Math Investigation eTwinning. Mission: Infinity...Affairs! Volume: Infinity | Page 10

Finally let us suppose that a whole bus company arrives at the hotel: they have infinitely many
buses, each with infinitely many seats. Can we also move all those infinitely times infinitely many
passengers into one hotel?
It turns out that we can, using a simple rule: the passenger sitting on seat number i in bus number j
gets room 2i 3j in the hotel. Here i and j can be any numbers, for example the passenger on seat 5 in
bus 2 gets room 25 32 = 32 × 9 = 288.
Of course we have to check that we don't happen to give two passengers the same 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. But 2i 3j is the prime factorisation
of the room numbers we assign to the customers, so different passengers will always get a different
room.
In addition, all rooms which have prime factors other than 2 and 3, for example rooms 5, 7, 10, and
so on, remain empty. As we walk down the infinite hotel corridor, only very few of the rooms will
be used. This is useful in case an infinite bus arrives in the middle of the night and you don’t want
to wake everybody up.

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LYKEIO SCHIMATARIOY, GREECE

- VALLAURI FOSSANO, ITALY

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