 I'm very delighted to bring up someone to introduce tonight's speaker. Our introducer tonight is not a mathematician. She's actually a Tony-nominated actress, and she also has done work on television. She's just wrapped up a five-year series on Showtime called The Affair, and she'll soon be headed off to California for her next project. But tonight, she's with us to introduce our speaker. Please join me in welcoming Kathleen Chalfant. Thank you, Cindy. As you heard, I'm an actor, not a mathematician. But since the job of the actor is not to actually know anything, but to convince you that I know something, you can judge me for the next little while because my job will be to convince you that I have been flinging around polynomials since my infancy. And I will introduce Brian Hopkins, who has in fact been flinging around polynomials at least since his infancy, maybe before that. Brian and I know each other, as you might guess, not from math, but from music. Both of us work with the fierce, beautiful, adventurous choir known as Cantori New York. Brian sings with them, and when they need people to talk in between the singing, they sometimes call on me. Brian and I have been involved with two world premieres in the last, I think it's about five or six years. So we've had a wonderful time getting to know each other through Cantori. Brian does as a chorister what he does as a mathematician, which is to say that he takes things that are very, very, very, very, very hard, and transforms them into objects of beauty and order. He's a pretty good piano player too, I've heard. And you should know that he has sung in the crypt of the Canterbury Cathedral. So I have a list of things that I know about Brian Hopkins that I think you might like to know. Oh, I know firsthand that he is a wonderful guide to the museum because I came last week for the first time and he took me around and I felt almost as smart as all those five-year-old kids who know everything. Brian is a married man from Texas. His husband's name is Michael. Michael is here. They've been married for seven years since 2013, but they've known each other much longer so there are no itches or anything involved. He is a professor of mathematics at St. Peter's University, but he could have pursued life as a poet. And actually maybe that's what he's doing, pursuing life as a poet. His students are very, very lucky. He is the recipient of the Heimo Award for Excellence in Math Teaching. You'll get a taste of that in just a few minutes and you should know that normally people from, oh, California or Massachusetts get that prize. Brian is the first person from New Jersey ever to get that prize. He was also given the Polio Award for Excellence in Expository Writing. He's practiced his teaching arts all over the place, from California to Bangkok, and the NYU School of Political Science as well. He has written, it says on his thing at St. Peter's University, 20 papers, but that's not true. He's written at least 30, well actually 31 because one was just accepted today. He has edited and contributed to three books. He was the editor of the College Math Journal, which I suspect all of you read from 2000 and try to get into from 2014 to 2018. And then there is the fact that in middle school, he wrote a stamp column. He joined other famous full-adolist musicians like John Lennon, Ron Wood, Freddie Mercury. And that brings us back to polynomials because today Brian will help us rationalize our postage with polynomials. I'm not quite sure how forever stamps fit in, but I think that might come up. He's going to show us the relationship between polynomials and making sure that luck is a lady tonight and maybe, just maybe, he can help us enlist polynomials to make November turn out the way we all know it must. So here he is, Professor Brian Hopkins, who's going to make his polynomials sing for you. Thank you so much, Kathy. It's a wonderful introduction. So what we're going to talk about tonight are dice and stamps and counting. And all of you have this fun little double die that came on your seat, right? So there's a normal dice and there's another one inside it. So this has the benefit of allowing you to roll two dice at once and have less chance of it falling on the floor when you do that. Someone still will, and that's okay, but just hopefully less problems picking things up than would have with two normal dice. So the way this works, when you roll that, you're going to add the number on top and the number that's on top on the inside. So it's like doing two dice at once. Everyone do that right now. Roll it and see what number you get. Add up the top number and the top number on the inside. Anyone get a 12? Any box cars? Anyone get a two? Okay. How many people got seven? A lot more. Okay. So this is sort of why dice are an interesting thing to use in gambling, because even though the range of numbers you can get go from two all the way up to 12, the frequencies, though, those are quite different. So seven's going to come up a lot more often than 12 does, or two does, for that matter. So how many of you have seen this table sometime in your life? So this is the beginning of every probability lesson. You have the two dice, one, two, three, four, five, six, and you make this table of 36 numbers looking at their sums. And then you sort of count how many times each one shows up. So we have the sums from two up to 12, and then the frequencies, how often they come up underneath that. So a two just comes up once, but there are six different ways you could get a seven. So that's why a seven is more likely. So one could ask, is there a way that you could have dice that have different numbers on them that ended up having the same sums with the same frequencies? Not clear why you'd want to, but it's a question. Can you do it? And before you think about this too long, there's a way you might think to do it that is kind of not an exciting solution. So you could slide all of one set of numbers down. So instead of one through six, make it zero to five. Then slide the other numbers up. Instead of one through six, make it two through seven. And then the numbers on the inside don't change at all. So those are different dice that give the same thing, but it's not very different in a way, if you will. Or you could say, oh, I'll move some of them down. I'll move one set down by point two and move the other set up by point