>> Amy Draves: Thank you for coming. My name is Amy Draves, and I'm pleased to welcome Jordan
Ellenberg to the Microsoft Research visiting speakers series. Jordan is here to discuss his book, "How
Not to be Wrong," in which he tells us how to use math to extend our common sense and to see
through to the true meaning of information we often take for granted. He is a professor at the
University of Wisconsin-Madison and lectures around the world on his research in number theory. He
has been featured on the Today Show and NPR, and his writing has appeared in numerous publications,
including Wired, The New York Times, and the Washington Post. He is the author of the Slate column,
"Do the Math." Please join me in giving him a very warm welcome. [applause]
>> Jordan Ellenberg: Thanks so much. It's great to be here. This is my first visit to Microsoft Research.
So on the way here, actually, I was chatting with the fellow next to me on the airplane, as one does, and
he is like, "Well, what are you traveling for?" I said, "Well, I'm giving a talk about a book I wrote." He
said "What's the book about?" I said, "It's about math," and then he said, "So what makes your book
different from all those other books by those math guys?" [laughter] and well, I hadn't really been
expecting the conversation to take an aggressive turn so early. [laughter] So I pretended to fall asleep,
as one does, and while I was pretending to sleep, I thought about what I should have said to this guy,
who is now somewhere else. So I will -- maybe I'll sort of say it to you. I mean, one thing that I try to do
in this book, "How Not to be Wrong," which you kind of can't do in a more standard journalistic 1200word-at-a-time format is to really trace out all the crazy connections between all the parts of math, the
way it is many things but it's also one thing, everything together in a big network, which I think is, for
people who do math, so much one of the characteristic features of the subject, it's underlying unity.
What this means is that the book kind of goes all over the place and is about a lot of different things, so
to kind of talk about this book project in a short talk is not really possible. I would have to say sort of
like ten seconds about a lot of different things. So maybe just for the first slide, I'll just say some of the
things I won't talk about in this talk, but we can talk about it after if you want to.
The Laffer curve and taxation. The triumph of mediocrity, which is sort of on old-school piece of
business physics that people were interested in in the 30s. The sort of questionable nature of public
opinion. The existence of God. There's actually a lot of that in the book, sort of more than I expected
there to be when I started writing it. The soup you get into when you start thinking about negative
numbers and their proportions. False assumptions of linearity of the Supreme Court, David Hilbert,
Derek Jeter, those three are sort of all one thing that are connected, and I could have put, sort of, the
theory of umpiring in there too, which is sort of the glue that draws that together.
But I figured that in this context it was probably better just to tell one story, and that's what I am going
to do. And even this story, I'll sort of only tell part of it. So the story I want to tell has to do with the
lottery and thing if I click on this it's not going to play a movie, but I am going to check what happens if I
click on. It. Okay, that works as a still picture too, actually, to tell the truth. So this is a picture of the
drawing from a lottery called the Massachusetts Cash WinFall.
Really the only reason to play the movie is because it's good to sort of enjoy the kind of comedy Boston
accents that you hear when you hear the drawing of the Mass lottery, but those of you who have been
in Massachusetts can just image it. So what this is a picture of is actually the very last Cash WinFall
drawing that took place in February of 2012, and the point of the story is to talk about why this was the
very last time that Cash WinFall was played. And the mathematical notion that I'm trying to explicate
with this particular story is the notion of expected value, so let me quickly sort of either tell or remind
you about that.
How does the lottery work? We all know. A ticket costs a certain amount, and you have a small chance
of winning a big prize. In this case, I made the prizes sort of not so big so we don't have to have large
numbers on the board. Maybe there's a one-in-200 chance that you'll win $300. How do we think
about how the lottery works? Well, if you play once, probably you'll get nothing. Maybe you'll get a
large prize, but if you play a lot, and people who play the lottery often do play a lot, let's say you play a
thousand times. Let's say you played for about three years. Well, if you have a one-in-200 chance of
winning, you'll probably win, okay, about five times out of those thousand plays, which means that you'll
take home about $1,500.
One way to think of that is if you had played a thousand times and won $1,500, you've won a buck fifty
per ticket. So this kind of relatively unobjectable computation underlies what we mean when we talk
about expected value. The expected value is simply the amount you expect to win per ticket. But I want
to point out this is one of these moments that occurs again and again where the mathematical use of a
word sort of pulls away from the ordinary English use, something that causes sort of so much trouble
when we speak with civilians, right, about mathematics, that the expected value is not at all the value
you expect the ticket of as the notation might suggest. You don't expect the ticket to be worth $1.50.
Indeed, the ticket cannot be worth $1.50. That's not even a possible value the ticket can have. It's
either worth nothing or it's worth $300 so if somehow we could go back to the dawn of mathematical
notation, we would probably call this something more like average value because that's really what it
represents. It represents the average over many plays, whether it's many plays by a single person over a
long period of time or many plays by many, many, many players at a single time for single drawing.
