# Infinite Hotel

talzamir
Posers and Puzzles 01 Oct '11 19:57
1. talzamir
Art, not a Toil
01 Oct '11 19:57
A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.

On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyone is given a new room by moving from room n to room n + 1,000, which vacates rooms 1 .. 1,000 for the new arrivals.

On the second day, an infinite number of new people arrive. How are they given rooms?

On the third day, an infinite number of buses arrive, each with an infinite number of people. How does everyone now get a room?
2. 02 Oct '11 13:26
Originally posted by talzamir
A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.

On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyon ...[text shortened]... mber of buses arrive, each with an infinite number of people. How does everyone now get a room?
No idea... ask Hilbert, I'm sure he knows.

Alternatingly odd and even, and in a plane-walking pattern, respectively.

Richard
3. 02 Oct '11 13:582 edits
Number the rooms a1; a2, b1; a3, b2, c1; a4, b3, c2, d1 etc

Guests move to &quot;a&quot; rooms, first bus to &quot;b&quot; rooms etc.

The power is in the etc.s
4. forkedknight
Defend the Universe
03 Oct '11 05:21
Silly problem, an infinite number of rooms can't possibly be fully booked.

But nor can an infinite number of guests possibly be accommodated.
5. wolfgang59
howling mad
03 Oct '11 07:17
Originally posted by forkedknight
... an infinite number of guests possibly be accommodated.
some guests will have to share ....
6. wolfgang59
howling mad
03 Oct '11 07:17
Originally posted by forkedknight
Silly problem, an infinite number of rooms can't possibly be fully booked.

.
Some guests will have more than one room.
7. talzamir
Art, not a Toil
03 Oct '11 07:50
If infinity and hotels together feel silly, this could of course be mathified, e.g.

Is it possible to pair up with positive integers...
a. all natural numbers?
b. all integers?
c. all rational numbers?
d. all real numbers?

where cases a..c are just about the same as the cases of adding a finite number of new guests, an infinite number of new quests, and infinity squared number of new guests. Correct answers given already by readers on those cases.
8. forkedknight
Defend the Universe
03 Oct '11 14:16
Originally posted by talzamir
If infinity and hotels together feel silly, this could of course be mathified, e.g.

Is it possible to pair up with positive integers...
a. all natural numbers?
b. all integers?
c. all rational numbers?
d. all real numbers?

where cases a..c are just about the same as the cases of adding a finite number of new guests, an infinite number of new ques ...[text shortened]... infinity squared number of new guests. Correct answers given already by readers on those cases.
a) yes
1:1, 2:2, etc

b) yes
1:0, 2:1, 3:-1, 4:2, 5:-2, etc

c) yes
A bit more complicated, but you can walk a table diagonally 1:1/1, 2:1/2, 3:2/1, 4:1/3, 5:3/1, 6:4/1, 7:2/3, 8:3/2 etc

d) no

Therefore, case a,b,c are all adding the same number of guests ðŸ™‚
9. wolfgang59
howling mad
04 Oct '11 02:271 edit
Originally posted by talzamir
A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.

On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyon ...[text shortened]... mber of buses arrive, each with an infinite number of people. How does everyone now get a room?
On the third day the buses are numbered as they arrive and the occupants go to the floor indicated by their bus number.

Each floor has an infinite number of rooms to accomodate the infinite number of passengers on each bus.

EDIT: The current guests all remain on the ground floor.
10. Palynka
Upward Spiral
04 Oct '11 12:40
Originally posted by forkedknight
Silly problem, an infinite number of rooms can't possibly be fully booked.

But nor can an infinite number of guests possibly be accommodated.
It's a countably infinite number, so bijection says otherwise.

And...here...we...go!

You put 2 numbered balls in a bag and...
11. forkedknight
Defend the Universe
04 Oct '11 18:001 edit
Originally posted by Palynka
It's a countably infinite number, so bijection says otherwise.

And...here...we...go!

You put 2 numbered balls in a bag and...
Just because you take take infinitely many guests and map a room to each guest doesn't mean you can fill a hotel with infinitely many rooms. There will always be more rooms. That's the definition of infinite.
12. Palynka
Upward Spiral
04 Oct '11 23:41
Originally posted by forkedknight
Just because you take take infinitely many guests and map a room to each guest doesn't mean you can fill a hotel with infinitely many rooms. There will always be more rooms. That's the definition of infinite.
If there is a bijection, then there are the same amount of rooms as guests. By definition.
13. Anthem
The Ferocious Camel
05 Oct '11 00:431 edit
Originally posted by Palynka
If there is a bijection, then there are the same amount of rooms as guests. By definition.
What mathematician ever defined "same amount" via the existence of a bijection? That sounds more like the definition of cardinality to me.
14. forkedknight
Defend the Universe
05 Oct '11 02:17
Originally posted by Palynka
If there is a bijection, then there are the same amount of rooms as guests. By definition.
And yet you can add 1000 more guests, and they all still have rooms.

Using the concept of infinity as though it is a number doesn't make any sense.
15. Palynka
Upward Spiral
05 Oct '11 07:282 edits
Originally posted by forkedknight
And yet you can add 1000 more guests, and they all still have rooms.

Using the concept of infinity as though it is a number doesn't make any sense.
A finite number of elements can always be added, but you need to change the mapping else there would be no available rooms.

What doesn't make any sense is your blatant disregard for set theory and cardinality. If two sets are linked by a bijection, they have the same cardinality, and sets with the same cardinality have the same number of elements.