.

It’s like having a never-ending line of boxes, you can always add more things, even if all the boxes are already full!