two. Yes, you'd get the same table inside. So the real question is, could you put other positive integer labels on the dice and end up with the same sums, with the same frequencies as normal dice? That's the challenge. Now I've had classes, capstones for senior majors and things like that, where the students are kind of trapped there for three hours. And I give them this problem. And then I just sit there like a spanks as they work on it. And they think it can't happen. They try something. They think it's going to happen. Then it doesn't work. And it just goes on and on. And I say, hmm, oh, okay. And then they slog through it. We don't have the time for you to suffer through all of that. So one thing that you do in mathematics often is you try to solve a smaller problem. And hopefully that will give you some guidance for the larger problem you care about. So how many people play role-playing games of some sort? So you may be aware that there are different kinds of dice. Not all dice have six sides. So there are, in particular, four-sided dice that exist. So let's think about those. So we'll have two normal four-sided dice with the labels one, two, three, four. And we get the same little table. It has now not 36 numbers, but 16 numbers inside. The number that comes up most often is five. So you can see the distribution down here. And now I'm going to ask the same question for that. Can I have other four-sided dice with positive integer labels that work out to give the table to be rearranged but the same sums with the same frequencies in them? Okay? So you see that in addition to your double die, you have a pad of paper and an opad and a pen. That's for the fact that you're going to be working on things. So this is the first time for you to actually try to do something. So see how you can do with this question. Play around with finding other different labels besides one, two, three, four for each die and try to get the same distribution of sums inside of it. And most of you are sitting near other people. Feel free, please, to collaborate. There's no reason for you to do math in isolation. So spend some time working on this and I'll wander through to see how you're doing with it. Feel free to talk to the people around you to bounce ideas off of and see how they're doing as well. Everyone clear on the problem, what we're trying to do. Nothing says they have to be different. Let's see. Oh, the thing is we're trying not to use it. So it has to be a positive thing. So zero is a positive. It would be nice if it were a positive thing. You have to be careful how you're... If the seven reasoning... I agree you can't have a seven. But does that same reasoning really work for six? Does it really work for five? So let me not finish things, but just bring together the one point I've heard several people make. If you start with the eight, this is a little challenging because there are lots of ways you can have two numbers add up to an eight. But if we start at the other end, we need to have a two. And how can you have two positive numbers that add up to two? Right, you have to have a one and a one. So whatever your dice... It says it... Right there. Okay? So one of these has to be a one. And that's... I either die. So that gives us the two. Right? So whatever dice you're coming up with, each one of them has to have a one. And then the next thing to think about is, okay, we need to have two threes. How can that happen? Right, so we need to have two twos somewhere. Now if we start to put a two on the top and a two down the bottom, we're leading towards the one, two, three, four that we were starting with, and we're trying to find something else. So does it work to have two twos on a single die? Can that play out? So let me just write down that part. If we have twos here, and then those are our threes, then a few things have to happen. The number down here has to be bigger than two, and also the number here next to the one, that can't be a two because that would give me another three. So this next number would have to be bigger than two. So if you haven't been down this little road, play with this. So everyone agrees that this is a feasible start? So see if you can develop this to something that gives, again, the sums from two up to eight and the frequencies is the one we started with. I think that almost works. So the problem is you have two fives, so that gives me two eights, and I only have one eight. Okay, let me see. I like that, yeah. So the tricky thing about an assignment like this is that it's very open-ended and people work in different paces and try different things, and some people find the solution and other people have not yet. So the difficulty for me then is that I have to do this or do we sort of forge ahead? And so is anyone going to be horribly offended if I say, let's go ahead, I'll show you how this works and we're going to proceed with the talk. Any violent oppositions? Okay, so we'll proceed. So some people found this. This is, in fact, possible. So if you label one of the die one, two, two, and three, and the other one, three, three, and five, you see that when you work out the addition table, there is a single two, two threes, three fours, the four fives there in the middle, three sixes, two sevens, and an eight. So it can happen. There it is. And with brute force and enough dedication and time, you could find this. Now for the six by six case, the original question, if it's longer, if it happens at all, and then of course, those people who do role-playing games know there are other kinds of dice that have even more sides. There are eight-sided dice and 12-sided dice and 20-sided dice and you would not want to brute force finding alternatives for those kinds of things. So our goal here is to come up with some other methods using some things you know subtitle of the talk. It's putting polynomials to work. So let's do a little background. So polynomials take up a lot of American high school education in mathematics. So you see something like this, x squared minus 11x plus 24 and you say automatically, what do I do? You factor it. Polynomials are made to be factored apparently. So does this one happen to factor? Anyone see it? Good. That's x minus 3x minus 8. So we spent a lot of time learning that for some reason. And then you go to physics class and you have polynomials that don't come anywhere near factoring. So you have things like this. So this is the description of the height of a ball. The minus 4.9t squared is the effect of gravity. The 10t is how hard you throw it off the building. 