On Mon, 9 May 2005 10:52:03 -0400, John Luedeman <[log in to unmask]>
wrote:

>Official solution:
>Scenario 1:  Move each tenant to the next higher room number.  In this
case
>room 1 becomes free for La Poisson.
>Scenario 2:  Move each tenant to the room with double their room number
>(e.g. tenant of room n goes to room 2n).  In this case, the odd numbered
>rooms are free so Herr Banach can assign waiting customer k to room 2k-1
>(e.g. the first in line goes to room 1, the second to room 3, the third to
>room 5, etc.)

My original solution (swear it was) was better than the "stock solution"
which I found on the web since my post, because I brought in Herr
Banach and the implicit mapping <though unnecessary to solve these
two, was more "general" and "advanced").  In addition, I contrasted
the physicist solution with a mathematicians solution, a la the
coffee pot problem.  :-)

You have a stove at "A" and a coffee pot at "B".  You want to make
a pot of coffee.  Both the physicist and mathematician would take
the pot from B to A.

                 A    <--------   B

Now change the problem to A, B, and C

                 A                B

                                  C

The physicist would take the pot from C to A, but the mathematician
would take the pot from C to B, thus reducing it to the first problem.


http://www.bbc.co.uk/dna/h2g2/A414523

================== excerpt from web link
Q: A guest arrives at reception, and asks for a room. Is it possible to
allocate him a room?

A: Yes, it is possible: get everybody to shift to the next room (i.e. the
person in room 1 moves into room 2; the person in room n moves into room
n+1). Since the hotel is infinite, there will always be a next-door to
move into. The newly-arrived person will take room 1.

Q: Twenty guests arrive at reception, and they ask for rooms. Is it
possible to allocate them rooms?

A: Yes, it is possible: get everybody to shift to the room twenty
positions down (i.e. the person in room 1 moves to room 21; the person in
room n moves into room n+20). The newly-arrived group will take rooms 1 to
20.

Now, you're thinking, this is a bit dodgy, isn't it? The hotel was full
already! Well, this shows how infinity can't be regarded as a number in
the normal sense, because infinity plus a finite number is still infinity.

But wait, there's more...

Q: A bus containing an infinite amount of guests arrive at reception, and
they ask for rooms. Is it possible to allocate them rooms?

A: Yes, it is possible: get everybody to move to the room which is double
the room number they have already (i.e. the person in room 4 moves to room
8; the person in room n moves to room 2n). The first person in the newly-
arrived group gets room 1; the nth person gets room 2n-1.

This is especially weird. Even in a full hotel, you can accomodate an
infinite amount of guests. With this case, we show that two times infinity
(or, in fact, any finite number times infinity) is still infinity.

But wait...

Q: An infinite number of buses, each containing an infinite amount of
guests arrive at reception, and they ask for rooms. Is it possible to
allocate them rooms?

==============  end excerpt from web link

>Are these e-mails teaching a fish in a school?

What e-mails?  Mine are read and sent from Brotha' Jeff's LISTSERV
webpage for SE <Seriously Engimatic> LIST.

La Poisson.

>
>John K. Luedeman
>Professor Emeritus of Mathematical Sciences and Teacher Education
>Clemson University
>864 882-6735 (H)
>864 656-5129 (O)
>
>
>-----Original Message-----
>From: SCUBA or ELSE! Diver's forum [mailto:[log in to unmask]] On
Behalf
>Of Reef Fish
>Sent: Monday, May 09, 2005 10:44 AM
>To: [log in to unmask]
>Subject: Re: [SCUBA-SE] World's Hardest Number Quiz (Part I) (was Re: Paul
>Erdos)
>
>On Mon, 9 May 2005 10:14:02 -0400, John Luedeman <[log in to unmask]>
>wrote:
>
>>Mine is 2. I co-authored with Ringeisen who co-authored with Erdos.
>
>I didn't know THAT!  Right under the mathematical leady roofs of Clemson,
>there were a Erdos Number 1 and Number 2.  Brawley is probably no less
>than Number 2, via Carlitz and other Number Theoriests.
>>
>>
>>
>>Sorry for the misread - I thought you were being mathematically precise.
>
>I was!  Just appended the mathematically precise statement with a
>NED-friendly remark which did not contradict the mathematical precision!
>>
>>
>>Now, a puzzle.  Hilbert's Hotel has two scenarios:
>>
>>1.    Hilbert's Hotel has a countably infinite number of rooms numbered
1,
>>2, etc.  In scenario 1, all rooms are full.  La Poisson shows up for a
>dive
>>trip and wants a room.  If all rooms are full, how can La Poisson get a
>>room?
>
>You probably have your "stock" solution, but here's La Poisson's original:
>
>Since an infinite-dimensional Hilbert Space is always a Banach Space,
>La Poisson simply asks for the manager of the Hilbert hotel, Mr. Banach
>to re-assign the room numbers by adding 1 more room to it.
>
>
>>2.    In scenario 2, all rooms are again occupied.  Now La Poisson shows
>up
>>with his countably infinite number of friends for the Hilbert Nedfest.
>How
>>can each be assigned a room?
>
>If you're a physicist, you would ask for manager Banach to add
>countably infinitely many rooms to the Hilber hotel in ONE
>re-assignment of room numbers, for the infinitely many NEDs.
>
>If you are a mathematical, you would note that this reduces to
>Scenario 1 if you ask for manager infinitely many times, to
>accommodate the NEDs 1 at a time.
>
>
>>John K. Luedeman
>>Professor Emeritus of Mathematical Sciences and Teacher Education
>>Clemson University
>
>I am curious as to what the "official" solution is.
>
>La Poisson.