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Hotel Infinity: Part Two Recap: You are the proprietor of a hotel with
an infinite number of rooms. One night all the rooms are full, but then the President
shows up -- The President of Space. She wants a room, too. What do you do?
Well, here's what you do: Get on the intercom, and announce to every room simultaneously:
"Announcement! The President of Space is here! We need to make room for her. Move to the
next room down the hall, immediately!" The guests all leave their old rooms, take a few
steps down the hallway, and enter their new rooms, in unison. When they're done, every
guest still has a room, but Room 1 is now empty, and ready for the President. Problem
solved. The President's stay gets your hotel some
good publicity, and you start filling your rooms every night. But you realize that, with
this new tactic, you never need to turn people away just because the rooms are all taken.
If four people show up each wanting a room, you can ask all your current guests to move
down four rooms, leaving rooms One, Two, Three, and Four available. If twelve people show
up one night, just ask your guests to add twelve to their current room number, and then
move to that room. Even if a hundred people show up at once, you can ask all your guests
to teleport a hundred rooms down the hallway. Once they do they'll still all have rooms,
and the first hundred rooms will be empty. It's an excellent system, and you're enjoying
the relief of knowing you'll never have to turn away a visitor. Enjoying another relaxing
night running your infinite hotel, with all the rooms full. But suddenly, a bus pulls
up in the parking lot. A very, very long bus. You can't even see the end of it, it's so
long. On the bus is an infinite football team. They
just won the infinite Superbowl, which is a pretty big deal, as you can imagine, and
they want to celebrate their victory with a night in your famous hotel.
When I say it's an infinite football team, what I mean is this: Their jerseys are numbered,
by the positive integers. One, Two, Three, and so on. For every positive integer, there's
a team member with that number on their jersey. There's a player 1000, a player 1000000, a
player A Googol. They're all milling around on that very long bus, but the coach (who
doesn't need a room herself) swears they're very organized when they need to be. The bus,
she says, is equipped with its own set of instantaneous loudspeakers and teleporters.
Once the bill is paid and the room assignments are figured out, each football player will
teleport straight to their assigned room, very simple.
But... every room is full, you tell her. She raises an eyebrow. "I thought you had ways
of solving this kind of problem," she says. You don't want to disappoint her or the victorious
players. If only there were a finite number of them, you could do it. If there were a
thousand, no problem. If there were a million, piece of cake. But making room for an infinite
number of players... can it be done? As it turns out, the answer is yes, it can.
In the next video we'll look at one solution, but see if you can come up with one of your
own, in the meantime. See you next time, in Hotel Infinity.