50 is how high you are when you throw the ball in the first place and the value where t equals 0 tells you when it hits the ground. And so you can't factor that to find the answer. So sometimes you have to learn something like the quadratic formula. So we're not going to visit that potentially dark memory. We're going to do other things with polynomials that are a bit simpler. So we'll need to remember two things. So how do you multiply powers of x? So what's x squared times x cubed? x to the fifth. Very good. So x squared means you have x times x and x cubed is x times x times x and you put them all together and you have five of them. So it's x to the fifth. And then when you take, when you have ratios what's x to the sixth divided by x squared? x to the fourth. Good, because that's really like six x's, two x's, you cancel them out and you have four left on top. Okay, those are the kinds of things we're going to do with polynomials. The connection we're going to make and we'll spend a little time trying to establish why this makes sense is I'm going to associate to a four sided die labeled one, two, three, four a polynomial. In particular x plus x squared plus x cubed plus x to the fourth. So you see that the value is one, two, three, four. Remember when you write an x by itself there's an implicit x to the first power. So there's like a one there that we usually don't write. So the exponents here are one, two, three, four and somehow that's going to correspond to the one, two, three, four on the die. So why is that? You have the benefit of being the second people to hear this talk. So there's a slide I've added to help drive this point home. It's not on this in the power in the beamer part but we have a document camera. So here's our table where we were adding up the numbers one, two, three, four. One, two, three, four. And my claim is that we could x, x squared, x cubed, x to the fourth with that somehow this is going to be the same somehow. So what I really want to do is I want to square the polynomial x plus x squared plus x cubed plus x to the fourth. And if you worked that out by hand you would multiply each of the things by x. And then you would go through and add up what happens when you multiply each of those things by x squared. Then you would go through and add together what happens with each of those things when you multiply by x cubed x to the fourth. So what's the difference between those two tables? The operator, one has x's and one doesn't. But this table is really just that table where I put an x underneath each number, right? So what this is showing is that when you're adding up the faces on the die that's encapsulated in the same way of adding the exponents when you multiply the powers of x. So for instance if you had a two and a four on the dice you would say okay two plus four is six. And that's going to be corresponding to saying I'll have an x squared from this first factor an x to the fourth there multiply them together, that's an x to the sixth. So now if I look at this polynomial when I square it and I bring all the common terms together so you see for instance there are two x cubes there are four x to the fourths so if I come back here and I actually write out what happens when I multiply x plus x squared plus x cube plus x to the fourth by itself it expands to be x squared 2x to the cube 3x to the fourth, etc. So where does that 3 come from? That 3x to the fourth came from doing the x in the first time the x cubed in the second the x squared in the first times the x squared in the second factor and then the x cubed in the first times the x in the second factor And those three things all simplify to x to the fourth. There are three out of them. That's why you have a coefficient of three. So the correspondence is that the sums are going to be recorded in the exponents of the expanded polynomial. And the frequencies will be the coefficients. So that's the magic connection between the dice and the polynomials that will help us do some of these things. So so far it's entirely analogous. It's like you just add a bunch of x's just to make it hard or something. But you will see that adding, putting into the polynomial context allows us to do some other things and find solutions to these problems more easily than what you just went through trying to find an alternative to the 1, 2, 3, 4. So to wit, we can think about what happens when you factor x plus x squared plus x plus x cubed plus x to the fourth squared. Now, if you haven't been in school in a while, you may not know that there are now these computer algebra systems where you can say, factor that thing. And it says, OK, here. So in the same way that you had a calculator before that would do addition and multiplication, there are now systems widely available. You've got a Wolfram Alpha online, for instance. And it will factor these kinds of big polynomials. If you were in high school and you showed someone the x to the eighth polynomial, they're like, ah, I've never seen that. I only see x squared. What is that? That looks too hard. But with this kind of computer algebra system, in fact, we're not trying to actually find the roots, it's really just this bookkeeping that it'll do for you. So if I start with this factorization, you see everything squared. Everything came up twice. If I made these two factors using one of each kind of thing, I come back to just the polynomial for 1, 2, 3, 4 twice. That's kind of where we started. So the question for finding the alternative dice, can I rearrange these factors in a way that's gonna give me dice with different labels? And in fact, you can. So what I'm gonna do instead of splitting things, I'm gonna put both 1 plus x squared in one of these, and then I'm gonna put 1 plus x squared squared, both of those in the other. And now when you expand those, the first one is x plus 2x squared plus x cubed. The second is x plus 2x cubed plus x to the fifth. Now what does that mean in dice context? Well, remember that's the invisible one. That's a two, and the coefficient two means there are two of them. And then the x cubed is a three. On the other one, we have a one. We have three twice, and we have a five. So that one, two, two, three, and the one, three, three, five