Tim Chartier on how to use math to win gold at the Olympics

Tim Chartier, co-author of Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms with Anne Greenbaum, explains how to take home the gold using math.

A free-surface simulation of the forces experienced when diving, provided by Speedo® in their press release for the Fastskin Racing System®

Math can help win gold in London!  From air passing over an athlete’s body, whether that person be running or biking, to water streaming along a swimsuit or the hull of a boat, many events benefit from numerical analysis and its role, in particular, in computer simulation. For example, aerodynamic research can improve a swimmer’s suit and shave off time that would otherwise be taken with added friction.  Such numerics can also inform a biker on a more efficient body position.

Such work involves computing numerical solutions of partial differential equations.  Two important stages occur in such work.  First, one must develop and utilize appropriate mathematical models.  If the model is too simple, its solution will not accurately reflect the real-world phenomenon.  In such a case, the swimmer could end up with a suit that isn’t minimizing friction with the water.  The second stage is solving the numerical solution to the model, which is performed on a computer with finite precision. As such, numerical methods that can efficiently and accurately solve the mathematical model are needed.

From sports science to the laboratory, modeling and numerics often complement each other, giving modern science a power, not possible without such digital resources.  As such, learning the strengths and limitations of numerical methods, often coming through mathematical analysis, enables one to appropriately utilize such tools and leverage them to explore today’s difficult and important problems.

So, as you watch the Olympics, keep in mind that the body mechanics and equipment we see were often informed by mathematics.  Such tools play an important role in training and the innovations that contribute to the feats we will witness in the coming weeks.

Math and the Olympics

After 69 days of traveling, the Flame crossed the River Thames on Friday to reach its final destination at London’s Olympic Stadium. There are many reasons to be excited about the Olympics, but here at Princeton University Press, we can’t help but think about the abundance of equations and mathematical modeling taking place during the summer games. From the design of the Olympic logo and the sports equipment, to the actual athletics, math is taking place everywhere during the Olympics. For example, watch for the swimmers who win a race by .001 of a second!

We’re not the only ones excited about it either. Cambridge University’s math education initiative, The Millennium Mathematics Project has been running the Math and Sport: Countdown to the Games for the last 18 months. Check out their website for fun activities that celebrate both math and the Olympics: http://sport.maths.org/content/.

As the games continue, we’ll be hearing from PUP authors, excerpting from our books and staying up-to-date with the math and science of the Olympics. Stay tuned.

Tim Chartier’s Bracketology 101 Webinar — the Math You Need to Pick the Winners

[Update for 2014 -- check out Tim's new tips: http://blog.press.princeton.edu/2014/03/17/top-tips-for-2014-march-madness-brackets-from-tim-chartier/]

Check out Tim Chartier’s webinars on the applications of linear algebra for more helpful hints on how to fill out your bracket.

Tim looks at how famous bracketologists have fared in the past (tip — Obama did OK in the 80th percentile, Dwyane Wade, not so much in the 43rd percentile) and provides step by step analysis of the math behind the Colley method and others.  He also guides viewers through a free java program that is available for download on his professional web site at Davidson College.


Bracketology 101: http://davidson.mediasite.mcnc.org/mcnc/SilverlightPlayer/Default.aspx?peid=bbac5c0483ad4659bea8adfc7258e3051d

Advanced Bracketology: http://davidson.mediasite.mcnc.org/mcnc/SilverlightPlayer/Default.aspx?peid=600e3c4c27e741edbc34cbf89a510b7a1d

You can find a link to the Java code for ranking on Tim’s blog: http://sites.davidson.edu/mathmovement/bracketology-101/

Listen to Tim’s talk on March Mathness at the MAA Distinguished Lecture Series: http://www.maa.org/dist-lecture/past-lectures.html

Tim Chartier is an Associate Professor of mathematics at Davidson College. His ability to communicate math both in and beyond the classroom were recognized with the Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member from the Mathematical Association of America. His research and scholarship were recognized with an Alfred P. Sloan Research Fellowship. Tim serves on the Editorial Board for Math Horizons, a mathematics magazine of the Mathematical Association of America. He also serves as chair of the Advisory Council for the Museum of Mathematics. Tim has been a resource for a variety of media inquiries which includes fielding mathematical questions for the Sports Science program on ESPN. He is co-author with Anne Greenbaum of Numerical Methods: Design, Analysis and Computer Implementation of Algorithms. His latest book, Math Bytes, includes a section on bracketology as well as other fun math and computing endeavors.

