The Infinite Hotel Paradox: How Can More Things Fit?

Imagine a hotel with infinite rooms. Every room is full. A new guest arrives and asks for a place to sleep. The manager doesn't need to build a new room! Instead, the person in room 1 moves to room 2, the person in room 2 moves to room 3, and so on. This creates an empty spot in room 1. Even though there were infinite people before, they all just shifted one step down the hall.

What if infinitely many buses arrive? Each bus has infinite passengers. You might think we run out of space, but we don't! We can move everyone to double-numbered rooms (2, 4, 6...) leaving all odd rooms (1, 3, 5...) empty. Then, we put the first passenger from the first bus in room 1, the second in room 3, and so on. It seems impossible because our brains think infinity is just a really big number that ends eventually. But infinity isn't an end; it's a never-ending list.

Infinity plus one is still infinity!

This puzzle was created by mathematician David Hilbert to show how infinite sets behave differently from finite ones.