And in particular, let me remark on the sort of fundamental inequality going on here. $1.50 is less than
$2. That's important. $2 is how much you are charged for a ticket and a $1.50 is, on average, how much
you're going to win. And the expected value is less than the cost and lotteries are usually like this. Let
me explain why. Because I may say, well, maybe don't care what the average is. I'm not buying a
thousand tickets, maybe I'm only buying one. But the state does care about the average right, because
the state is placing lots of those lottery bets on the other side, thousands upon thousands. And if the
state is, on average, handing out $1.50 for each $2 it takes in, then the state is making money, which is,
after all, the point of the lottery.
A lottery which, on average, gave money from the state to the players would not be doing what a lottery
is supposed to do, which is raise revenue. So let me show you. This is very typical of real-life lotteries,
let me show you the prize matrix for a real lottery. And this is sort of the basic prize structure for
Massachusetts Cash WinFall, the game we started with. And what these numbers literally are doesn't
matter so much. I just wanted to show you that I did look them up.
When you can check is that the expected value of this two-dollar ticket, I mean you would be delighted
if it were $1.50. It's, in fact, much less. It's about $.80. So this is what you would call a pretty bad play
for the player and a pretty good play for the state. So here's the situation. Massachusetts had this
lottery. The jackpot probability was set quite low. Nobody one won for a year. People were getting
demoralized and depressed. People were not playing, and that's bad if you're the state. I mean, you
really can't take their money if they don't play.
So the state decided to change the rules. Here's the rule. They introduced a thing called the roll down,
which worked like this. They said, well, the jackpot is getting bigger and bigger. Maybe that should be
enticing, but when nobody wins people get depressed and demoralized. Instead let's say when the
jackpot hits a certain threshold, when it goes over $2 million, there's going to be a roll down. That
means if nobody wins the jackpot, instead of money just kind of piling into the pot, that money rolls
down to enrich the lower tier prizes to make them worth more money, those prizes that you actually
might feel you have a much better chance of winning. Maybe I'll look back for a second. You know, for
instance, if you match four out of six, which you have a one-in-800 chance of winning a prize like that.
Maybe you've never won but you probably know somebody who's won if you know a lot of people who
play the lottery. That's sort of somehow within the realm of realistic possibility.
This was Massachusetts's plan to make a lottery that seemed like a better deal for the player. And it
worked. It worked too well, because what Massachusetts had done inadvertently is made a lottery
which actually was a good deal for the player, which is not -- let me show you what the WinFall payoff
matrix looked like on February 7, 2005. It looked like this: That four-out-of-six prize, which used to be
worth $150 on a normal, non-roll down day, on this day, with the extra enrichment of all that jackpot
money, was worth almost $2,500. That's pretty good. If you have a one-in-800 chance of making about
$2,400, that is $3 worth of expected value right there for your two-dollar ticket. And, in fact, the
expected value of the two-dollar ticket on the roll down day was about five and a half dollars.
I mean, [laughter] you're allowed to get out and check your phones and see if any of these games are
currently being played. They're not; I checked. So why do I know the exact value of payoffs on that
particular day, February 7, 2005? It's because I got them from the following document, which is
probably a little hard to read here, but it's not so important that you can read it. Let me just tell you
what it is.
This is a 27-page report from Gregory W. Sullivan, who is the inspector general of the Commonwealth of
Massachusetts, writing to the state treasurer to explain just what the heck had happened with their
lottery game.
I have to say it's kind of an amazing story, and I'll tell part of it today. This is definitely the only state
fiscal oversight document written by a municipal bureaucrat which will make you ask, "Does somebody
have the movie rights to this?" It's really pretty amazing. [laughter]
What happened? Well, you can kind of imagine what happened. What happened is that pretty quickly
after the introduction of this game, which they had borrowed -- they borrowed this game from a game
in Michigan, also called WinFall which had closed down. Pretty soon after the introduction of this game,
people started to catch on to what this game actually was. So let me tell you a little bit about the sort of
main characters. The main group I you about is a group of players called random strategies. These were
a group of MIT undergraduates led by a fellow called James Harvey, who, by an incredible stroke of good
luck, was in his senior year writing an independent study project about the expected value of lottery
games when this game was launched in January 2005.
As sort of part of this project, he was like, well, let me compute it for each game offered by the state of
Massachusetts, and then, you know, presumably, his pencil fell out of his hand.
And he got a bunch of friends together, which, at MIT you can gets a bunch of friends together who can
compute expected value, and these guys got together and started making large bets. So, in fact, the
reason that we know the payoffs for that exact date, February 7, 2005, is that that was the day where
James Harvey and Yuran Lu bought $14,000 worth of lottery tickets at Star Market in Cambridge right by
MIT, which was their first big play.
Another group was the group led by a guy called Jerry Selby, who was a retired engineer from Michigan.
He sort of had about thirty of his family members in on this. I actually mentioned Michigan a minute
ago. Do you guys remember why? Why did I mention it?
Yeah, so this was the guy -- this was like the veteran. This guy had been doing the same thing in
Michigan for two years until they closed down the game. And then, I mean, you can image his
incredulity and joy when, six months later, he read that another state was opening the same game. And
immediately he and his wife got in their car and drove to Deerfield, Massachusetts, in the corner of the
state, which I think is geographically the closest Massachusetts convenience store to the state of
Michigan. And they started playing. And then there was a whole nother group called the Dr. John
lottery club. Biomedical researchers, bought a lot of tickets down in Quincy.
