Hotel Hilbert - 1, 2, 3 ... Infinity

On a dark desert highway — not much scenery

Except this long hotel stretchin’ far as I can see

Neon sign in front read “No Vacancy,”

But it was late and I was tired, So I went inside to plea.

The clerk said, “No problem. Here’s what can be done —

We’ll move guest in room N to the next higher one.

That will free up the first room and that’s where you can stay.”

I tried understanding this as I heard him say:

Chorus:

“Welcome to the Hotel In...finity —

where every room is full (every room is full)

Yet there’s room for more.

Yeah, plenty of room at the Hotel In...finity —

Move ‘em down the floor (move em’ down the floor)

To make room for more.”

I’d just gotten settled, I’d finally unpacked

When I saw 8 more cars pull into the back.

I had to more to room 9; others moved up 8 rooms as well.

Never more will I confuse a Hilton with a Hilbert Hotel!

My mind got more twisted when I saw a bus without end

With an infinite number of riders coming up to check in.

“Relax,” said the nightman. “Here’s what we’ll do:

Move to the double of your room number:

that frees the odd-numbered rooms.”

Repeat Chorus

Last thing I remember at the end of my stay —

It was time to pay the bill but I had no means to pay.

The man in 19 smiled, “Your bill is on me.

20 pays mine, and so on, so you get yours for free!.”

~ Lesser, Lawrence Mark (2001) "Hotel Infinity," Humanistic Mathematics Network Journal: Iss. 24, Article 4. Available at - https://scholarship.claremont.edu/hmnj/vol1/iss24/4/

To know more about this fascinating thought experiment, please read - https://archive.nytimes.com/opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/