Tuesday, 19 August 2014

Logistical problems in the infinite hotel

There's a weird mathematical thought experiment to illustrate the properties of a countable infinity that goes as follows: you are the manager of an infinite hotel. That is, you have rooms numbered from 1, 2, 3 and so on, up and up, never ending, and right now, every room is full. When you get one more guest, even though every room is full, you can make room for them by moving everyone up one room and putting the new guest in room 1. The guest that was in room 1 goes to room 2, room 2's guest goes to room 3 and so on. Nobody has to leave and everyone still gets a room. This keeps working no matter how large a group arrives at once, as long as there's a finite number of them.

If an infinite number of guests arrives at once, you can still make room for them by advising every current guest to double their room number and move in there. The guest in room 1 goes to room 2, room 2 goes to room 4, room 3 to room 6 and so on. The new guests now check into the odd-numbered rooms and there's still enough space for everyone.

What I'd like to talk about, in a silly way, is the logistics of running a hotel like this. When these guests arrive, it's kind of a pain to move people. You could call up each room individually and ask them to move, but you'd never finish. You could just have the first guest relay the message to the next one, so room 1 tells room 2 to move to room 3, room 2 tells room 3 to move to room 4, and so on. Probably the best communication method is an infinite PA system so that you can address all the rooms at once.

The next part is what I always thought would get me down about staying in an infinite hotel. When that infinite group of new guests arrives, it's not so much trouble for the guest in room 1 to move next door to room 2, nor for room 2 to move to room 4. The guest in room 50 might be a bit miffed at having to move all the way up to room 100, but what about the poor saps up at room 1,000,000 and above? You'd probably still be on the move when the next call came over the PA to move up. So you start at room 1,000,000, and the manager calls out "infinite new guests, everyone please double your room number and move up there". You pack up your things and start walking from 1,000,000 to 2,000,000. Before you pass the door for room 1,005,000, the same call comes again, so now your new room is 4,000,000. It just gets further away all the time. Beyond a certain point, you're better off turning back to the check-in desk and announcing yourself as a new guest. In fact, that's probably where all these "new" guests keep coming from.

Lastly, every time you hire a new cleaner, you never see them again. They'd start cleaning at room 1, move on to room 2, then room 3 and so on down the line, never finishing until they just quit. Realistically, you probably need your guests to clean up after themselves.

Mokalus of Borg

PS - Past a certain point, though, it would be impossible to verify.
PPS - Plus, on average, you'd probably spend all your time travelling to evict guests whose credit cards were declined.

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