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Hotel Infinity: Part Three Recap: You are the proprietor of a hotel with
an infinite number of rooms. One nice thing about this is that, even when all the rooms
are full, you can still make room for new guests by rearranging your hotel's current
occupants. You can make room for four new guests, or twelve, or a thousand -- any number,
as long as it's finite. But can you make room for an infinite number of new guests?
This question arises when an infinite football team arrives, wanting to celebrate a major
victory with a night in your hotel. How can you possibly make room for them?
Well, this is how. Get on the intercom and announce to every room at once: "Announcement!
The winners of the Infinite Superbowl are here! We need to make room for them. Please
take your current room number, and double it. That's the number for your new room, so
please move there at once. (Thank you for your time.)"
Your guests do what you asked. The person in Room One moves to Room Two, the person
in Room Two moves to Room Four, the person in Room Three moves to room six. The person
in Room One Million moves to Room Two Million, the person in Room A Googol moves to Room
Two Googol -- fortunately for them they can take the teleporters and be there in an instant,
because it would be a very, very long walk otherwise.
When they're done, every guest still has a room. But now all the guests are staying in
rooms with even numbers, and so all the odd-numbered rooms are empty.
Now you just need to get the football players into the odd-numbered rooms. So you have the
coach tell them this: "Everyone, take the number on your jersey. Double it, and then
subtract one. That number is your room number, so go there. Enjoy your stay!" So Player 1
teleports to Room 1, Player 2 teleports to Room 3, Player 3 teleports to Room 5, and
so on. They all go simultaneously, and when they're done, every single person has a room
to themselves -- original guests and football players alike.
The team enjoys their stay, they tell their friends about it, and soon your hotel becomes
very popular among infinite sports teams. Some nights two inifinite teams even show
up at once. But it's not a problem -- in that situation, you just ask your current guests
to multiply their room numbers by three, instead of two. Then rooms 3, 6, 9, 12, and so on
are full -- every room that's a multiple of three, in other words. Tell the members of
one team to take their jersey numbers, multiply by three, and subtract one, and they'll end
up in rooms 2, 5, 8, 11, and so on. Then tell the members of the other team to multiply
their numbers by three and subtract two, and they'll end up neatly filling the remaining
rooms. Very convenient. It runs smoothly for quite some time. Until
one night, that first victorious football team shows up again. You're happy to have
them, but the coach pulls you aside. "I'd heard you could accommodate multiple teams
at once," she says. "I've been doing some recruitment, so we've got a few other teams
along with us. I hope you have room for them, too?"
"No problem!" you tell her. "We can handle any number of infinite teams. Ten of them?
Fine. A hundred? Bring them on over. We never turn anyone away, at Hotel Infinity!"
She shifts uncomfortably. "Well... when I said we had a few other teams, that might
have been a bit of an understatement..." Suddenly another bus pulls up in the parking
lot. And another, and another. You look past them, only to see bus after bus pulling up
to your hotel, a field of infinitely long buses, stretching out to span the horizon.
"My recruitment may have been a bit... overzealous," says the coach. "I've got an infinite number
of teams, and each team has an infinite number of players."
"They're very well organized, though!" she assures you. "The teams are all named after
numbers -- there's Team One, Team Two, Team Three, and so on. For every positive integer
there's a team with that number, and I've had them pull up in order in your parking
lot." "And on every team" she says, "There's a Player
One, a Player Two, and so on. We're all fully equipped with instantaneous loudspeakers and
teleporters. Just tell me where they need to be, and we'll make it happen."
"So," she asks, hesitantly, "Do you have room for them?"
Do you? You're not sure. You only have one infinite hotel -- is it possible to make room
for an infinite number of infinitely large teams? Or will you have to turn a customer
away, for the first time? Next time we'll answer this question, but
think about it on your own, in the meantime. If you think it can be done, then how? If
you think it's impossible, then how can you be sure of that? Try to find an answer, one
way or the other. See you next time, in Hotel Infinity.