# Hilbert's Hotel

(draft)

Can you think of an intergalactic soccer match in which both teams have infinitely many players?

I know, this sounds weird, but hang on for a moment ...

## The players

Each player carries a number on his shirt. And for each number greater $0$ there is a unique player. So we have all players numbered as
$$1,2,3,4,\ldots$$

## The hotel

The players of team A arrive at the hotel and want to check in. Each room has a number and for each number greater than $0$ there is a unique room. Thus, the hotel has rooms numbered as
$$1,2,3,4,\ldots$$

To keep it easy, the hotel manager assigns to player $1$ room $1$, etc. Clearly, there is exactly one room for each player (and, conversely, for each player exactly one room).

At this stage the hotel is full.

## Making space in a full hotel for a new guest

Now the manager of the soccer team arrives. As we have noticed, the hotel is full. How can the hotel manager make space for the new arrival?

The hotel manager will tell each of the players in which room to move. Which is the new room that the hotel manager assigns to player $n$?

(There may be more than one answer.)

## Making space in a full hotel for infinitely many new guests

The next day, the opposition team arrives. Now the hotel manager needs to find infinitely many empty rooms in a full hotel. What does he do?

## Postscriptum

One may think of taking this even further. As you can imagine, an interestellar soccer tournament will also have infinitely many teams. Can the hotel accommodate infinitely many teams (each with infinitely many players)?