So all of these guys were playing that game like mad until, rather late in the game, on typical a roll down
day, 80 to 90 percent of all the tickets being purchased in the lottery were being purchased by a
member of one of these three groups [laughter].
Each group was typically making plays of about 200,000 tickets at time. The end is a bit anticlimactic.
The end of this story looks kind of like this. It looks like a front-page story in the Boston Globe explaining
what was going on with the Massachusetts State lottery.
This is called "The Game with a Windfall for a Knowing Few." And as soon as it this story runs, the game
doesn't end immediately, but basically the game is up, because for this to work, it does actually require
that people are playing not on the roll down days. It requires that the sort of general public has some
confidence in the game. Running a lottery at times a very psychological enterprise.
You have to convince people to play, and once people get the sense that the lottery is somehow not
quite on the level, then it's all over. So the last drawing I showed you takes place just a few months
after this expose by the Boston Globe takes place. So that's the basic story. I tell it in more detail in the
book. And actually let me just say one thing that was amazing that happened. I was talking about this
stuff in a bookstore in Palo Alto, gosh, I guess just two nights ago, and Yuran Lu came to the bookstore
and I met him.
I mean, you don't often get a chance for, like, a character in your book to, like, show up at the reading.
And it was very weird. It would be like if somebody from Narnia came up and was like, "Let me tell you
how it really went down." [laughter]
So that was extremely amazing. So anyway, yes, they are real people, I can now verify. Anyway, what I
want to talk about is two mathematical puzzles that this story presents us with. One is an easy answer.
One is a more complicated, mathematically richer answer. Puzzle one is how can anybody possibly get
away with this? I mean, remember the state lottery knows who wins, right, because they have to pay.
So they know that, like, oh, interesting, like a thousand winning tickets were, like, sold at the same store
and that happened last time too and also the time before that.
I mean this is not -- it somehow doesn't take Sherlock Holmes to sort of figure out that something
nonnormal is going on in the play of the lottery. But this puzzle, which sounds hard, actually has an easy
answer.
The answer is that the state did figure it out, and I think it's quite safe to say that the state knew almost
from the very beginning exactly what was going on with this lottery.
In fact, one reason we know that is that we know that James Harvey went to the state lottery office
immediately upon doing this computation and said, "Hey, given the way this lottery is structured, my
plan is to got as many people to loan me money as I can and then buy tens or hundreds of thousands of
tickets and make lots of money. Is that legal?" Because, you know, you guys probably know kids who
went to MIT or you went to MIT. These are kind of good, like, rule abiding, good student types of
people right, so. Okay, maybe people disagree with that .[laughter] Anyway, well, this guy was like that
anyway.
And we don't actually know what happened in this conversation on the lottery side, but it must have
been something like, "Go ahead. Knock yourself out" because a week later he and his friends make their
first play.
So that puzzle creates a subpuzzle. If the state knew about it all the time, why did they allow it to go
on? Like what kind of casino allows a player to come in and beat the house week after week for a period
of 7 years? And they end up taking, by the way, about $3.5 million in winnings from the state over the
course of this whole thing.
Well, I will explain this with a very sophisticated diagram. This looks like this. See, let me explain how
the lottery looks from the viewpoint of the state. A two-dollar ticket is sold and Massachusetts takes
$.80 out, and that's the state's revenue from the lottery ticket. That $.80 is going to go to pay police
officers and fund schools and pay the salaries of state university math professors and other absolutely
critical functions of state government.
That's what -- and the other $1.20 is going to go back in prizes. The state doesn't keep that. So what
that means is that the state doesn't care who wins, doesn't matter. The state only cares how many
people play the game. If these guys are coming in and buying hundreds of thousands of extra ticket,
that is a win for the state.
So one way to think of it is the flow of money here is not really from Massachusetts to random
strategies; it's from the other players to random strategies with Massachusetts taking a cut at every
stage. So really, I think the way this was played out in the newspapers, in the Globe and other places,
when this was being revealed was that random strategies and the other big bettors were somehow
beating the house. But what were they actually doing? They had found a bet that had a positive
expected value and they were making it again and again and again. They were making it in such large
volumes that they were very reliably winning money. So if that's what you're doing, then you are the
house. That's who random strategies was, and the regular lottery players were like the players at the
casino. So who is Massachusetts in this scenario? Who's the state? Well, I think the easiest way to say
it is that the state is the state.
I mean, here's another very sophisticated picture, which I think is a pretty much exact analogy
[laughter].
If random strategies is the house, and the regular lottery players are like players, well, what does the
state do? The state kind of puts its hand out and collects a certain percentage of each transaction, and
that's what states are good at. Statements don't like to gamble. States like the collect taxes. And that,
in essence, is what Massachusetts was doing. They had sort of -- you could think of what they were
doing with this game as sort of licensing a virtual casino and then collecting a tax on each bet.