that you spent a while working out kind of falls out from just playing with the factors of the polynomials, okay? So we can use this same technique for the original question about the six-sided dice that believe me can take hours to work out by hand. But instead what we'll do, we'll factor it and just play with how we can group these. Now this becomes a little delicate because we have a minus sign when it factors. You see there's that one minus x plus x squared term. That would be a problem if in our final polynomials we had negative coefficients. That would say I have negative two faces that are labeled three. That's not a good thing for physical dice. So we need to be careful to group those factors in a way that when we expand them that there aren't any negative coefficients. But we can do that. So here's a way. And this is really the only other way you can do it, showing that's a little bit more detailed than we'll get into. But it works out. And you get one is corresponds to one, two, two, three, three, four, and the other corresponds to one, three, four, five, six, eight. And if that seemed to slick, we can actually check this. So I can put this back in the tables. So there's the one, two, three, four, five, six against itself, and then on the right hand side is down the side one, two, two, three, three, four, and across the top one, three, four, five, six, eight. And if you look inside this table, they're in different places, but they're still one, two, two, three, three, fours all the way down. Everything works. And these have a name. These were discovered by a guy named George Stickerman of State in New York. And they were made known by Martin Gardner, wrote a popular column in Scientific American about mathematics. A lot of people got into math because of Martin Gardner columns. And so he talked about this. And then after this, there were spate of other extensions that people came up with in the math research journals and student journals and things. So we'll do a few of those. Just to get a feel for it. So while you have this special piece of plastic in front of you, we're gonna do a different operation on the double dice. So what I want you to do now, when you roll it, you're gonna add the number on top to double the number on the inside, okay? So roll your die right now and tell me what you have. You have a six on top and a five on the inside. So she had a six on top, five on the inside. The operation says you write down a what? 16. 16, so you do six plus two times five. So you're gonna record that result as a 16, okay? So just spend some time rolling your weird double die and write down the numbers you get. And so what are the ranges you can get? And is this the same as rolling three dice and do you think there's a way we could rewrite this using some of the polynomial stuff to get two different dice and give the same result? So just spend some time working with this and getting a feel for this operation on the double die. So this is also kind of an open-ended thing. So let's see how we're doing. So what's the range of values you can get with this operation on the double die? Three to 18. And that's the same as with three normal dice, right? You can go for three up to 18, right? So are they equivalent? Do they come up with this? No, people are shaking their head now. So we can check that by doing the frequency tables or we can just, we'll do instead the polynomials which kind of do it all for us. So what's really happening with the double die is you can think of it, I saw some people making this note. The doubling of the inside part is basically relabeling that inside die. So instead of being one, two, three, four, five, six, it's really, we're pretending it's two, four, six, eight, 10, 12. So when I wanna work out the polynomial, I'll use that instead. And I multiply it down and I get that polynomial, right? So what's interesting about this is that a lot of the numbers happen just three times, an equal number of times. And then as you get out towards the edges, some things happen just twice or just once. Now with three normal dice, there are a lot more things that can happen, right? So if we remember we were doing tables before. So if you're really doing three dice, you'd have to have a stack of tables. You'd have to have like a box of numbers or something because it's six by six by six. There are 216 possibilities. If you added up all these coefficients, down here they would add up to 216. And up here they just add up to 36. That's already pointing at a difference between what's going on. And you see that there's just one way to get a three, lots of ways compared to a way to get a 10 or 11. So even though these have the same range from three to 18, they have different frequencies, different probabilities. And then to answer the other question, just to practice this idea of coming up with different dice that have the same outcomes. If I take these two polynomials instead of expanding it, I factor it, I get that list of factors. And then to find other dice that do the same things, I just have to carefully tease those apart into two groups and expand them and see what happens. Carefully again, because they're more of these minus signs now. So when I multiply things together, I have to make sure that the combinations don't have any negative coefficients because remember those don't work for reality of actually labeling dice. I can't have negative number of faces labeled four or something. There are actually several ways you can do this. Here's one. So you make one group there and then you put the other factors in and you get these two polynomials at the bottom. So again what this means, the first one says our labels are one, two, three, seven, eight, nine on one die and the other is two, four, five, six, seven, nine. And to be more convincing perhaps that the numbers work out, you can just make those two tables. And you see that both of those tables have a single three, a single four, two fives, two sixes, et cetera, all the way down the same frequencies. So to finish up this topic about dice, let me just show you some of the other things you can do. People with role-playing game backgrounds know that they're 12-sided dice. So they actually end up being seven different ways you can label a pair of 12-sided dice and have the sums and frequencies the same as two normal 12-sided dice. So lots of different ways