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.)

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

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.