The inspector general estimated that Massachusetts drew about an extra -- I think in total they drew
about $120 million in revenue from this game over its life span. And I think it's fair to say that if you
walk away with $120 million, you are not the one getting scammed. [laughter]
That's one puzzle which I think, as I said, I think the answer is pretty simple. I couldn't quite get -- if I
were a real reporter, I probably would have -- I mean I called the Massachusetts lottery PR people and
they said, "Of course, we didn't know any of this was going on." Obviously that's a lie, and if I were a
real reporter I could probably prove that, but I'm not a real reporter. I am a math professor, so I will just
report that they say that it's not true. But it is true. Here's the other puzzle. So there actually is a
description between random strategies and the other two teams and it's the reason why I focus on
random strategies. In the book and it's because of at the actually did do something rather differently
than other teams. The other teams used what's called the quick pick machine. I don't know if you guys
know. Are you guy a lot of lottery players? Who plays the lottery? Oh my god, a lot of liars work here.
[laughter]
I am going to pretend that you guys don't know what the quick pick machine is and tell you that is a
machine what will just print out a random ticket for you with random numbers, which, if you're going to
buy 200,000 tickets, is an extreme convenience and labor saving device. But the guys from random
strategies did not do this. They filled out their tickets themselves by hand, 200,000 tickets. Why?
That's the puzzle, especially because if you know how expected value works, you know that the
expected value of every ticket is exactly the same, so that the expected value of 200,000 tickets -- it
doesn't matter which tickets they are. The expected value just is what it is. So how could it have
possibly been worth their while to take the incredibly tedious task on of filling out all these tickets by
hand? That's the puzzle.
I am going to take the rest of the time and just talk about the answer to that question. Something that
we do in math when we're trying to understanding something that's kind of mysterious is we take a
problem that faces us and try to replace it with something much smaller and simpler. Hopefully when
we do that, we don't throw out the features we're actually interested in.
Hopefully we keep them but on a smaller scale. So what I am going to talk about is a as much smaller
lottery which, for reasons I was unable to figure out, is often called the Transylvania lottery.
In this lottery, instead of 46 different numbers like in the Cash WinFall, there's only seven different
numbers. And instead of pulling six balls out of a cage, like you do in Cash WinFall, you only pull three,
and that reduces the total number of possible jackpot combinations to a very small, manageable
number. In fact, it's so small that I've written them all on this board. Here are all ways, the 35 ways, of
choosing three numbers out of seven. So for the binomial theorem fans, that's the binomial coefficient
called seven choose three or 35.
And let me also kind of simplify the prize structure of this game so that we can kind of see how to think
about it. In the Transylvania lottery, there's only two kinds of prizes. There's the jackpot. That's what
you get if you pick -- if you hit all three numbers. And if you only hit two out of the three, you get a
lesser prize. Let's call it deuce to remind us of the number two.
So for instance, if the jackpot winning combination is 137, if your ticket says 137 then you have the
jackpot, right? You have it exactly right. If it says 127, you have a deuce, and if you have 145, that
overlaps only one place with the winning ticket, and so you get nothing. So this is how this lottery
works.
What do we do if we're making a big play in this game? We don't have to buy 200,000 tickets to make a
big play in the rather small Transylvania lottery. In fact, just to sort of set ideas, let's say a big player just
buys 7 tickets. Let me show you what it would look like. Here's a computation. I think I won't do this on
the screen.
But I will just tell you that the expected number of deuces if you buy seven tickets is actually 2.4. It's
between two and three.
If you look at the what happens if you choose seven tickets at random and see how many of these deuce
prizes you expect to win, here's what it looks like. So you can see just as you might expect, the most
likely outcomes are that you get either two or you get three.
But you might get completely shut out. It's not that likely and you might actually do great and happen
to have, like, lots more deuces among your seven tickets. So what I want to show you now is a different
way of choosing the seven tickets, one that is not at all random but one that is chosen quite carefully.
So here's my seven tickets. These are the seven tickets I buy when I play the Transylvania lottery and
maybe just to sort of show you what is special about these, this will be like the audience participation
portion of the program.
If one of you guys can be the cage and give me like a random set of three numbers between 1 and
seven. That would be great. You can just shout it out. I don't care where it comes from.
>>: 429 [laughter]
>> Jordan Ellenberg: I am going to give somebody else chance [laughter] 127. I heard 127.
Let's look and see how I did, okay I look -- I hope I win the jackpot, no, I didn't okay too bad. Out of my
luck. But let's see. I do have that 123, so I get a deuce from that.
I have a 247, so I am going to get a deuce there, and I have 167. I get a deuce from there. And then I
think everything else is a no-prizer, right? So I got three deuces. Now another.
456 okay. Once again, I fail to win the jack potting. I see a 145. Let's see if you guys can count as fast as
me. I see a 346. That's another deuce and 256. Okay, that's a deuce too, and then I think that's it.
Okay. Anybody else? We're not going to do it for that long. We got to do a few to get the idea. What is
it with you guys and the 9s? [laughter] If it was an 8, I would understand. Anybody else. 156. I had 156
okay. So I have got a deuce with the 145. And I've got a deuce with the 167. Oh, 256 is a deuce again
for that one. And then I think that's it.
Okay. So what happened here? You guy threw out three random numbers, and each time I got three
deuces. That's no coincidence. In fact, there's something quite remarkable about this set of numbers.
I has the property that no matter what numbers are drawn, there's only two things that can happen.