March Mathness — Choose Your Brackets Wisely via ESPN

9,233,372,036,854,775,808 = number of ways to fill out a 64-team March Madness bracket after all teams have been seeded.

147,573,952,589,676,412,928 = number of ways to fill out an expanded 68-team March Madness bracket.

How will you pick your winners?



Looking for a better method to fill out your brackets? Read up on Kenneth Massey’s criteria and system here: http://blog.press.princeton.edu/2012/02/29/march-mathness-the-massey-method/

Don’t forget to join in March Mathness by submitting your brackets to the Princeton University Press sponsored March Mathness group on ESPN: http://games.espn.go.com/tournament-challenge-bracket/en/group?groupID=186896&entryID=4680711

Figures from Who’s #1?

March Mathness — The Massey Method

In this post for March Mathness, Kenneth Massey whose popular ratings (http://masseyratings.com) help rank the BCS teams each year, offers an overview of what goes into filling out his brackets for March Madness.



Advice on filling out March Madness brackets from Kenneth Massey


I’m a college basketball fan, but to be honest, I don’t watch many games during the regular season. Since my personal expertise is a function of ESPN highlights and commentary, I’ve learned to trust the math more than my own feelings about a matchup.

All season I compile a monster list of the various computer rankings for college basketball: http://www.masseyratings.com/cb/compare.htm

By following the results, I have a pretty good idea about which teams are over/under rated by the media and which ones are coming on strong or fading into tournament time.

Once the pairings are announced, I usually fill out a bracket based on my limited first-hand knowledge of the teams, the impressions I have from following the rankings, and maybe some “gut” intuitions. For that particular bracket, I don’t do any additional analysis, or even look at the numbers–I just rely on what’s already accumulated in my brain.

Now let me describe how I fill out my more analytical brackets. I have two different strategies, but both of them start with estimating the probabilities that each particular team will advance past each round.
In this post, I will describe that process. In a later post, I’ll describe how I use those probabilities to actually fill out my bracket picks.

I’ve been doing computer ratings for years, and have experimented with many mathematical models, one of which is described in Who’s # 1?. The model I currently use, and post on masseyratings.com, is proprietary, but I will list some of the pertinent aspects of it.

1) Margin of victory matters, but in an intelligent way. There are diminishing returns for blowouts, and adjustments are made for the pace of the game. For example 60-45 may be more impressive than 100-80.

2) Winning is rewarded, especially on the road. Even if the margin is small, a team gets a bump by winning games against good competition.

3) Schedule strength is implicit in all the equations. Everything is measured relative to the opponent, so there is higher reward and less risk for playing tough opponents.

4) The model has a decaying memory of early season games. The team in March is different from the team in November.

5) Games between mis-matched opponents are not as important as games between well-matched opponents. There is a lot more information in a #18 vs #23 matchup than there is when #18 plays #230.

6) My model produces offensive and defensive ratings for each team, as well as homefield advantage estimates. From these, it is possible to predict the distribution of final scores for a hypothetical matchup between any two teams.

After the ratings are computed, I use conditional probability to effectively account for every possible scenario of how the bracket could “play out”. For example, if team X makes it to the Sweet 16, who are they likely to face? According to the seedings, some teams have easier paths of advancement. I can compute the probability that each team advances past a given round, the expected number of rounds a team will win, and ultimately each team’s probability of winning the championship.

The great thing about probabilities is that you are never “wrong”. For example, last year my calculations showed that UConn had an 86% chance of winning the first round, a 54% chance of advancing to Sweet 16, a 29% chance of advancing to Elite 8, 12% chance of advancing to Final 4, 5% of playing in the championship game, and a 2.3% chance of winning it all.