of doing it, not just the one we saw for four and six-sided dice. So suppose you're playing and you need to roll two four-sided dice and you don't have them. Well, what you can do is flip a coin and relabel an eight-sided dice die and it's gonna be exactly the same thing. Or you can do all these weird combinations. If you wanna roll a six-sided die and an eight-sided die, that's gonna work out to be the same as the four-sided die with those labels and a 12-sided die with those labels. So you can find all these different alternatives. At this point, I missed the overeagered five-year-old boy who was here the afternoon at this point said, that's so cool. So I know you were thinking it, but he said it out loud. So now let's go on to talking about stamps. And I realize we may need to do a little cultural background here. So even if you get paper mail still, which is not so common, it probably doesn't have a stamp on it. It probably has this mirrored kind of thing. Well, that's not so interesting mathematically. Or if you do see stamps, if you try to buy a stamp for the past nine years, they're forever stamps. So a forever stamp is good no matter when you mail it. The postage rate changes next year. You can still use your forever stamp to mail a letter. And last year, when I wrote the column that Kathy mentioned in middle school about stamps, stamps were about important things, about political figures or things that happened, or architects or whatever. Now they seem to make stamps about everything. So last year, they made Sesame Street stamps. And luckily, they included every mathematician's favorite muppet, the count. So you can go get forever stamps that have the count on them. And by the way, forever stamps started in 2007 when the postage rate was 41 cents. So if you'd had the foresight, you could have bought stacks and stacks of 41 cent forever stamps. And now you could sell them and you have had a 2.5% annual percentage increase in your investment. Not very good to compare to socks, but probably better than your bank account. So what we're talking about are stamps that have denominations. So the person who made the poster for this talk dug up these stamps. So on the left-hand side is a 4 cent stamp from 1901 showing off this brand new technology called an automobile. And then the 7 cent stamp came from when Alaska became a state. If you happen to own any of these 4 cent stamps, you should donate them to the Museum of Mathematics. They're worth hundreds of dollars each. So now we're gonna spend some time coming up with what we can do with 4 and 7 cent stamps. So the first question is a double question. So can you make the current first class postage rate just using 7 cent and 4 cent stamps? So I'm better than this. Who knows what the current postage rate is? 51 cents. 55 cents. 55 cents. So the question is, can you make 55 just using 4s and 7s? So 12 4s and 1 7 is one answer. Anything else people can think of? Five of each. Five of each. Those both work, right? So sometimes there's more than one way of doing these. So certainly 12 4s make 48, add the 7 you get 55. Or you can think of the other one as being five times the combination, five times 11 is 55. So the next activity, you have four sheets of papers, so there's more to do. What are the values you can make with 4 cent stamps and 7 cent stamps? And kind of start from the bottom. So can you make three cents? No. You can make four. Can you make five? So start down there and work your way up and see what you notice. That's a scowl. Yeah, so you'll get a whole bunch at some point. Right. Well, so is there a point, so think about this one. Some things you can't get. And if you can get everything from some point on, what is that point? So are you kind of comfortable with where you are on this? Okay, so I've seen a lot of people work this out. So there are certainly lots of things you can make. So here's a smattering of some of the small things you can make. I've written this, you'll understand the order I put these in in a second. You can make any multiple of four, right? You can make seven, you can make 11, which is four more than 11, et cetera, 15, 14, and then you hit a point where you have four in a row, right? You can make 18, 19, 20 and 21. And at that point, I think you can see, we'll explain why you can make everything higher than that. Because all I have to do is just add four to each one of those things. It'll just keep adding up, right? So after doing 18, 19, 20, 21, whatever those combinations were, I just add a four-cent stamp to each of them. I have 22, 23, 24, 25, 26, 27, 28, 29, et cetera. It just goes on and on and on, okay? So working this out may have felt a little ad hoc. So you might wanna go back and think about tables like it do with the dice. So what's tricky about the table though, is that we have this innocuous little dot, dot, dot that goes off to the right and down the bottom. And dot, dot, dot's a powerful thing, but it has a little dangerous, right? So it's just going off forever. So some of the things we just talked about, to see that you can get all the numbers from 18 on up, you just find 18 there in the third row, 19 in the second row, 20 in the first row, 21 in the fourth row. And then, just panning off to the right from each one of those, to see all the other numbers higher than those. The other thing is that there's a lot of redundancy in this table. So this next picture I've highlighted all the numbers that show up two or more times. So in a way, there's this table, even though it goes off in all directions, this sort of doesn't need to at some level. So what we wanna do is think about the polynomials now with the stamps. Now, with the dice, it was kind of a direct thing because the dice only have so many faces and we just make the polynomial corresponding to the faces. Here we can have as many force and stamps as we want. So the polynomial's gonna be one plus x to the fourth plus x to the eighth plus x to the 12 plus dot, dot, dot. So I'd like to have a better way of dealing with that than having to have the dot, dot, dot that goes on forever. So what I wanna do is multiply that by one, minus x to the fourth. And why do I wanna do that? It's because it works. So it's a little hard to motivate, but you'll see that it works out very nicely when you do this. So when I multiply that long thing by one minus x to the fourth, when I multiply through