This cache is not at the posted coordinates. To get the coordinates, solve the puzzle below!
Story
Welcome to David Hilbert’s Grand Hotel, where there’s always vacancy, no matter how many guests check in!
How can this be? The trick is that the Grand Hotel has infinitely many rooms! They are numbered with natural numbers 1, 2, 3, 4, 5, . . . and so on, but the numbers never end. You will never find the last room.
Currently, every room in the hotel is filled, but that still doesn’t mean there’s no vacancy! It is, unfortunately, a bit of a hassle to accommodate new arrivals, but it can always be done.
Suppose a new guest arrives. How do you think we do this? Well, since all the rooms are already filled, we can’t just tell them to go to room ∞ + 1 . . . they’ll never find their room! (They would have to go past infinity first, which is impossible.) What we can do, though, is reassign everyone to the next room over. So if any given guest is in room n, they will move to room n + 1, and this will give everyone a room close by that they can move to, along with freeing up room 1 for the new guest! There’s always room at the Hilbert Hotel!
Suppose now that an infinitely long bus arrives, with an infinite number of guests each seeking their own rooms. How can the hotel possibly accommodate this? But there’s always room at the Hilbert Hotel - and the answer is simpler than you might expect! In this case, if any given guest is in room n, the receptionist will simply move them to room 2n. This moves all current guests to only even-numbered rooms, freeing up the equally infinite odd-numbered rooms for the new guests from the bus! True, some guests may have a long elevator ride, but since no one has to go past infinity, everyone will be able to get to their room eventually.
As the hotel gains popularity, this situation becomes even more dire. Now, it’s a regular occurrence that, at some point every night, infinitely many of these uncannily long buses arrive with their infinitely many passengers! At first, the hotel found a way to accommodate them using prime numbers, since there are also infinitely many primes. First, any current hotel guest in room n would be moved to room 2n (so, for example, the guest in room 9 goes to room 512). Next, the buses would be assigned odd prime numbers (the first bus would be 3, the next would be 5, the third would be 7, then 11, 13, 17, 19, and so on). Let pb represent the odd prime number assigned to bus b. Assume that, inside the buses, each passenger is assigned a seat number 1, 2, 3, 4, . . . , and so on; let s represent a passenger’s seat number. Every bus passenger was then assigned to room pbs so that everyone, whether already checked in or arriving on one of the infinite buses, would have a room at the Grand Hotel.
The biggest problem with this method, though, was that it left a lot of rooms empty. It was an absolute nightmare for the cleaning staff, who had to walk so far in between rooms that actually needed cleaning that most of their time on the clock was spent just trying to find those rooms! Not to mention, they had to go to higher and higher numbers to clean—which, when you’re already cleaning infinitely many rooms, just isn’t fun. And the hotel realized it was losing money on the infinitely many unoccupied rooms!
In response, the receptionist devised a new method for accommodating the hotel guests. This time, they decided to use a mathematical concept called triangle numbers to their advantage—and in fact, they don’t know why they didn’t think of this before, since the hotel was built like a big triangular prism, with room 1 at the top, rooms 2 and 3 on the next floor down, then rooms 4, 5, and 6, then 7 through 10, and so on. (It’s a magical hotel that grows into the sky as needed and extends infinitely into the ground . . . I don’t know how it works exactly, it’s just a thought experiment.) The numbers that show up along the rightmost edge—1, 3, 6, 10, 15, 21, . . .—are the triangle numbers.
To accommodate the infinitely many new guests, the receptionist first realized that they could treat the already present guests as having arrived on bus 0. That way we can use the similar formulas for everyone! From there, everyone in the hotel (b = 0) in room n gets reassigned to the triangle-numbered rooms on the rightmost column of the hotel, as follows:
Those in the buses (b = 1, 2, 3, . . . ) in seat n then get assigned to all the other rooms, as follows:
This method worked much better for the hotel, as they could start predicting the arrival of an infinite number of buses, and then simply assign existing hotel guests to the triangle numbers every night, leaving all other numbers available for the buses. Furthermore, this method filled every room with one, and only one, guest.
