I know you’re good at thinking because you play chess. Let’s see how your problem-solving skills hold up to infinity itself. Try to figure out the answer to some of the challenges in the Hilbert Hotel story below and let me know how you did.

As an unemployed math expert, you eagerly accept a job as hotel clerk and friendly concierge at the Hilbert Hotel. Your assignment is simple: guide guests to their rooms. Fortunately for you, what makes Hilbert Hotel so convenient is that there is no shortage of space. There are an infinite number of rooms for your guests, labeled 1, 2, 3, 4, etc.

On your first day, a guest named Charlie shows up. He pays for a room and awaits his room number. How do you assign him his room number?

Answer (highlight to reveal): Literally any way you want. It makes sense to go in order, so how about you assign him room number 1. But any number would have worked.

The next day, Charlie packs up and leaves. Now all the rooms in the infinite hotel are vacant once more. But by 3:30 pm, a bus pulls up with a lot of guests. A man walks out and explains that in fact they are a vacation group with an infinite number of guests. Each guest has a unique label, with there being a Guest Number 1, Guest Number 2, etc.

Aware that you won't be able to individually tell each guest individually what room number they will receive, you decide to go get your loudspeaker from the storage room.

How do you go about assigning rooms this time?

Answer (highlight to reveal): You have an infinite number of rooms, labeled 1, 2, 3, etc. There are an infinite number of guests, labeled 1, 2, 3, etc. Simply say on the loudspeaker: "Go to the room whose number matches your guest number!"

Now all the rooms in your infinite hotel are full.

However, just before dinner time, a taxi pulls up and another client shows up. "My apologies for being late," she says. "I'm part of the same vacation group as the bus that arrived earlier today."

"No problem," you respond. "What's your guest number? That's what your room number will be."

"Oh," the guest starts hesitantly. "You see, all the others got their numbers before I joined them. I don't have a number."

You realize that there aren't any empty rooms. What do you do?

Hint 1 (highlight to reveal): Pretend this is Guest Number 0.

Hint 2 (highlight to reveal): Luckily, guests don't mind moving rooms—They're used to the chaos of infinity.

Answer (highlight to reveal): Use your loudspeaker (or some other public address system) to announce that every guest should move to the room whose number is exactly one greater than their current room. Then, you assign Room Number 1 to the late guest.

The next day, another bus pulls up. This one, too, transports an infinite number of guests, labeled 1, 2, 3, and so on. Their leader requests a room for each of them.

All your rooms are full, and now you must make room for another infinite number of guests. How do you assign rooms to these new guests?

Hint (highlight to reveal): Did you know there are infinitely many even numbers?

Answer (highlight to reveal): First, you ask all the guests that already have a room to move to the room whose number is exactly double their current room number. Then, you ask all the newcomers to go to the room whose number is exactly one less than the double of their guest number. This assigns the even numbers to the guests who already have a room, and the odd numbers to the ones who don't have a room yet.

Well done making it this far! Although your clients had to change rooms quite often, they are pleased with the hotel service and leave good reviews as they leave. Now all the rooms are vacant once more.

The next day, a convoy of buses show up. A man steps out from the first bus and explains that his group involves an infinite number of buses, each containing an infinite number of guests.

"Are each of your guests labeled with a unique number?" you ask hopefully.

"Within each bus, yes," he replies. "But Guest Number 1 exists in Bus 1, Bus 2, and indeed in every one of the infinitely many buses."

How will you assign these people their rooms? (This one is harder.)

Answer (highlight to reveal): There are multiple ways to do this, all of them pretty difficult to spot. Imagine a grid, where horizontally there are the guest numbers and vertically there are the bus numbers, so that each square represents a unique guest. You then draw a line, starting at guest 1, bus 1 and zigzagging upward so that it systematically touches every square. The order the guests appear in the graph is their room number. You would somehow project this grid to let guests know the system, or you could provide a mathematical formula to derive the room number from the guest and bus numbers.

The infinite number of guests from their respective buses move in and then leave on the next day.

Your boss is pleased with your week's work and raises your salary.

Next week, a single bus pulls up with an infinite number of guests. Unfortunately, these ones aren't labeled 1, 2, 3, etc. What's special about these guests is that they each filled out a survey and assigned each of the rooms in your hotel (every single one of the infinite number of rooms) a 1-to-10 star review. Each guest represents a unique combination of reviews for each of the rooms, and there is no combination that is not represented by a guest.

These reviews are the only identification each guest has. How, using this, will you assign each guest his or her room?

Hint (highlight to reveal): You should be suspicious here.

Answer (highlight to reveal): Unless you cheated (like assigning the room number to each guest in the order they got out of the bus), you weren't able to find a method to assign rooms to these guests because in fact there is no solution. It has been mathematically proven that no matter how you arrange the guests, some will be left out. There are more guests than there are rooms in the hotel.

You're probably wanting to ask me, "Wait, but I thought we had an infinite number of rooms? And that means that there are always more available."

Here's the thing: Not all infinities are the same size. This Hilbert Hotel paradox illustrates it.

The rooms in the Hilbert Hotel line up neatly as 1, 2, 3, 4, etc.—that's the infinity of the natural numbers, sometimes called countable infinity because you can count them (at least, you can start).

But the guests in the last bus are stranger. Each one carries a complete set of reviews for every room, like a secret code of infinitely many digits. These ones don't nicely line up in a list the way the room numbers did. They spill over into a bigger infinity, the infinity of real numbers (which includes decimals, fractions, and numbers with endless non-repeating digits). This is sometimes called uncountable infinity.

Uncountable infinity is strictly larger than countable infinity. No matter how clever you are, you can't fit all those guests in the hotel. That’s why mathematicians say infinity comes in different sizes, written with the Hebrew letter ℵ (aleph). The smallest infinity, ℵ₀ (aleph-null), is the number of rooms the hotel has. Larger infinities include ℵ₁ (aleph-one), ℵ₂ (aleph-2), etc. 🙂

Did you actually read all that? Thanks! I hope I didn't bore you too much. All this is 25% of the reason why I chose my current username.

SharpAleph-2