Either I win the jackpot -- that's certainly possible -- if it's one of my seven numbers that is drawn, but if
it's any of the other 28 numbers, I get exactly three deuces.
Now, why is that good? Let me explain from a financial point of view why that's good. Then I'll explain
from a geometric point of view what's going on. The reason it's good from a financial point of view is
that by choosing these numbers I've instituted what's called a hedge. I've hedged away my risk.
The expected value doesn't change, and yet I've eliminated the possibility that I will get either zero, one,
or two deuces. Either I get the jackpot or I get three deuces. Now, for the expected value to get the
same, if I get rid of my risk, it means I have to give something up at the top, right? It means I give away
the possibility of really making a lot of money on each play. But in this kind of game, in this kind of
context, that is usually considered a good. Given options with the same expected value if that expected
value a positive, you usually want to diminish your risk and the reason that you want to -- well, imagine
if you your business model is that you are asking everybody in your MIT dorm to give you the money
they have and you're going to go buy lottery tickets with it.
If you lose all that money and then you come back and say, "But statistically speaking, we're likely to
come out ahead in the long run if you give me more money," [laughter] that is sort of, for sociological
reasons, not such a winning play. [laughter] So if you're sort of playing with somebody else's money, if
you're relying on borrowing, it's really quite a good thing to hedge away the risk if you can, and this is
sort of the key to the puzzle. By choosing the numbers yourself, by not allowing it to do it randomly, the
expected value is the same but you can make the risk completely go away.
Now let me show you why it works with a picture because it comes from a somewhat unexpected
source. So what is this? It looks like a triangle, but what I want to say is that this is not a triangle. This is
the plane. Okay, what do I mean by that? Maybe it's better to say it is a plane, but it's a funny kind of
plane. It's a plane with only seven points. They are marked and numbered here. It's a plane which has
only seven points and also which has only seven lines. Most of them look like lines on the page, but one
of them does not. One of them is this little circle thing, but I am going to kind of aggressively say that
those seven numbers, those black circles or points, and all those little curves I've drawn are lines.
Then you will note the seven tickets I bought precisely correspond to the seven lines. So like here's is
123 along the bottom. That's one of my tickets. Here's 167, which is another one of my tickets. The
circular one is 256, which is one of my tickets. And if you went through each one and looked at what
points are on it -- there's three points on each line -- and looked at the seven tickets I bought, you would
see that they exactly matched.
Okay, well, anybody can draw a picture. Why do I say this is the plane? Because to a modern geometer,
geometry is defined by axioms. In other words, we say what are points? What are lines? Well, points
are things which behave the way points behave, and lines are thing that behave in the way that lines
behave. So in particular, why do I say these are points and lines? Well, what are the rules of points and
lines? It is that any two points are contained in a single line and any two lines intersect in a single point.
If you sort of contemplate this for a while -- you can check it exhaustively if you wanted to you -- will see
that those things are literally true. If I choose any two points in this diagram, there is a unique line
between them. In other words, they behave just as lines are supposed to behave. And if you choose
any two lines in this diagram, there is a single point where they meet. Now, by the way, that is actually
not exactly the way that lines behave in our usual plane geometry. What's the difference?
Does anybody see the subtle difference?
>>: [indiscernible]
>> Jordan Ellenberg: Yes. In regular geometry like we learned in the ninth grade, we don't say any two
lines intersect in a unique point. We say any two lines intersect in, at most, one point. If they're parallel
they intersect at zero points, otherwise they intercept in exactly one point. Okay, so that's kind of
disgusting, right? Rules should not have exceptions.
Rules are much nicer if they're like literal rules, so this geometry is, in fact, much better than Euclid's
original one. It's an example of what's called a projective geometry. So this is where in the book I kind
of veer off into talking about Florentine painting and how people there sort of started to understand
that when you paint a picture lines that are, in real life, parallel on the painting actually meet at a point
called the vanishing point.
These guys, as much as I love math, the painters were sort of ahead of us on this point and understood
much better how to make a geometry in which every pair of lines met and there was no such thing as
parallel lines. And that's the kind of geometry this geometry is.
In fact, maybe I'll just mention that it sort of takes an even farther course afield -- and I won't talk about
it in this talk, but it talks about the relationship of all this stuff to the dawn of information theory and
error correcting codes and sphere packing, which I mention just in order to say that I get to talk about
the work of a colleague of mine, Henry Cohn, who is at the other campus of Microsoft Research over in
New England, who is, like, one of the world's champion sphere packers. I'm sure a lot of you guys know
him. So Henry appears in this book, and actually I talked to him a lot in the course of writing this
chapter.
So what does the geometry have to do with the magical lottery properties I just said? Well, why is that?
I can guarantee that I get three deuces no matter what the jackpot numbers was? I think it was -- the
last one we did -- 156. I think that's what it was. 156. Let's look at it. Here is 1. Here is 5. Here is 6.
How did I know it was going to win exactly three of those prizes? Well, I wonder. The question is: Am I
going to get a deuce from the 1 and the 6?
Do I have a ticket with a 1 and a 6? I do because there's a line through 1 and 6. And that's the ticket I
have. Do I have of a ticket with a 5 and a 6? I did because there's a unique line through those two
points. That's a ticket I have, and the same for 1 and 2, through 1 and -- which one didn't I say? -- 1 and
5.
