As we walk down the infinite hotel corridor, only very few of the rooms will be used. In addition, all rooms which have prime factors other than 2 and 3, for example rooms 5, 7, 10, and so on, remain empty. But 2 i 3 j is the prime factorisation of the room numbers we assign to the customers, so different passengers will always get a different room. This is because 2 and 3 are prime numbers and because prime factorisation is unique – meaning that if two numbers are different, they have a different prime factorisation. Of course we have to check that we don't happen to give two passengers the same room. Here i and j can be any numbers, for example the passenger on seat 5 in bus 2 gets room 2 5 3 2 = 32 × 9 = 288. It turns out that we can, using a simple rule: the passenger sitting on seat number i in bus number j gets room 2 i 3 j in the hotel. Can we also move all those infinitely times infinitely many passengers into one hotel? The hotel is completely full, yet everybody gets a room.įinally let us suppose that a whole bus company arrives at the hotel: they have infinitely many buses, each with infinitely many seats.
Infinity plus infinity is still infinity.Īn infinite bus arrives at the Hilbert. Since there are infinitely many odd numbers, there is enough space for the infinitely many new guests.
Notice that now all the odd numbered rooms are empty. The guest in room n will move into room 2 n. Thus the guest in room 1 will move into room 2, the guest in room 2 will move into room 4, and so on. Instead the hotel does something very clever: they ask their guests to move into the room with twice the number of their current room. Now the previous method won’t work anymore: you can’t ask the guests to move up infinitely many rooms – they would never arrive in their new room.
Let us suppose that the hotel is full again, and that infinitely many new guests arrive. This wouldn't have happened on an infinite ship!Įxtract from 'Murder Ahoy!', based on the Miss Marple stories by Agatha Christie On this ship, there are only finitely many rooms – therefore, if somebody new arrives, one officer is left without room.