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Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Evelyn lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, currently enjoying very beautiful spring mountains, which my guest and my cohost can see behind me in my Zoom background. And this is my co host.



Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. My wife and I are off to California this weekend. So, you know, she's she's a book artist, and there's a there's a big biannual, every two years is biannual, right? Or is that semiannual?




EL: Maybe?




KK: Who knows?




EL: Biennial is also a word. Maybe they’re the same word? [Editor’s note: dictionary.com says biannual can mean the same thing as biennial (every two years) or semiannual (twice a year). The Oxford English Dictionary, on the other hand, says biannual means twice a year and biennial means every two years.]




KK: Every two years. Except that this two years is this two years is three years, because two years ago was, well, anyway.




EL: Right.




KK: So yeah, so it's called CODEX, and she is exhibiting there, and I am tagging along because I like Berkeley.




EL: Fun!




KK: And we're going to do stupid things like spend too much for a meal at Chez Panisse.




EL: That sounds great.




KK: Yeah, we're really looking forward to it. So anyway, but let's talk math.




EL: Yes. And we are excited today to be talking with Tien Chih. Tien, would you like to introduce yourself?




Tien Chih: Hi, my name is Tien Chih, and I'm currently an assistant professor of mathematics at Montana State University Billings, which is a comprehensive teaching school in the middle of Montana. But later on this fall, I will be joining the faculty of Oxford College at Emory University. Oxford is college separate from the main campus of Emory that's a small liberal arts spin off, so students can do a small liberal arts experience with the first two years of their undergraduate degree at Emory and then go to the main campus to finish, so I'm really excited for that.




KK: Yeah. And that's that that's a little bit outside of town, right? So Emory itself is in Decatur, correct? And the Oxford College is where? I know it's not right there.




TC: Oxford, Georgia, about a half hour or 45 minutes or so east, I think.




KK: Okay. Nice.




EL: Yeah. Well, I am excited that we got you while you're still in Montana, because I just love having guests who are also in the Mountain Time Zone because that they don't think I'm, you know, a layabout because I never want to do anything before 11am.




KK: This is a common problem we have.




EL: Yeah, I am excited, I'll be going to Glacier, or I'm working on planning a trip to Glacier National Park for this summer, which I know isn't actually that close to Billings, because Montana is enormous. But I think it will be very beautiful. Some of the pictures that you post sometimes are really beautiful with the scenery up there in Montana. I'm sure, maybe you're a little behind us on the spring timeline, but similar mountain beauty.




TC: Yeah, I went to graduate school, actually at the University of Montana in Missoula, which is much closer to Glacier. So while I was a grad student I managed to go up there a couple of times, and you're in for a really good time this summer.




EL: Yeah, I'm excited.




KK: Excellent.




EL: Yeah, well, let's dive into the math. What math would you like to talk about today?




TC: Okay, so my favorite theorem is not exactly a theorem, or is a theorem, depending on your point of view. But my favorite math concept that I'm going to talk about today is mathematical induction. There are a couple of reasons why I chose this. One is that I am a combinatorialist/graph theorist. In our field, we don't have a lot of big foundational theorems or theories. In our line of research, we tend not to build skyscrapers, we tend to sprawl. And so because of that there aren't like these big foundation things, so it's hard to point at, like, one theorem and say this is a key theorem in our discipline, but the idea of induction is always present in our work, and especially in my work. Another reason I like induction is because I do a lot of math outreach kind of things. I'm involved in the Math Circle community quite heavily. I run a student circle here at MSU Billings. And mathematical induction is one of those things that almost all students, even children, intuitively understand. The actual mechanisms and formal logic, of course, is not something most people are familiar with. But this idea of if you have something that's true, and you know, you can do this thing, and it's still true, that this means that it just keeps being true, that’s something that students inherently just grasp right away, especially if you compare it with visuals. So I think there's a lot of concepts in math that are like this, that are technically kind of difficult results to articulate, but intuitively, everyone already understands them to some extent.




KK: Right.




EL: Yeah. And you mentioned visuals. So what are some of the visuals you might use to describe induction?




TC: So one of the classic induction proofs that you would give as an example or an exercise in an intro to proofs class is the proof that the sum of the first n odds is n squared. And very often with as presented, that's done with the usual algebra, and the flip, and then the 2k+1 and all of that. But you can easily show that if you take a square, and you take a two by two square, you have to draw a one by one by one extra L around the original square to get the two by two square. And then you get the three by three square by drawing a two by two by one L around the two by two square. And then you just say, Okay, you keep doing that, right. And most students or people without formal mathematical training will recognize, oh yeah, that just keeps happening. But the “keeps happening” is induction, right? That's what we don't say out loud, but that's inherently in this reasoning.




KK: That’s a good visual, because as you say, the algebra — I mean, so when we teach our sort of intro to proofs course, this is where students really get their first taste of induction. And I think it's kind of cold, right? So you say, okay, yeah, so it works for the base case: check. You know, so 1=1. All right, we understand that. And then you say, assume it's true for k and then show it's true for k+1. And I think students learn this mechanically, but I'm never sure that they really grasp what's going on.




EL: Yeah, well, and you if you've learned it by manipulating, you know, like k+1, and, you know, multiply that out to square it or something, that sometimes does remove you from actually thinking about the concepts. I mean, it's an important way to be able to work, but if you have pennies that you're arranging on a desk, or cards, or something in a square, that can be a lot more like, yeah, look what's happening. Here's the next odd number, here we go. Yeah. So just maybe to back up for a moment. Maybe we should actually, state, you know, what induction is? Because it is I mean, it as you said, it's an idea that a lot of people will intuitively feel is correct, but might not have have actually seen as you know, this like packaged, you know, with a little definition bow on the top kind of thing.




TC: Right. So the idea of induction is you need two prerequisites. One is a statement, at least one typically integer value for which that statement is true. So you have a statement, let's say capital P, at an integer k, and we verify that P(k) is true. And then we couple that with an argument that shows that anytime something is true for, say, an n, it must also be true for n+1. So P(n) implies P(n+1), then starting at k, the statement is true for all n greater than or equal to k, so for k and then k+1, and then the implication gives you k+2, and so on. And then you keep going.




KK: And so the visual that I sometimes use with students is you know, it's imagine you have an infinite line of dominoes. If you knock down the first one, they're all going to fall down, right? Which isn't exactly correct, but it's a reasonable visual.




EL: There’s this TV show — not to derail totally, but I'm about to — I think, I don't remember if it's called Domino Wars, or Domino Masters, or something like that it [Editor’s note: It is called Domino Masters, and mathematician Danica McKellar is one of the hosts]. We were in a hotel room flipping through channels, and we saw this and they make these domino things, and in fact, sometimes not all the dominoes fall down when you push the first domino because there's some sort of problem in the the domino line of implications there. But in mathematics, the dominoes are all set up at you know, just the right distance or angle that they do fall down.




KK: Ideal dominoes, right?




EL: Yes.




TC: Although sometimes the dominoes, you can find arguments that that end up skipping a step in some of the dominoes. And so I don't know if you've ever seen these induction, like non-proofs. But I give this one as a challenge to my discrete math students: All cows have the same color.




KK: Yes.




TC: Yes.




EL: I don't think I — so this sounds familiar, but I don't remember what it is.




KK: Well, I’ll let Tien tell us, but I have a story about t