So the fact that through any two points there's a unique line is exactly the same as the fact that for any
two numbers on the jackpot I have that ticket that guarantees me a deuce. So it's precisely this
geometric property that gives this particular choice of tickets its wonderful feature. And that same table
I showed you before, in the case of this choice of seven tickets, looks completely different.
You have an 80 percent chance of getting three deuces, and if you don't, you're guaranteed the jackpot.
So you absolutely cannot lose. That's -- I guess I have a slide for this. This is exactly because of the
comment that I made about every two lines intersecting at a single point and every two points
contained in a single line. So this was for the simple problem, not for the real Cash WinFall. What about
the actual Cash WinFall? Unfortunately I mean, it's not so easy to make a geometry that exactly
matches 6 ball us out of 46 choices, but I wrote these guys a note. Don't you do something like this?
And they wouldn't quite tell me. I was able to make contact with them, but they were like, "Well, we
had a math guy, like, he had an algorithm. We don't know if he wants to talk to you," blah, blah, blah.
So I got kind of annoyed and then I got kind of obsessed with, like, trying to figure out what these guys
did. And in the end, I came to across a paper from 1976 by a guy called R.H.F Denniston in the field
known as combinatorial designs. And using this very beautiful construction of his, I was able to hook up
a way of buying about 220,000 tickets which gave you a near-guarantee that you would win five out of
these, five out of six prizes. So it had the expected value that all of these -- that any choice of this name
tickets would have, but it reduces the risk of losing almost to zero. So I don't know if this particular
combinatorial design is what they used or if they used something like this at all, but if they didn't, I think
they should have done.
So I'll stop there. This is a picture of the book, although I guess you guys can see it behind you, and I'll
take questions. Thank you very much. [applause]. You had a question, I'm sorry, during.
>>: I was going to ask you how did you come up with the geometry for those same numbers? Even that
would be helpful to understand.
>> Jordan Ellenberg: Oh, so that's one of most -- that's like sort of the beginning of the theory of
combinatorial designs, and that picture is called the Fano plane. It was from Gino Fano from the end of
the 19th century, who was one of the first people to really think about geometry in this axiomatic way.
It's a point of view we sort of think of as associated with Hilbert but Fano is one of the first.
There's a quote in the book. It's in Italian. I can't remember it in English either, but it basically says -- he
sort of very forcefully says, like, well, when I say a point and a line, I mean things that obey the axioms,
and we have to sort of free ourselves from this tyranny of thinking about what things look like on the
page and work in the sort of completely abstract realm of, like, which things obey the rules of points and
lines.
And that was what enabled him to make this kind of progress. So for the fans of finite fields in the room
-- and I'm sure there are some -- another way to describe it is that it's the projective plane over the finite
field of two elements. That's what we would call it today. I'm not sure if Fano himself thought of it that
way, but that's the way a modern kind of number theorist or geometer would refer to it.
Yeah, in the back.
>>: -- the most when you were writing the book?
>> Jordan Ellenberg: What surprised me the most? I mean I knew that I liked to go on and on, but I
didn't really understand how much. [laughter] Because when you start to research this stuff, everything
connects to something else and you're like, oh my god, that is crazy. How can you not talk about this?
So I submitted a proposal when I sold this book originally that had 18 chapters. And then at some point,
I had written three of them and I had about 300 pages, [laughter] so I had to have kind of a hard
conversation with Penguin where I said, "Do you more want an 1800-page book that you'll get, like,
seven years from now, or do you more want me just to write, like, five of these chapters?" And they, I
think thankfully, everyone involved, with their good commercial instincts said, "We just want you to
write five of the chapters. I think that will be fine." So that's -- yeah. I mean, when you sort of go into
the history and go into the connections between the different parts of the math, you just fall down, like,
infinitely many rabbit holes. And then inside each of these infinitely many rabbit holes is, like, another
exciting rabbit, like, running away down another corridor. So I think that's what I found surprising.
There's a lot of stuff. This is a good example of something. I didn't even know this story when I
proposed the book. I sort of came across it while I was doing something else, and I was, like, oh my god,
I have to write like 75 pages about this.
>>: I understand where you were going with the story, but what's the broader message of your book?
>> Jordan Ellenberg: Well, I think the broader message of the book is that for people who have been
trained to think of mathematics as this kind of separate system, right, that takes place in its own world,
that doesn't hook up with the ordinary things that we do, I think it's a very wrong way to think of it.
Like, I want to show how mathematical thinking is not separate. It's tied up with all the other kinds of
thinking that we do. Another way to say it is it's not something that we do instead of thinking. It's sort
of a formalized and strengthened version of our thinking, because we'll come back again and again to
explaining something that, to me, is part of mathematics. And I'll certainly explain it mathematically and
do computations and blah, blah, blah, but in the end, it's common sense, but it's a strengthened version
of common sense.
Yeah?
>>: We have a question from Michael on line. He says, "What do you consider to be the most exciting
new area of mathematics being dropped today?"
>> Jordan Ellenberg: So I mean, it's like picking from your children. I'll just say one thing. What's
tremendously exciting, especially for somebody like me who comes from number theory, is what's going
on in, like, machine learning and what you might call the new statistics.