By the nature of randomness, it is not really surprising that underdogs occasionally win. Even a dominant #1 overall seed rarely has more than a 25% chance of winning the entire tournament. That’s what makes the event so exciting–nobody knows what will happen.

After all the probabilities are computed, I proceed to fill in my picks. Don’t I just pick the teams with the highest probabilities? Not exactly. I’ll address that in a subsequent post.

Celebrate Major League Baseball’s Opening Day by Reading about Baseball in Blue and Gray

Today is THE day baseball fans. Major League Baseball is back in action. Over at the New York Times, they are celebrating by looking back at the early days of baseball. Specifically, they have posted an article from Princeton University Press author George B. Kirsch on baseball during the Civil War.

Compare Kirsch’s description of “spring training” and “opening day” in 1861 to the great hullabaloo today:

In late March and early April 1861, ballplayers in dozens of American towns looked forward to another season of play. But they were not highly paid professionals whose teams traveled to Florida or Arizona for spring training. Rather, they were amateur members of private organizations founded by men whose social standing ranged from the working class through the upper-middle ranks of society. There were no formal leagues or fixed schedules of games, although there were regional associations of clubs that drew up and enforced rules for each type of bat and ball game. Contests between the best teams attracted large crowds (including many gamblers), and reporters from daily newspapers and weekly sporting magazines wrote detailed accounts of the games.

While much has changed in American baseball since 1861, what hasn’t changed is the anticipation, excitement and pure sport of the game. Unfortunately, this spirit wasn’t enough to hold the reality of the Civil War at bay according to Kirsch. He writes:

As military action between the North and the South loomed, sportswriters highlighted the analogy between America’s first team sports and warfare. Yet they were also aware of the crucial differences between play and mortal combat. In March 1861, The New York Clipper anticipated the impending crisis:

God forbid that any balls but those of the Cricket and Baseball field may be caught either on the fly or first bound, and we trust that no arms but those of the flesh may be used to impel them, or stumps, but those of the wickets, injured by them.

But three months later sober realism replaced wishful thinking. A Clipper editor remarked:

Cricket and Baseball clubs … are now enlisted in a different sort of exercise, the rifle or gun taking the place of the bat, while the play ball gives place to the leaden messenger of death. Men who have heretofore made their mark in friendly strife for superiority in various games, are now beating off the rebels who would dismember this glorious “Union of States.”

Click over to read the complete article and peruse the Disunion feature at the New York Times. Disunion is tracking, day-by-day, the course of the Civil War in America through terrific articles from experts in a variety of fields. While there is certainly a lot of military history, the editors are also focusing on cultural and social issues (like baseball!) which make for truly compelling reading.

This Week’s Book Giveaway

Can’t get enough baseball? This week’s book giveaway,  Baseball on the Border:  A Tale of Two Laredos by Alan Klein, will help fill the gap.

From 1985 to 1994 there existed a significant but unheralded experiment in professional baseball.Baseball on the Border For ten seasons, the Tecolotes de los Dos Laredos (The Owls of the Two Laredos) were the only team in professional sports to represent two nations. Playing in the storied Mexican League (an AAA affiliate of major league baseball), the “Tecos” had home parks on both sides of the U.S.-Mexico border, in Laredo, Texas and in Nuevo Laredo, Tamaulipas. In true border fashion, Mexican and American national anthems were played before each game, and the Tecos were operated by interests in both cities. Baseball on the Border is the story of the rise and unexpected demise of this surprising team. Anyone with an interest in baseball will be enlightened & entertained by this informative book.

“Read this book, enjoy the characterizations of the players, managers, and administrators … listen to the crowd cheer for their home town heroes, and pause to think, as Klein paints the picture with a masters stroke, of what this [book] can tell us about transnational relations and the impact of sport.”–Patricia A. Adler and Peter Adler, authors of Backboards and Blackboards

“The book is very well written. . . . It contributes greatly to the literature on the cultural basis of sport, to our understanding of the manner in which cultural inventions reflect national identity and processes, and substantiates an important insight to the idea that sport may provide a window to ongoing social change.”–Carlos Velez-Ibañez, American Anthropologist

Everyone who LIKES us on our Facebook Page is automatically entered in our weekly book giveaways.