There’s always room at the Hilbert Hotel!
Puzzle
With infinitely many people arriving every day, there’s bound to be some famous people at the Grand Hotel now and again. Here is a record of some famous people that arrived one day:
Catherine the Great, Empress of Russia from 1762 to 1796, arrived at the Grand Hotel on bus 226 in seat 1559, and was assigned a room accordingly.
Ellen Degeneres, American comedian and television host, and first openly gay lead character on an American network television show, arrived at the Grand Hotel on bus 392 in seat 12694, and was assigned a room accordingly.
Jerry Seinfeld, American stand-up comedian and actor specializing in observational comedy, arrived at the Grand Hotel on bus 574 in seat 1246, and was assigned a room accordingly.
Lord Frederick North, former Prime Minister of the United Kingdom from 1770-1782, arrived at the Grand Hotel on bus 847 in seat 2087, and was assigned a room accordingly.
David Hilbert, German mathematician and philosopher who devised the thought experiment of the Paradox of the Grand Hotel, arrived at the Grand Hotel on bus 1023 in seat 1862, and was assigned a room accordingly.
Ricky Montgomery, nonbinary American singer-songwriter whose songs have gone viral on TikTok, arrived at the Grand Hotel on bus 1472 in seat 1993, and was assigned a room accordinly.
Queen Lili’uokalani, only queen and last sovereign monarch of the Hawaiian Kingdom from 29 Jan 1891 until it was overthrown on 17 Jan 1893, arrived at the Grand Hotel on bus 1917 in seat 4692, and was assigned a room accordingly.
Mae West, American actress and screenwriter active from 1907-1979, arrived at the Grand Hotel on bus 2247 in seat 1653, and was assigned a room accordingly.
Mansa Musa, ninth Mansa of the Mali Empire from c. 1312 - 1337 and known for his wealth, generosity, intelligence, and good character, arrived at the Grand Hotel on bus 2255 in seat 9566, and was assigned a room accordingly.
Alecia Beth Moore-Hart, American singer-songwriter known professionally as P!NK, arrived at the Grand Hotel on bus 2496 in seat 15, and was assigned a room accordingly.
You can validate your puzzle solution with
certitude.
Another Quick Note
By the way, this whole puzzle and lesson hinge on work by David Hilbert and Georg Cantor about how not all infinities are the same—that is, there are infinities with different sizes. Mathematicians use the number ℵ0 (“aleph null”) to describe the size of the set of natural numbers, i.e., the number of rooms in Hilbert’s hotel, or the number of seats on one of those buses. This is slightly different from ∞, which is technically more of a limit as the real number line goes off to infinity. However, no matter which infinite subset of the natural numbers you use, the size will still be the same ℵ0—there’s exactly that many odd numbers, even numbers, prime numbers, triangle numbers, and so on.
ℵ0 is what we call “countable infinity”—you can assign counting numbers to the elements that make up the infinite set! But there are bigger infinities, as well, and this doesn’t just mean they have more numbers in them. In fact, even if you double the number of elements, ℵ0 still stays the same! There are as many positive and negative numbers as there are just positive numbers! This is because we can still count them just as quickly—we can still fit them all into Hilbert’s Hotel. Even the number of rational numbers (numbers that can be written as fractions) is ℵ0, and they can all fit in the hotel!
Where it gets tricky, though, is when you try to consider the number of real numbers. The simple reality is, there are a lot of irrational numbers on the number line. I’m going to tell you without proof or explanation that, even just between 0 and 1, the number of real numbers is
Notably, ℵ1 > ℵ0, which means there are more real numbers on the continuum between 0 and 1 than there are natural numbers. But there are as many real numbers between 0 and 1 as there are on the entire real number line! ℵ1 is also the number of irrational numbers, transcendental numbers, and complex numbers. I should note, though, that ℵ1 is more commonly written as 𝖈 or |ℝ|.
We better hope that a ferry with |ℝ| people on it doesn’t arrive . . . because in that case, there won’t be room at the Hilbert Hotel!