Because this is an area where, you know, if you do number theory, you're working in an area that people
have been mining for literally 2,000 years, right? It's a very old subject. There's a huge learning curve,
right? Everything that's easy has been done already. And you have an idea, and you get excited about
it, and you tell somebody about it and they're like oh, that's great but people tried this in 1930. You
know, this happens a lot in number theory. In machine learning, you talk to these people, and it's
tremendously exciting. It's a brand-new field that's being created under your feet as you go. Everything
is up for grabs, including, like, what the field should be called, what's it's about, who should be doing it,
et cetera, et cetera.
When you have an idea and you tell people they'll say, like, "Oh, that's a good idea, but somebody did it
like six months ago." And I think for somebody who works in a very old field like me, there's a wonderful
appeal to that kind of stuff. And I think just the fact that it's a field that we don't know what it's going to
look like five years from now, ten years ago, I think is tremendously exciting.
Yeah in the back.
>>: When I was looking up the book online, there was a tag line that said, like, the Freakonomics of
math. Were you influenced by Dubner and Levvitt, the Freakonomics guys? Is that something you
follow?
>> Jordan Ellenberg: So I signed this thing that said that this could be distributed outside Microsoft and
yet -- so it's somewhat risky to me to reveal that I have not actually read that book [laughter]. Yet I'm
billing my own book as the Freakonomics of math. Let me defend myself for doing that. So while I
haven't read Freakonomics, I have read a lot of their columns and, like, their articles and stuff like that
that make up the book. And what those guys have done -- and not just them by the way, but Paul
Krugman and other sort of outward facing economists as well, it's not just that they've written their own
books. It's that they've taken ideas which previously were kind of inside the academic silo of economics
and they have made them so popular that they've become part of the common discourse, right, of
people now. It's ordinary to read the paper and read something about incentives or read something
about diminishing returns or something like that.
You read that not just in the business section or in just something that is an economics column, but
anywhere in the newspaper, right? I think all of us -- and I'm just one person doing this, but all of us
who are in the project of outward facing mathematics, want to take some of these mathematical
notions like expected value, like regression to the mean, like linearity, like formalism, like some of these
things I talk about in the book, some of the big themes in the book, things that all mathematicians sort
of know what they mean and know the sort of breadth of their explanatory force and kind of make it
safe and normal for people who are not mathematicians and not writing in the science section of the
paper to use those notions s in everyday life.
So I think that's the sense in which I think my project is the same kind of project as the Freakonomics
project. Yeah.
>>: I have to ask this: How long prior to our flight should we be at the airport? [laughter]
>> Jordan Ellenberg: Oh, yeah. So there was a sort of a weird kind of game of telephone with British
newspapers where somehow it came to be common knowledge that apparently I'm an expert in airport
studies. So this refers to section of the book where I talk about a famous old dictum of the economist
George Stigler, who said that if you never miss an airplane, you're spending too much time in the
airport. I'll stand by that, but what is being expressed there -- am I allowed to write on this, actually?
Can I do this? It's just this picture.
Here's the picture. This is everything you need to know. There it is. That's the explanation. No, so
what does it mean? It means that, in terms of your overall benefit of your strategy, if you leave in the
morning for a flight that's at night, you're clearly wasting a lot of your own time, right? There's such a
thing as arriving too early, and there's such a thing as arriving too late where you're essentially certain to
miss your flight. So somehow there is an optimum somewhere in between, where there's some nonzero
chance of missing your flight. I mean, it might be small but it's not zero, and that's exactly what Stigler is
saying. But I mean, I think somehow that got translated into people thinking I was going to say, like,
"Here is the exact number down to the minute." And obviously, that's going to vary depending on your
personality, depending on context. Actually one thing I learned from what people said about that is that
there are people who actually love hanging out at the airport. I didn't know that. I thought that had
negative utility for everyone, [laughter] but apparently there are people who are like, "But then you get
to shop at the duty-free shop." The duty-free shop is terrible compared to a regular shop. I don't really
understand, [laughter] but I learned that a lot of people, it's like the highlight of their week to, like, go to
the duty-free shop. And so, yeah, that's all.
It's the same thing that I say in the book as, you know, if your government doesn't waste any money,
you're spending too much money fighting government waste. [laughter] Yeah.
>>: -- along the same lines that he wanted to know who really did win the Florida election in 2000.
>> Jordan Ellenberg: Oh, so this is -- so I talk a bit about Bush vs. Gore. Some of you in the room are too
young to remember this, terrifyingly enough. [laughter] There were these two guys who wanted to be
president [laughter]. Anyway, actually, that ->>: Three.
>> Jordan Ellenberg: -- would have been a great moment for mathematicians to stand up and say that
that question doesn't really have an answer. We don't -- we really cannot know who got more votes. I
think that's fair to say. And so I write about that in the book as an example of the fundamental
formalism of our legal system, that I think it would have been in -- I mean, people probably had more
pressing things to worry about, but if it had been considered an opportunity for math education, it
would have been a nice time to sort of stand up and say, "This is a formalist moment where we accept
that we don't actually know who got more votes." And we say there's a procedure, and whatever the
procedure says, that's who's president, but we don't pretend that somebody won or somebody lost in
the sense of, like, actually got more votes.