Baseball on the Border: A Tale of Two Laredos by Alan Klein

March Mathness explained by Vickie Kearn and Tim Chartier

Just in time for the first Sweet Sixteen games, math editor Vickie Kearn speaks with Tim Chartier about how math and Google is used to predict the bracket winners. Enjoy this exclusive dialogue below.

Vickie Kearn: When I was at the University of Richmond, I only went to one football game in four years. That was not a program that got a lot of attention then. However, the Spiders had a great basketball team, as they do now. Because of their great success, I have been watching as many games as possible and following the brackets with a huge amount of energy. A few years ago, we published a book by Amy Langville and Carl Meyer, Google’s PageRank and Beyond: the Science of Search Engine Rankings, and part of the “Beyond” is how you use math to rank sports teams. It is really quite fascinating.

Amy and her students at the College of Charleston and Tim Chartier and his students at Davidson College use mathematical algorithms to rank teams and they are doing fantastic on their brackets this year. I asked Tim and his student, Lucy McMurry, a sophomore who plans to declare a major in math and a minor in Spanish, how they used their math background to make sense out of a bracket with 68 teams. How do you pick who will be #1?

Tim Chartier: There are a variety of techniques that can be used to rank items. Sports teams are often ranked by winning percentage. Elections are often won by the person who gains the highest number of votes. Search engines often use techniques from linear algebra. Amy and Carl’s book discusses important aspects of Google’s PageRank algorithm that makes it suitable for ranking webpages and how it is scalable to analyzing billions of items. For our brackets, we rank sports teams also with linear algebra. PageRank uses a stochastic matrix that is built from underlying probabilities based on a model of a surfer randomly traversing the web. Our method builds a linear system, Ax = b, which while still a linear system, has different properties than a stochastic system like PageRank’s.

Lucy McMurry: To begin, we acquire all of the data about the 68 participating teams including when each game was played, the score of each game, and whether or not it was home or away for each team. From this point, it is up to the student how he/she wants to implement the code. For example, an away win could be worth more than a home win and a game played later in the season could be worth more than one played at the beginning. Once the code is set, a ranking is produced using all of the data. From here, we assume that the higher ranked team will win each game. Thus, the top ranked team will win the tournament.

VK: When you found out on March 13 who would be playing in the tournament, how did you go about setting up your brackets and selecting who you predict will win on April 4?

LM: I personally don’t know very much about basketball and hadn’t followed the entire season. Therefore seeing who would be playing on March 13th didn’t really affect how I wanted to structure my code. I based my code on what seemed reasonable to me from an outsider’s perspective, which resulted in me creating a code based on when in the season a game was won.

TC: This is quite true. Fundamentally, the students think mathematically about their models, which allow all the students to participate. I’ve seen some students fiddle with their models when they have a particular team they want to see perform well or actually poorly. In many cases, this doesn’t help. While there are a number of different ways one could measure success, we submit our brackets to the ESPN Tournament Challenge, which allows us to compete against each other and millions of other brackets.

VK: Is there more than one way to rank the teams?

TC: Yes. Some people pick the winning team based on which team’s mascot they prefer! And yes, there are also multiple methods for doing this mathematically. In college football, the Bowl Championship Series (BCS) uses several methods to rank the teams. One such method is the Colley method is a linear system which is based on wins and losses and there is also the Massey method which can integrate the scores of games. One can also adapt Google’s PageRank algorithm.

LM: In fact, I’m using, as many of my fellow students are, the Colley method. However, we use recent research by Drs. Chartier and Langville that allows us to model momentum. By using these different models and ideas, we are all able to come up with original codes that can produce very different results.

VK: Suppose you love math but don’t know anything about basketball. Will your rankings still predict a reasonable winner?

LM: I love math, but as I mentioned earlier, I do not know very much about basketball. However, I am tied for first placed in our class pool along with three other students and am currently in the 91st percentile nationally! Therefore, I think I can safely say that my code predicted a very reasonable ranking of the teams.

