Fun with Infinity

[Nick]

A man walks into the Infinity Hotel. He asks at the desk if there are any available rooms. "No, sir, I'm sorry but we have no vacancy". The man says, "But I really need a room!". "Very well, we'll have everyone shift over one room, and you can have the first room".

Discuss.

[zero]

What, did Hilbert's Hotel finally find a competitor?

How about the man returning a few months after his visit, because he was so pleased with the extraordinary service of finding a room for him despite there being no vacancies whatsoever when he arrived that fateful day. In fact, he told everyone he knows about the place. Since he's a popular guy, he knows as many people as there are positive integers. They all come with him to request a room. Again there's no vacancy, but -- hey! -- no problem.

All the current guests are simply moved into the room with double their original room number, freeing up infinitely many odd numbered rooms to accommodate all their new guests.

[Nick]

What, did Hilbert's Hotel finally find a competitor?

How about the man returning a few months after his visit, because he was so pleased with the extraordinary service of finding a room for him despite there being no vacancies whatsoever when he arrived that fateful day. In fact, he told everyone he knows about the place. Since he's a popular guy, he knows as many people as there are positive integers. They all come with him to request a room. Again there's no vacancy, but -- hey! -- no problem.

All the current guests are simply moved into the room with double their original room number, freeing up infinitely many odd numbered rooms to accommodate all their new guests.

Is that what it's called? I find thinking in terms of infinity fun.

[zero]

Good! Then here's a puzzle for you.

Every last one of these infinitely many guests also appreciated the benefits and comfort of Hilbert's Hotel so much that they each return at the same time a month later, and coincidentally they each bring their own infinitely large set of friends with them. These are very friendly people. The hotel clerk in charge of rearranging rooms to accommodate the guests is recovering from a severe hangover (after attempting but failing to ingest an infinite amount of alcohol the night before), and requests your assistance. What do you do?

[Nick]

Good! Then here's a puzzle for you.

Every last one of these infinitely many guests also appreciated the benefits and comfort of Hilbert's Hotel so much that they each return at the same time a month later, and coincidentally they each bring their own infinitely large set of friends with them. These are very friendly people. The hotel clerk in charge of rearranging rooms to accommodate the guests is recovering from a severe hangover (after attempting but failing to ingest an infinite amount of alcohol the night before), and requests your assistance. What do you do?

I'm going to say... I give each person and all their friends the same infinitely large room.

[zero]

Hilbert's Hotel has infinitely many finite rooms. You'll have to build your own to get infinitely large rooms. In the meantime, there are several different ways to give all these guests what they want. Suppose you start by freeing up all the odd-numbered rooms like before (where each current guest is moved from Room N to Room 2N). Now you have infinitely many returning guests, each with infinitely many friends lined up behind them. They all want single rooms, please. It's OK if there are empty rooms remaining afterwards, so long as you can easily assign a separate room to each and every one of them.

My favorite method starts by assigning each of the returning guests to a room with an odd prime number, then all their friends get rooms numbered with powers of the same prime. In other words the returning guests take all the prime numbered rooms (except Room 2, which is occupied by the guest formerly in Room 1). For any particular returning guest assigned to room P, she will find all her friends in rooms P^2, P^3, and so on.

It's more of a challenge to find an easy way to do this that keeps the hotel full, so there are at the end no unoccupied rooms and the "No Vacancy" sign stays on outside (not that anyone seems to pay it the least attention). Alternately, for extra credit -- or extra headache -- plan ahead for the following month, when all these guests suddenly decide to come back all at once, still grouped in the same infinitely many infinite sets of friends, only this time they each bring infinitely many cousins, too (all new people). How do you give them their room numbers now when the hotel is, as usual, completely booked up?

[Keiji]

Good! Then here's a puzzle for you.

Every last one of these infinitely many guests also appreciated the benefits and comfort of Hilbert's Hotel so much that they each return at the same time a month later, and coincidentally they each bring their own infinitely large set of friends with them. These are very friendly people. The hotel clerk in charge of rearranging rooms to accommodate the guests is recovering from a severe hangover (after attempting but failing to ingest an infinite amount of alcohol the night before), and requests your assistance. What do you do?

Use the pairing function.

Enumerate each guest in the initial set from 1 to infinity.

For each of those initial set guests, enumerate each of their friends also from 2 to infinity (where the initial sets all have 1 in this field).

Let i be the number of the guest from the initial set, and j be the number of their friend (or the 1 mentioned above). Compute the pairing, and assign that number as the room number of each person. If the hotel wasn't empty, use the doubling technique to fit them in.

This method will indeed ensure that there are no empty rooms, unless the doubling technique was used with the hotel initially partially filled.

For n recursive sets of guests, simply use no pairing for n=1, the square pairing for n=2 (the ordinary pairing), the cubic pairing for n=3, and in general the nth order pairing for n ∈ ℤ, n > 0.