Short Wave - Choose Your Own (Math) Adventure

Episode Date: October 18, 2022

Ever read those Choose Your Own Adventure books of the 80s and 90s? As a kid, Dr. Pamela Harris was hooked on them. Years later she realized how much those books have in common with her field: combinatorics, the branch of math concerned with counting. It, too, depends on thinking through endless, branching possibilities. She and several students set out to write a scholarly paper in the style of Choose Your Own Adventure books. Dr. Harris tells Regina G. Barber all about how the project began, how it gets complicated when you throw in wormholes and clowns, and why math is fundamentally a creative act. See pcm.adswizz.com for information about our collection and use of personal data for sponsorship and to manage your podcast sponsorship preferences.NPR Privacy Policy

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time to course correct and land in one of two locations. The toxic jungle to the west or the sunken sea to the east. What's it going to be? Read onward or turn to page five and find out. So remember those Choose Your Own Adventure books of the 80s and 90s? You know, the ones that had you exploring faraway planets or dark dungeons. Well, for Dr. Pamela Harris, they're more than just nostalgia. She remembers reading her very first one as a kid and getting hooked in.

instantly. I think by like the fifth page or something like this, I already had to make a choice about where the story would lead. And it was something like, do you want to stay and fight the dragon, go to page 73, or do you want to run away, go to page 75? And so all of a sudden I had to make a decision about what was going to happen to the future of me as this character. And I was hooked. I think I read a couple of them and I kept on dying. So I was going to gave up. Pamela is a math professor at the University of Wisconsin, Milwaukee. She specializes in combinatorics, the field of math that's all about counting.

And she says, thinking through her research is all about envisioning these endless branching possibilities. You wonder what will happen if I do this instead of this other thing. And so you're always kind of at the fork of this road where you're making a decision about what is the next thing you'll think about. So a few years ago, Pamela was working on a research project specifically on parking functions. These are problems in combinatorics that are all about sorting hypothetical cars

into hypothetical parking spots. So suppose there are N cars driving down a one-way street with N parking spaces, where N is any whole number, like 2 or 5 or 10,000. But each of the many drivers have different preferences for their favorite spot. So one at a time, they sort themselves based on their preferences. If all the cars can find a space to park, that's a parking function. They're not really used for actually parking cars, but they shed light on lots of important questions in math and computer science.

So Pamela was working with a group of math students on the outcomes of different parking functions. And they kept coming to me and saying, well, what happens if we do this thing? Or what happens if the cars move this way? Or what happens if the street looks like this? And then I was like, I don't know. I don't know. You should write that down. Write that down.

Write that down. And all of a sudden, by the end of the summer, we just had a document with all of these other possible projects that we could work on. That's when she realized there's actually a lot of overlap between the kinds of math problems she works on, like these parking functions, and her childhood fascination. And I thought, you know, this reminds me of these books where you have to make a choice and then it leads you down this new avenue. So with some of her undergrad research students, Pamela wrote a paper called Parking Functions, Choose Your Own Adventure.

In it, the reader is sent down a spiral of different parking scenarios. Like what if the car only can move forward, or move backward, then forward? Or what if every family decides to host a barbecue for their friends on the same day? Or what if a bunch of clowns need to pile into each clown car to go to the circus? Or what if the parking lot is now riddled with worm, holes and the cars can teleport. The reader is making choices about the mathematics that they want to learn more about. And as Pamela wrote this Choose Your Own Math Adventure, she found more and more ways to turn

these parking functions into stories. I think if you can't tell a good story with a math paper, then that's not a math paper I want to read. And there's a plot involved, right? There's characters, parking functions. There's adventures they go on, these theorems. And then hopefully there's some room that other people can join on its adventure. And so that's what I consider your open problems, problems that we don't yet have an answer to.

And so for me, that's a recipe for a good choose your own adventure problem. Is there storytelling that happens in mathematics when you're like looking for problems? It mirrors to me the process that I take when I think about creating new problems, you know, in this idea of just tweaking something a little bit to see if new results appear. And so all of a sudden, you end up finding these like beautiful, discoveries that were sort of unexpected by just continuing to ask a lot of questions. And for me, the asking a lot of questions mirrors the making a lot of choices in that adventure. Today on the show, we dive into this choose-your-own parking function adventure.

