Our theme for February is actually a challenge: write a piece without using first person pronouns (I, me, we, etc.)

Let’s say that you own a hotel. You name it “Hotel Infinity.”

It’s a very large and successful hotel –  yugggggge – even. In fact, it’s so yuggggge that there are an infinite number of rooms in your hotel.

This is great for business. How great, you ask? Well, it’s sold out. No vacancy. There are an infinite number of people in your infinite number of rooms.

Then John comes into your lobby. He needs a room. How do you get him a room? (Spoiler alert: answer below.)

Answer: Have each guest add one to her room number. John can take Room 1.

No sweat, right? All good. You’re just in time for your next task.

Suppose John owns a bus company. It’s called “Infinity Bus Company.” And it’s a very successful bus company. So successful, in fact, that John can fit an infinite number of people on his bus. And sure enough, he does.

And all of them want a room in your hotel. How do you get a room for them?

Answer: Have each guest multiply his room number by two. The passengers can fit in the odd numbered rooms.

But that’s not all, no. Because, as his business’s name may imply, John does not own only one bus. He owns an infinite number of buses.

Now suppose one day, business is very good for both you and John. Your hotel is full of an infinite number of people. And each and every one of John’s infinite number of buses is filled with an infinite number of people. How do you get them all a room?

Answer: Have each guest multiply her room number by two. This will leave the odd numbered rooms open. Assign each bus with a different prime number, starting with 3, 5, 7, 11, etc. This will be the base to which each person’s seat number is an exponent. For example, person 5 on the first bus would be in room number 3^5 = 243. Person 13 on the third bus would be in room number 7^13= 96,889,010,407. Told you it was a yugggggge hotel.

Yes, there is a fourth level to the problem. It involves “Infinity Car Ferry” service – an infinite number of people on an infinite number of buses on an infinite number of car ferries. I think you can guess where it’s going. You can find the answer here, on the Wikipedia page for Hilbert’s paradox of the Grand Hotel.

But this problem is my favorite because it represents the very best that the study of mathematics has to offer.

It’s a great story. It’s impossible to explain this problem without smile on your face. It’s accessible but not easy. It’s not something most people have ever thought about – and the solution is just within reach.

It requires connecting some dots. To solve this problem, you need to be able to mix and match some basic operations with infinity, some principles of prime numbers, and a little countably infinite set theory.

And most importantly, it requires creativity. To a third grader memorizing times tables, math can be mechanical and rote. Heck, it can feel that way in college. Somehow, math often gets a rap as the least creative subject of study.

But mathematics at its best is a pursuit into the unknown, equipped with a few tools with which you are familiar and a solution that requires an innovative use for them.

It’s erasing two pages of a proof because you went down a rabbit trail – and you’re likely about to start down another. And it’s the moment when the wheels stop turning and it all finally clicks.

Life is often about finding creative solutions for unprecedented problems. Perhaps the telling of “Hotel Infinity” gets us one step closer to that infinite goal.

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