by one, I just get the same thing back. One plus x to the fourth plus x to the eighth plus x to the 12th. When I multiply it all through by minus x to the fourth, you see I get minus x to the fourth, minus x to the eighth, minus x to the 12th, minus x to the 16th. So then when I add those two rows of numbers together, the one is there, everything else cancels out. So if I divide both sides by one minus x to the fourth, I get this strange thing that one plus x to the fourth plus x to the eighth plus dot, dot, dot equals this fraction. One over one minus x to the fourth. This is strange, right? So this is saying this fraction with this little x in it, x to the fourth, equals this infinite positive polynomial. It's a very strange thing we're saying. So in general, this is called a geometric series. So if I just have anything are raised to powers and I add them together, it simplifies in the same way to one over one minus r. And it's the title of this slide. Who's ever heard of Zeno? So Zeno was a pre-Socratic philosopher, best known for his paradoxes against motion. So if I'm standing here and I wanna walk to that corner, let's say that that's two away right now. So before I get there, I have to walk a distance of one, halfway there. And then before I get to you at a distance of one now, I have to walk half of the way. So I go half. And then I have to go half the way still and that's gonna be a fourth. Then I have to go half the way still and that's an eighth. And so Zeno's argument is I never get there, right? I always have half of the way still to go. And of course we know, yes, you do get there. And he knew too, you can walk someplace. But the idea is that the paradox is that if you think about it in this way, it doesn't quite make sense. You can actually ever get there because you're adding together infinitely many things. How could you possibly add together infinitely many things and get a finite number? If we put this in the context of the geometric series, it plays out like it needs to. So here's the one, the one-half, the one-fourth. Those are all powers of one-half. And if I put that in the formula, I'm looking at one divided by one minus the half and that gives two. And that's exactly what it needs to do. So this was a challenging point for the ancient Greeks. David Richardson, who talked here in the fall, thinks that Archimedes was really close to figuring this out, but he got killed by a Roman soldier too soon. So calculus might have been developed a long time before and it'd be a different world now. But instead we had to wait quite a long while for Newton and Leibniz to come around and make more sense of this. So if we think about this in terms of symbols, which is how we'll do it, it's all fine. If you think about plugging numbers in, it starts to get really kind of strange. So if we put in R equals one, the left-hand side is I'm adding together infinitely many ones and the right-hand side, I'm looking at one over one minus one, one over zero. Ooh, you're not supposed to do that, right? It explodes. So that's strange. If I put in two, I'm looking at one plus two plus four plus eight, that's an even bigger infinite number somehow. And the right-hand side, I have one over one minus two, which is negative one. So don't think too much about that. We're not gonna be plugging in numbers. We're just thinking of these as ways of keeping track of variables. For right now, it's just kind of a shorthand, although we'll see later it's gonna be a powerful tool for some other things. So if I wanna think about the stamps again, to see what four and seven stamps can give me, I just have one over one minus six to the fourth times one minus six to the seventh. So that one minus six to the fourth is one plus six to the fourth plus six to the eighth, dot, dot, dot. One minus six to the, one over one minus six to the seventh is one plus six to the seventh plus six to the 14th, dot, dot, dot. And you have your favorite computer algebra system work all this out for a while, and you get this. And this four, seven, eight, 11, 12, 14, 15, 16, 18, and every higher number, those are exactly the things we found before that we can make with four and seven cent stamps. So we're getting the work you did by hand for free from the polynomial. Now you notice that some of these have higher coefficients. So that's two x to the 28th and two x to the 32nd. That's because there are two different ways of making those values. I wrote out here the two different ways you can make 32 before you found two different ways of making 55, right? And that's all recorded in the same polynomial. So some people in the room have heard about this recently in a class. Sort of the more interesting question here is where are the numbers you cannot make? And this is important enough, it has a name. So if I have positive integers A and B that don't have a common factor, that's an important point. The largest value you cannot make with A cent stamps and B cent stamps is called the Frobenius number. Frobenius was an 1800s mathematician who thought about this. So we just worked out that the Frobenius number of four and seven is 17. That's the biggest number you can't make. So let's see if we can get the polynomials to tell us this number. So what I'm gonna do, what's one over one minus x? Well that's just one plus x plus x squared plus x cubed. It's all the powers of x. So that's gonna be all the numbers. And I'm gonna take away, I'm gonna subtract the numbers I can make with four and seven cent stamps. And you work this out, and this is that list we saw. One, two, three, five, six, nine, 10, 13, 17, and then there's other stuff. Maybe some numbers came up more than one, so when we subtract it, they're gonna have negative coefficients. I have a minus x to the 28th, minus x to the 32nd, and that's not as good. So we wanna come up with an adjustment that stops at x to the 17th. It gives us just the list of numbers we cannot make. So the next slide's a little technical, but what we can do is instead of doing one over one minus x to the fourth times one minus x to the seventh, in the numerator I'm gonna replace that with one minus x to the 28th. And the heuristic for that is that by subtracting that x to the 28th, I'm taking away the redundancy of doing four sevens versus seven fours. Because what you see when you work out this polynomial, all those higher coefficients are gone. There's