Lesson 1 · Infinity

Hilbert's Hotel

A hotel with rooms that go on forever. Every room is full. But somehow… there's always space.

Scene 1

The hotel is full. A new guest arrives. 🧳

Welcome to the Infinity Hotel! It has rooms numbered 1, 2, 3, 4, 5… going on forever. Every single room has a guest. "Sorry, we're full!" says the manager. But wait — a new guest just walked in. Can we still fit them?

Try it: tap the pink button. Watch what happens to room 1.

Scene 2

An infinite bus arrives. 🚌♾️

Now imagine a giant bus pulls up — with infinite new passengers, all wanting rooms. Moving everyone up by 1 only frees one room. We need to free infinite rooms. What if everyone moves to twice their room number?

Try it: watch how every odd-numbered room becomes free. That's infinite free rooms!

The Mind-Blow

Infinity is weird 🤯

∞ + 1 = ∞

∞ + ∞ = ∞

If you add 1 to infinity, you still get infinity. If you add infinity to infinity, you still get infinity. That's why a full infinity hotel can always make room.

This puzzle is real math, invented by a mathematician named David Hilbert over 100 years ago. Grown-ups still find it amazing.