That's why I talk about Derek Jeter and the sort of famous dictum of Bill Klem, the great National League
umpire who said of the question of, like, you know, do you ever miss a ball or a strike, he said, "It ain't
nothing till I call it."
The umpire's decision makes reality in the same way that our legal system, for better or for worse, a
Supreme Court decision creates the reality. It doesn't observe reality. Yeah.
>>: So what does mathematics have to say about the probability of your running into a subject from
your book in a bookstore or some other probable coincidence?
>> Jordan Ellenberg: Well, I did e-mail him actually. When I told the story I made it sound like he just
showed up to make the story a bit better, but if you're going to press me on it, [laughter] yes, I e-mailed
him to say I was going to be in Palo Alto. [laughter] Yeah.
>>: Where or does your power of mathematical thinking actually ever really sort of let you down?
>> Jordan Ellenberg: So one thing I try to do in the book, I mean, I think, of course, out in the world, all
of us who work in quantitative fields feel that in general the world is not quantitatively inclined enough,
right? We can't help but feel that way. At the same time, there is kind of on the other side, a danger of
what you might call math supremacy and feeling like in the end, we'll sort of achieve some kind of
apotheosis where we can sort of sit and compute about stuff instead of thinking about stuff.
What I think it's -- to the extent there's a danger, it's important to push back on it and sort of talk about
the limits, and I do that. I mean, that actually especially comes up in the parts about religion where
there's, like, such a long tradition of people kind of desperately and wistfully hoping that mathematics
will help them find the way out of the conundrums of faith.
And I think every attempt like that has completely, like, fallen down and failed. Like, I think that is not a
question about which math gives an answer one way or the other. I mean, lots of people have said that
it did. At least on my assessment, all those attempts are failures, and I think that's sort of one place in
the book, which is returned to several times, where you see that we have to have math with us all the
time but not only math.
>>: I had a really bad experience with math growing up as a kid. It was only after getting out of school
where -- it haunted me to such an extent that I decided to tackle it and take it on. I now have an 8-yearold son ->> Jordan Ellenberg: Oh, me too.
>>: Oh, okay. And I don't want him to have the same challenges that I had when I was a kid. I'm curious
to know what your thoughts are on math education and how it might be improved. It seems like there's
a lot talk about that lately.
>> Jordan Ellenberg: Yes. I talk about that a little bit directly in the book, but here's what I would say.
Here's one thing that makes it so hard. I think when I started out teaching and learning how to teach, I
was, like, "I am going to find out the best way to do this and do it," right, because that's how we
approach a lot problems that we have to attack.
And over the years what I've come to feel is that one of the fundamental challenges of teaching is that
kids are really different from each other, like really different, and there is not a best way. So sort of
something that we do that's going to engage one population of kids is going to completely bore another.
And so what you hated in school, there are probably other kids who liked, and something that would
have worked well for you probably would have turned off somebody else. So I think as teachers, the
best that we can try to do is to kind of broadcast on all channels and do things in a lot of different ways
to maximize the chance that, like, each student finds that thing that hooks into them, that thing that
enables them to engage with mathematics. I mean, it's a phenomenally hard challenge, and I think -- I
mean, one reason I hedge is because as somebody who teaches in a university, our teaching conditions
are so completely different from somebody who's teaching 8-year-olds. They are under the constraints
of knowing that their students are going to be taking, like, a test that has high stakes for both them and
for the teacher and for the principal and for the school.
So it's easy for me to say, oh, they should do this and they should do that, but they are under very series
constraints. And I think -- well, maybe one thing that's not talked about enough -- we talk a lot about
what the teachers should do, but in a world in which the teachers' actions are so highly constrained by
the tests, I think it's important to remember that the people who write the tests have a massive amount
of responsibility on their shoulders in that system to create tests that test what we want to test. I mean,
that's one thing.
I think that does matter. Are we going to talk about -- I'll take one more.
>>: Is there a good book for number theory for 8 and 9-year-olds? [laughter]
>> Jordan Ellenberg: Oh, I mean, boy that really depends on the 8 and 9-year-old. [laughter]. I mean, I
think when I was that age, I'd feel like the Bible and the Torah and the Koran of my life was Gödel,
Escher, Bach, which I did not understand, but I think what's amazing about that book is that it has so
many different parts of so many different levels of difficulty but all of it is deep, the easier parts and the
difficult parts. So I feel like at least for me that was like a huge portal into like what mathematics really
was. And when I was 9, I probably just read the parts with the talking animals, but it was fascinating.
Each time I would come back to it, and each time I could read more and get more of the parts that I
couldn't. So I would say -- that's not particularly about number theory, but I would say that book. And
then, of course, the Martin Gardner books. I mean, like, nobody has -- even though those books are fifty
years old, like, nobody has really equaled what he did in the way of math exposition, and there's like god
knows how many. There's like 25 volumes of his Scientific American columns. I actually just bought two
of them for my 8-year-old, so I'll report back as to whether he likes them.
>> Amy Draves: Thank you [applause]
>> Jordan Ellenberg: Oh, thank you.