no longer two x to the 28th, it's just x to the 28th. It's no longer three x to the 56th, it's just x to the 56th. So that adjustment makes it show that you only get the numbers that you can make, and then you don't record how many ways you can make it. It either shows up if you can make it or it doesn't if you cannot. So now I do the same thing. One minus one over x gives me all the numbers. I subtract this adjusted thing that gives me the numbers I can't make, or the numbers I can make, and then that gives me the result is exactly the polynomial and the exponential in the numbers that you cannot make. Okay, so that's our list. One, two, three, five, six, nine, 10, 13, 17. Now the one we care about the most is the 17. So I don't even want to do this much work. I don't want to have to work out that whole polynomial. I really just care about what's the biggest exponent that comes up in this polynomial. What's called the leading term. So I'm gonna do one step towards simplifying this polynomial. So you know often though, when you work with fractions, sometimes your job is to make them find a common denominator and make one fraction out of it. You may have heard that language. So I'm started doing that. And all I need to do at this point is notice that on the top, the polynomial, the biggest exponent that comes up is x to the 29th. Because I have an x to the 28th there, I have to multiply one minus x when I combine things. On the bottom, x, x to the fourth, x to the 12th, x to the seventh gives x to the 12th. And there's a lot of stuff after that, but I don't care. So I have an x to the 29th, an x to the 12th. What happens when you divide them? What do you do? Subtract. 29 minus 12 is 17. So the thing that we need to know for this to work, as we saw this for this case, is that when I simplify these two, when I simplify this ratio of polynomials, it ends up being a finite polynomial. That's not something I'm gonna attempt to convince you of. We'll just sort of take that for granted right now. Let's do another example just to drive this home. So instead of four and seven cent stamps, imagine you have three cent stamps and $2 stamps. 200 cent stamps, right? So if you're looking at all the things you can make, for a long time all you can get are multiples of three. All the way up to 198. Then you can use your $2 stamp and get more numbers. You still get all the multiples of three, that's the 201 going on. But you can also get $2 and 203 and 206 and 209, et cetera. And then finally, you get high enough and you can use two of the $2 stamps and now you can get everything. So what's the largest number that I cannot get? So it's a little easier if I actually draw in the numbers we don't see. So this is the same thing, but I've added in brown the numbers that we don't make. So all the way up to 200, all I get are the multiples of three. I don't get one, two, four, five, et cetera. Then starting down here, I don't get 199, but I get 200, 201. I don't get 202, but I do get 203, 204. So now what's the biggest number I don't get? 397, right? The biggest brown number there is 397. And this works out the same way in the polynomials. So if I do exactly the same thing, but instead of using four and seven, I use three and 200. I mean, again, think about your high school self, 200? X to the 200th power? What in the world are you doing? But we're doing this just kind of do some bookkeeping in a sense. So I work this out, and on top, I have a polynomial whose highest term is 601. And smaller stuff, I don't care about the smaller stuff. And on the bottom, I have X to the 200, X cubed and X to that's 204. And again, when you have a division, so again, we have to assume this is gonna work out nicely. But if it does, what's 601 minus 204? 397. So now if you believe this, you have two examples, we have a general result. The Frobenius number works out to be for A and B, relatively prime is another way of saying they don't have any common factors. It works out to be the product AB minus A minus B. Nice little formula. And here's why it works. It's exactly the same thing we've done just with the A's and B's instead of the numbers. So I make this fraction, remember what this thing is doing? This is recording all the numbers minus the numbers I can make with A and B. So it's gonna give me an expression with the numbers I can't make. It's a ratio on top, the highest exponent is AB plus one. On bottom, the highest exponent is A plus B plus one. I take the difference, I get AB minus A minus B. So this was done by Sylvester in 1882, another example of something where the person for whom the problem is named is not the person who actually solved it. Lots of those in mathematics. And so I'll finish this little chunk with explaining that this was just for two numbers. You could also ask about three numbers. So there was something called the McNugget problem for a while. At one point McDonald's sold chicken McNuggets in boxes of six and nine and 20. And what does a mathematician do? They say, oh, what numbers can I make and not make with those? And if we had more time, I would let you work on that. We don't, so I'll just tell you. You can't make 43 McNuggets using those combinations unless you throw things out. If you do that, then that's a boring problem. But from 44 and higher, you can make everything. So the Frobenius number for six, nine and 20 is 43. What's strange though is that for three values, it's a much harder problem. I mean, for a given three numbers, you can figure it out. But there's nothing general, like the nice formula that we had for two numbers. And it's not just that someone hasn't found it yet. People actually proven that there cannot be a nice simple formula in A, B, and C that gives you the Frobenius number for those three numbers. So math can be kind of depressing that way. Not only do people find things out, they also can prove that things cannot be done. It's not that you haven't tried hard enough, it's just impossible. And right. Lots of McNuggets bought to figure this out. So the last thing I wanna talk about are a concept in math called integer partitions. It's a very naive thing. I'm just gonna ask about how I can add numbers together. So I'm gonna just look at all the different ways I can take positive numbers that have a fixed sum. So all the partitions of four are listed there. I can have four by itself. I can do three plus one. I can do