TC: Lucy and three other students that are currently in the lead are performing better than over 5 million other brackets! Kelly Davis, who you will see in the video, is one of the students leading the class. Daniel Martin, who is also interviewed in the video, is in a pool outside the class and is currently in the 96th percentile. Interestingly, some students in the class pool tried very novel approaches to modeling momentum and a few such methods are performing quite poorly, with one such method ranking in the 1.5 percentile!

VK: Amy Langville’s students received quite a bit of publicity in the past because their predictions were so good. Have you experienced a bit of fame from your predictions?

TC: Yes. The media has covered our brackets for the past three years as we’ve done well each year.

LM: Just last week, Derek James of Fox News Charlotte came to our class to film Dr. Chartier talking to us about our brackets. Many of the students in the class were able to email their parents to watch Fox News and get a glimpse of us in class. You’ll see me in the news segment but I definitely have a good portion of any eventual 15 minutes of fame left! The interview concentrated on Dr. Chartier as well as a few students from my class discussing their brackets and the theories behind their code.

TC: In fact, I helped Derek create a bracket using our methods. I asked him to break the season into as many intervals as he wanted. For instance, suppose he chose 3. Then, he would weight the games during each interval. Suppose he chose weights of 1/2, 3/4 and 1. Then, all the games in the first, second and last third of the season would be worth 1/2, 3/4 and 1 game, respectively. He also gave weights to home and away games. In the end, he had a personalized bracket that is tied with Lucy’s! The winner of the class pool gets a prize from Ben and Jerry’s in Davidson. Derek isn’t eligible as coming for 15 minutes to class and talking during the lecture doesn’t qualify! We have great fun watching the brackets unfold and seeing how our modeling performs.

Vickie: To learn more about what Derek James learned in class, watch his interview below.

Derek James, Reporter FOX Charlotte-WCCB. Used with permission.

VK: In 2009 we published Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football by Wayne Winston and this is another great source for you veteran bracketologists. Also, if you go to waynewinston.com you can see all of Wayne’s calculations and odds of each team winning in a particular round. For example, for the Sweet Sixteen odds he gives the University of Richmond a .84% chance of winning the championship and Ohio State a 29.6% chance of taking home the trophy. Although my heart is still with Richmond, I am going to go with Duke for a repeat. All the rankings and calculations I have done give them less of a chance of winning than other teams, so they are a mathematical longshot. However, I added a little luck factor into my calculations. Wayne is going with Ohio State. Check back in a few weeks and we will let you know how we did.
Good luck to all!

The Green Bay Packers are the winners…at exercising collaborative governance!

This article from Jack Donahue and Richard Zeckhauser, co-authors of the forthcoming book Collaborative Governance: Private Roles for Public Goals in Turbulent Times, was posted at CNN.com just before the Superbowl this weekend.

According to the post, The Green Bay Packers and their fans, have more than the Superbowl to celebrate. The team also holds a distinction of being an excellent example of collaborative governance — a system where public and private enterprises work together toward a common goal.

As Donahue and Zeckhauser write, “The Packers are an anomaly in America’s famously mercenary sports world — a nonprofit with unabashedly public purposes. Nearly 90 years ago the people of Green Bay hard-wired into the team’s charter some key provisions that survive to this day: The team must be community-owned, with no single owner holding more than a sliver of the shares. (There will be some 111,000 owners with skin in Sunday’s game.) ”

While their articles is about The Packers, there are many other examples of successful collaborations including Chicago’s Millennium Park, NYC’s Central Park, Coast Guard-led port security initiatives, and numerous charter schools. Why are public-private collaborations on the rise?

According to Zeckhauser and Donahue, “When government’s will is divided and its wallet is depleted, the appeal of collaboration with the private sector expands.”

But this type of collaboration isn’t always the answer. As the authors note, “Public-private collaboration [doesn't] always benefit the public at large. Our research has unveiled triumphs aplenty, but no shortage of crack-ups and catastrophes.”

Read the complete article here at CNN.com and pre-order a copy of Donahue and Zeckhauser’s new book Collaborative Governance.