Warning, the reading of this paper will send you down many winding roads towards new and exciting research topics enumerating generalized parking functions. Buckle up. I'm Regina Barber, and you're listening to Shortwave, the Daily Science, and math, podcast from NPR. Before the grand adventure begins, before we park any of these cars, we need to back all the way up, out of the parking lot entirely, and discuss what is this thing called combinatorics. So combinatorics is the art of counting.

We're concerned with finding the number of objects that satisfy a certain property. For example, if you have coins, say you have a few, quarters in your pocket, nickels, dimes, pennies. And I say to you, okay, in how many ways could you make 37 cents out of the money in your pocket? So you might start thinking, well, I could use a quarter, I could use a dime, and I could use two pennies. There you go, 37 cents. But maybe you could use 37 pennies. Or maybe you could use, you know, three dimes, a nickel, and two pennies. So those are different ways in which you could add to 37 cents. So these are the kinds of problems that I'm interested and thinking about. Within commentatorics, there are all kinds of questions you can ask.

Like how to distribute favorite candy or how to solve a Rubik's Cube in all different ways. Or how to sort cars into a parking lot. Let's suppose that we have two cars and they're trying to enter a one-way street. The one-way street has only two parking spots and they happen to be numbered one and two. Now, let's suppose that you have, you're the person driving car number one. and you have a preference for one of those spots. Maybe you prefer the second spot. Well, listen, you're the first person to enter the street.

You're going to get exactly what you want. So you'll drive to spot two and you'll park there. Now, I have an option. I have no idea what you've done. So if my preference is the first parking spot, I'm going to get lucky. I'm going to park exactly in the first spot because you didn't take it. So the preference list, which is your preference, spot two,

followed by my preference because I was a second car, spot one. That list is what we would call a parking function because both of us were able to happily park on the street. Exactly, we were able to park. Now, let's consider a slightly different scenario. So in the case where we both prefer the second spot, you enter the street, park in the second spot because you're the first car, and now I enter the street wanting that same spot.

I'm not allowed to back up because I'm on a one-way street. You took spot two, so I have to. to exit the street without parking. And so the preference list is actually not a parking function. But parking functions can get a little bit more complicated. Like when there are a lot more cars and a lot more spaces, or when a car's gear shift doesn't work and you can't go backwards, or if everyone starts driving electric scooters and more than one of those can fit in a parking spot. This paper has many endings. And can you name some of your favorites? Yeah, I love the clown scenario where all of a sudden it's the clowns trying to get into the cars and you could have many, many clowns into a

single car. And that one just makes me giggle every time because I can like see, you know, all the clowns trying to fit into the particular car. Since Pamela finished the paper in 2019, she supported about 50 research students who have worked on parking function projects. And it has led to just an abundance of new mathematics has been developed based on the, the, the, the work that these students set forth in having all of these open problems. Now, she's hoping to write a whole math textbook in the Choose Your Own Adventure format, and she's hoping more than just mathematicians read it. I hope it's read by people who are curious about what it means to be a mathematician.

I think there is a misconception. The mathematics is dry, that all mathematics is already known. It's already in this textbook. And the reality of it is that mathematics is a creative endeavor, is very very, very very much like creating a piece of art. And so I hope that through these kinds of books or chapters or articles, that people get a glimpse of that, that creativity that goes in to developing new mathematics. Is it important for you to rethink how math textbooks are written? Like what can be gained by reimagining it in this way or in another way? Yeah, I think primarily is the opportunity

that it gives for people to engage with mathematical content outside of an educational setting, right? Mathematics doesn't just have to be something that you do and you have to do very fast with, you know, you have two minutes on the clock, go solve all these little equations. Mathematics can be a thing that you can just have fun with, that you can play with, that you can create with. You have a way of bringing joy and whimsy into mathematics. Like, where does that come from? First off, thank you.

That is so ridiculously kind. I don't know. I think I honestly just love math. I think that's really what you're seeing. The joy in the discovery, when you understand that you are the first person in the world to solve something and that it was a question that you got to ask and that you got to develop,

how can I not be happy that that's what I get to do with my professional career? Pamela, thank you so much for coming on our show. Math is pretty daunting, but you make it really fun and, like, comforting. Thank you. Awesome. I'm super happy that you invited me. Today's episode was produced by Margaret Serino, edited by Gabriel Spitzer, and fact-checked by Britt Hansen. Stacey Abbott was the audio engineer. Giselle Grayson is our senior supervising editor. Brendan Crump is our podcast coordinator.

Our senior director of programming is Beth Donovan, and the senior vice president of programming is Anya Grenman. Thanks for listening to Shortwave, the Daily Science podcast from NPR.