two plus two. Two plus one plus one. Or one plus one plus one plus one. And I don't care about the order. So one plus two plus one isn't different. So there are just these five different ways of getting partitions of four. So your last task is to work on partitions of nine. I mean lots of those. But you're gonna look at just some special ones. So this half of the room, your job, is to count how many partitions of nine there are where all the parts, all the sum ends are odd numbers. And this side of the room, your job, is to write out count all the partitions of nine where no number is repeated. So it doesn't have to do with even an odd at all. You can use any number as you want. You just can't repeat them. Right, so if we think back at our examples of four. So this side of the room is gonna be happy with three one and all ones, right? Cause that's the things that are made of odd numbers. You're gonna be happy with four and three one and nothing else, because everything else has repeated numbers. So work on that with the people around you for partitions of nine. So let's finish up. So how many different things is the odd group getting? So I'm hearing different numbers. Can I see your 10? That one isn't good, cause of two. And we don't count the order as being different. Yeah, yeah, yeah. Now that brings them down to eight. And how many did the group doing just no repeated parts? I hear, I see a seven, I hear an eight. There are more than three, I guarantee you. So here they are. It ends up there are eight of them over here that are made of odds. Is that okay? And then there are eight over here that have no repeated parts. Okay, so I'm nefarious like this. Maybe I picked an example where they happen to match up, that's just a fluke. But no. So actually this happens all the time. And so our final thing will be explaining why this always happens. So the first thing I want is to bring in our friends the polynomials to do the work for us. So there's a way of doing partitions with polynomials and it's really something we've already talked about. Partitions are like having stamps of every denomination you ever wanted. You have one cent stamps and two cent stamps and three cent stamps. So it's just gonna be this same thing where I divide by one minus x to the whatever except I do it for all of them. And so when you expand all of this, you see we have a five x to the fourth. Now where did that five x to the fourth come from? Well, one of them came from the x to the fourth way out here in the fours that I didn't write out is in the dot, dot, dot part. One of them came from doing x cubed times the x down there. One of them is just staring at us, the x to the two plus two, the two twos. One is this x squared and those two ones. And one is further down this little dot, dot, dot four ones. So you see that these terms which all simplified x to the fourth and make five of them correspond exactly to those partitions we listed in the first partition slide. So this is how the polynomials encapsulate the partitions. So the theorem, we're going back in time, now this is from Leonard Euler in 1700s, he proved, he was the first person to really think about partitions seriously, that the number of odd part partition of n equals the number of distinct part partitions of n. And we have all the tools now to understand why this is true. So for our final tour de force, we're going to understand why this proof works. So I'm going to use generating functions of course, these polynomials. If I want to look at the odd part partitions, I'm only going to put in the odd numbers down here, one minus x, one minus x cubed, one minus x to the fifth. And when you expand that out, you see that you end up with eight times x to the ninth and that eight corresponds to the eight odd part partitions that this half of the room found. For distinct part or parts with no repetition, it's even easier. I don't have to use the geometric series at all because either I have a one or I don't. I have a two or I don't. I never have two twos or three twos all the way down. There are no dot, dot, dots there. And so when you expand this, again you get eight times x to the ninth and that eight comes from those eight partitions you found where nothing was repeated. So here's the proof. We can prove this now. So there is that generating function we talked about for odd part partitions. So just one minus x, one minus x cubed, one minus x to the fifth. Now, I can multiply the top and bottom of a fraction by the same thing and it stays the same. May seem a little strange to do but I can multiply top and bottom by one minus x squared and one minus x to the fourth and one minus x to the sixth and all the way down. You may ask me why I'm doing it but it's legal to do, right? But now, what can we do with one minus x squared? How does that simplify? Right, I sort of disparaged factoring before but now we're gonna use it. That factors as one plus x times one minus x. What about one minus x to the fourth? Right, so it's the same difference of squares for me that's one plus x squared times one minus x squared. Of course, I'm gonna do the same thing with one minus x to the sixth that becomes one minus x cubed, one plus x cubed. So I've made this a lot more complicated. What are you doing, right? So this thing that had started out here now has all the stuff going on but what happens? I have lots of things that are on the top and the bottom in the numerator and the denominator. So there's a one minus x on both the top and the bottom. There's a one minus x squared top and bottom. There's a one minus x cubed top and bottom. There's a one minus x to the fourth out there in the dot, dot, dots. There's a one minus x to the fifth out in the dot, dot, dots. So when I cancel all of that, what am I left with? Just the one plus things and all of those together is exactly what we said was a generating function for distinct part partitions. And that's the proof. So this is sort of the poster child for generating functions and the power they have and how they can do really elegant things. This is what they call the proof from the book, the very elegant way of explaining why this is true. And there's a whole field of study of partitions that starts from this and goes on to this day but now you have a good foundation for it but we're at the end of the talk. So I appreciate your time. One more hand for our speaker. Thank you.