Financial Times highlights FIVE Princeton books in 2010 Nonfiction Round-Up

Five Princeton books were recently featured in Financial Times‘ Nonfiction Round-Up for 2010! Here’s what FT had to say about them:

Business & Economics

Banking on the Future: The Fall and Rise of Central Banking, by Howard Davies and David Green

The best assessment yet of the role played by the leading western central banks – the US Federal Reserve, the ECB and the Bank of England – in the run-up to the financial crisis and beyond, from two former insiders at the top level of UK policymaking.

Zombie Economics: How Dead Ideas Still Walk Among Us, by John Quiggin

A critical look, from a left-leaning perspective, at some of the defining intellectual fashions of the past three decades. Quiggin is a writer of great verve who marshals some powerful evidence.

Fault Lines: How Hidden Fractures Still Threaten the World Economy, by Raghuram G Rajan

A high-powered yet accessible analysis of the financial crisis and its aftermath, Fault Lines was awarded the FT/Goldman Sachs Business Book of the Year. Rajan, a University of Chicago economist, was one of the few who warned that the crisis was coming and his book fizzes with striking and thought-provoking ideas.

Science & Environment

Honeybee Democracy, by Thomas D Seeley

The year’s most enchanting science book. Seeley, biology professor at Cornell University, distils the insights of 40 years studying and keeping bees. He focuses on the astonishing “democratic” process that takes place when a swarm of thousands of bees leaves an overcrowded hive to find a new home: how scouts evaluate potential sites and advertise their merits, how a final choice is made, and how the swarm navigates to its new nest.


Gaming the World: How Sports Are Reshaping Global Politics and Culture, by Andrei S Markovits and Lars Rensmann

A very readable guide to the recent globalisation of sport by academics who understand both US and European sports. Packed with examples, from David Beckham to Kobe Bryant, the book explores the tension between sport’s globalisation and the fact that most teams still arouse the greatest emotions in their local areas.

Definitely a great year of non-fiction books! Congratulations to all authors mentioned!

The other creationist debate — who founded modern baseball? Baseball historian George Kirsch responds to Bud Selig’s comments regarding the Doubleday myth

For the uninitiated, the Doubleday myth is part American folklore, part sports history. The story goes like this — Civil War hero Arthur Doubleday founded modern baseball on a field in Cooperstown, NY. This version of history is appealing for a variety of reasons, but most historians now agree that it quite simply isn’t true.

In this article at the History News Network, Kirsch writes that debunking this myth is about as popular and difficult as “telling the truth about the tooth fairy or Santa Claus.” Case in point, the Commissioner of Baseball Bud Selig, recently stated, “From all of the historians which I have spoken with, I really believe that Abner Doubleday is the `Father of Baseball.’ I know there are some historians who would dispute this, though.”

So, why does this myth endure and continue to garner support from people who, frankly, should know better? Kirsch chalks it up to two reasons:

  • “baseball’s continuing association with American nationalism.” It is no mistake that Kirsch starts his article clashing with a tour guide at the Gettysburg battlefield.
  • and, “the enduring power [of] a very appealing story” that favors the “field of dreams” in Cooperstown over the urban setting of the inner city of NYC.

Click over to read Kirsch’s complete article.

Andrei Markovits at New York Law School, October 19

The New York Law School Institute for Information Law and Policy and the New York Law School Sports Law Society is pleased to present:

Professor Andrei Markovits

Professor Markovits is currently Arthur F. Thurnau Professor and the Karl W. Deutsch Collegiate Professor of Comparative Politics and German Studies at the University of Michigan, Ann Arbor. Professor Markovits, a world-renowned scholar, will speak about his latest work entitled, Gaming the World: How Sports Are Reshaping Global Politics and Culture (Princeton University Press, 2010).

Date: Tuesday, October 19, 2010

Time:  6:00 pm- 8:00 pm

Place: Faculty Commons, 2nd Floor, 185 W. Broadway, New York, NY 10013

Please RSVP to naomi.allen@nyls.edu by October 8, 2010