Fun Fact Friday
In Part 1, we tried to think about infinity as a number and things got pretty fucked up (mathematically), pretty quickly. In Part 2, we found that we can fit an infinite number of things into a finite space. The concept of infinity being sort of a “size” or “distance” is perfectly reasonable, but as we’ll see shortly, some weird things are possible when the size is infinite. But first we have to talk about the first math we learn when we’re little babies: counting.
Counting things is pretty fundamental. You learn at a young age to count your marbles[1], or how many bites of broccoli you have left before you can leave the table, or how many empty wine bottles are in front of mommy. Next you learn to count by 2’s, 5’s, 10’s.
Maybe you even learn to count by powers of 2. Maybe you got a lot of wedgies in school. I wouldn’t know.
ANYWAY, the point is, even a small child could count to infinity, given enough time on a car ride and some ear plugs for the parents. Counting happens to be useful when we think about infinity.
Let’s say you just opened up a brand new hotel with infinite rooms numbered 1, 2, 3, …, ∞. Ignoring the initial investment needed to build such a hotel, you are ready for business! As the shrewd businessperson that you are, you know that having as many people occupy rooms as possible will maximize your revenue. So you put an ad on during Super Bowl LXI (which will be Eagles vs. Patriots, just to make it more nerve-wracking for you) offering free rooms to everyone if the Eagles win.
On the surface this sounds silly, since you’re trying to make money, but you also have a huge Eagles fan shop in the hotel so you figure the merch sales will more than make up for the lost revenue for the rooms[2].
Turns out a LOT of people saw your ad, and after the Eagles win the game on a tush-push in the 3rd overtime, you have a bus carrying infinity Eagles fans show up to claim their rooms. As each fan stumbles off the bus[3], you assign them a room number corresponding to the order in which they leave the bus. Good job! Your infinite hotel is now filled with an infinite number of insanely drunk Eagles fans[4]!
Uh oh, someone missed the bus because they had to spend the night in the stadium jail and now they want a room. But the hotel is full and you can’t go back on your promotion for fear of a false advertising lawsuit[5]. So what do you do?
Welp, if you just tell everyone currently in the hotel to move one room up (room 1 goes to room 2, room 2 goes to room 3, etc) you now have an empty room 1 for Chuck T. from Philly to occupy. Since the hotel is infinite, you can repeat this process every time more stragglers show up for their rooms. When 𝑘 guests show up for rooms, you can just move every guest from room 
to room 
.
Yeah, but now it seems like we’ve got real problems because another bus shows up with an infinite number of Eagles fans looking to claim their rooms. As they start to pile off the bus, you panic. You can picture them rioting around your hotel. No street light, parked car or mounted police officer is safe.
Luckily, you have a nerdy partner who somehow wasn’t able to talk you out of this venture, we’ll call him “Samg”.
Samg[6] quickly realizes that there are an infinite number of even numbers and an infinite number of odd numbers. So we’ll use that to our advantage and tell everyone in the hotel to take their room number and move to the room corresponding to 
. Room 1 moves to room 2, room 2 moves to room 4, room 3 moves to room 6 and so on.
Since will always be an even number, that leaves all the odd numbered rooms open for the rowdy busload that just arrived. The first person off the street lamp (we’ll call them 
) goes to room 
(room 1) and each additional 
person does the same thing (
ₚ
). Problem solved!
Interestingly[7], you can repeat this process to infinity…if an infinite number of infinitely full buses show up, you’ll always be able to accommodate them. There are a bunch more complex ways to figure out these scenarios, but I don’t want to bore you with too much math[8].
Is there a moral to this story? No. Other than: definitely don’t do this. While it seems like these were pretty simple-ish solutions to the whole fitting-infinity-post-Super-Bowl-victory-Eagles-fans-into-your-magical-hotel problem, this would be the least of your (and Samg’s) worries.
[1] To this day, every time I count them I come up short. It’s like I’ve lost some of them
[2] I wouldn’t recommend this strategy
[3] The party went pretty late the night before
[4] You have a LOT of insurance against property damage, I hope.
[5] This is probably the least of your concerns at this point
[6] It rhymes with “ham gee”
[7] For me, at least
[8] LOL TOO LATE