Macmillan Academy The Scholar Lab

Maths · KS3 · ~40 min

The Infinite Hotel

A hotel with infinitely many rooms is completely full. Can it still take more guests? Welcome to one of the strangest ideas in mathematics — Hilbert's Hotel.

The challenge

The Grand Hotel has rooms numbered 1, 2, 3, 4, … going on forever. Tonight every single room is occupied. There are no empty rooms at all.

Work out a method for each of these situations, and explain why your method always works:

  1. One new guest arrives and asks for a room. Can you fit them in?
  2. A coach arrives carrying infinitely many new guests. Can you fit them all in?
The catch: you can't just say "use the last room" — there is no last room. Your method has to be a rule that works for every guest at once.

How to get started

  1. Picture the rooms as a never-ending list. Ask not "where is there a gap?" but "can everyone move by a rule?"
  2. For one new guest, try asking every current guest to move along. Where could they all go?
  3. For infinitely many new guests, you need to free up infinitely many rooms at once — think about a rule based on each guest's room number.
  4. Write your rule as clearly as a set of instructions a receptionist could follow.

Stuck? Open a hint

Try the challenge first — then reveal these one at a time.

Hint 1
For one guest: ask the person in room n to move to room n+1. Everybody still has a room (there's always a next one), and room 1 is now free.
Hint 2
For infinitely many: ask the person in room n to move to room 2n. Now every even room is taken by an old guest — and every odd room is empty.
Hint 3
There are infinitely many odd numbers (1, 3, 5, 7, …), so the infinitely many new guests each get an odd-numbered room. Everyone fits!

Take it further

Now the real test: infinitely many coaches arrive, each carrying infinitely many passengers. Can you still fit everyone? (Hint: prime numbers, or a zig-zag through a grid, can help.)

This puzzle is about countable infinity. Mathematician Georg Cantor proved some infinities are actually bigger than others — the infinity of decimal numbers can't be fitted into the hotel at all. Look up "Cantor's diagonal argument" and see if you can follow it.

Explore more puzzles like this at NRICH.

Produce & share

Make something: Write a clear explanation (a page, a poster, or a short video) of how you fit first one and then infinitely many new guests into the full hotel — and have a go at the infinitely-many-coaches extension. The test of a good explanation is that a younger student could follow it.

When it's ready, email your work to Mr King. The best pieces may be featured in the student showcase.

What you'll practise

Reasoning about infinity, thinking of operations as rules (functions and mappings), and constructing an argument that covers every case — the heart of mathematical proof.