WCoM Donates Statistics Textbooks

The Worldwide Center of Mathematics recently had some winter weather make its way into a stock room and slightly damage a number of Introduction to Statistics: Think & Do (Stevens).

The damaged books will not be sold and instead will be donated.

The recipient(s) include: Siem Reap at Life and Hope Association and Phnom Penh at People Improvement Organization. Both organizations are in Cambodia.

Water-damaged books prepared for donation.

Problem of the Week

Happy Election Season! Check out our problem of the week about the 1948 Presidential election! Let us know how you did in the comments, and make sure to vote!

Solution below the break.

Problem of the Week

Check out this week's Problem of the Week!

Solution transcript and video below the break.

Everyday Math: Sports

Game Time: Math Style

Known for underdog upsets and nail-biting game winners, the NCAA Basketball Tournament takes the cake for the most exciting annual college sports event. Nicknamed March Madness, the tournament begins with 64 teams, but after 6 rounds of games and plenty of Cinderella stories, the field is narrowed to only one National Champion. March Madness rallies the obvious sports fanatic, but also makes it easy for anyone to stay informed and enjoy the thrill that is collegiate sports. You may be thinking, "Okay.. but what does this have to do with math?". Staying with the trend of Everyday Math, the tournament inspired us to look into some key aspects of mathematics that are incorporated into various sporting events.


We'll start by taking a look at our inspiration- March Madness.

College Basketball

     Before the tournament tips off, it is estimated that more than 40 million Americans flock to ESPN or other sports-based websites to fill out their own tournament bracket. In all, it is estimated that about 70 million brackets will be completed. Bracket-filling strategy is a personal preference. Some stay loyal to their favorite college, selecting them to win-it-all, even though it isn't probable. Others hope to shock their competitors and pick underdogs, crossing their fingers that a rare bracket will do the trick. On the other hand, some people pick top seeded teams, leaning towards favorites. 

According to USA Today, the odds that you randomly select a perfect bracket– that's guessing every game correctly– are 1 to 9.2 quintillion. 

       More serious betters may look at the mathematics behind the tournament. Experts broke down mathematical models that can be used to simulate the tournament. The tournament can be thought of as a series of coin flips rather than games. However, instead of there being a 50/50 chance of flipping heads or tails, the odds are stacked differently depending on the teams playing. The NCAA has given each team a seed and has published data regarding the results of each of these seed numbers in the past. For an example, when #5 seeds and #12 seeds play against each other, the #5 seed wins 65% of the time. This data can be used to manipulate your bracket choices. This isn't the only way to mathematically simulate the games. Some use a point spread given by Vegas betting odds and convert these point spreads into winning percentage. The Bradley-Terry model converts computer rankings into probabilities. Of course, no method can perfectly account  for the twists and turns that occur in the tournament. Every year there are upsets that result in ruined brackets and one lucky guy who decided to take a chance on the underdog. The favorite this year is Kansas, and CBS Sports calculated the chance of a Kansas victory to be 16%.

Baseball

    Busy swinging the bat and keeping their eye on the ball, baseball players aren't consciously thinking about the mathematics that make their sp ort possible. However, hitting a home run or even making it to first base is centered heavily on angles, velocity, and energy. The speed of a pitch or of a ball after it's hit can be found using a specific equation. H_o is the height from which the ball is thrown, α is the angle at which the ball is thrown, v_o is the speed at which the ball is thrown, and x is the distance that the ball travels. From the graph below, it can be seen that hitting the ball at a 45 degree angle will cause the ball to travel the farthest. 

Distance baseball will travel

Graph showing range with different α's.
Black graph α = 30^o, blue graph when α = 45^o, red graph when α = 60^o

Projectile motion of a baseball

It can be argued that any sport involves similar physics. So why was baseball one of the sports chosen to examine with a mathematical lens? Mathematical approaches to managing baseball teams have surfaced throughout the sport. In fact, the term sabermetrics specifically describes the way in which statistical analysis is applied to baseball records. The term was coined from the acronym SABR (Society for American Baseball Research), and is used to evaluate and compare baseball players. Sabermetrics takes an emotionless, objective approach to baseball. It aims to answer only questions that can be proven with facts. Perhaps one of the most celebrated proponent of sabermetrics is Billy Beane, who inspired the movie Moneyball. Beane was the General Manager of the Oakland Athletics and used statistical analysis to lead the A's to a winning season. He looked at the risk factor of each player he brought into the program, and examined who was worth the money. He looked for players who may not have carried a famous name, but could still contribute to the team. In other words, he stretched the dollar and looked for economically smart choices. This is one example of how sabermetrics and analyzing stats is prominent in the sport of baseball.

Soccer

     Soccer is the world's most popular sport, and every day millions of people around the world take the field– from professionals to small children. However, I doubt any of these players take the time to examine the geometry that makes a soccer team successful. The basic shape of soccer is a triangle. the players on the field are connected by imaginary triangles, that build upon each other to diamonds and other shapes. Why a triangle? It allows for the most passing lanes and provides an option in every direction. The best teams in the world are known for being able to move the ball around the field in these triangles. 

     Free kicks provide another use for the application of geometry. Defenders line up in a wall, in hopes to impeded a direct path to the goal. The wall is set up at a specific angle and the person kicking the ball tries to bend the ball behind the ball. However, the amount of people that stand in the wall is dependent on where around the goal the ball lies. While the goalie isn't thinking about math when he/she sets up the wall, the logic behind it is definitely mathematical. The chart below shows that as the ball moves away from the center of the goal, less people are needed to defend the wall. When the ball is in the middle of the goal, the shooter has a larger angle, so the width of the wall needs to be greater. 

Everyday Math: Math in the Movies

Coming soon to theaters near you: Mathematics

You may not run to the nearest theater to watch a mathematician prove a theorem, but math may be starring in your favorite movie without your recognition. This installation of Everyday Math features big screen hits that would not be possible without mathematics. Usually a math story line is intertwined with a romance, mystery, or comedy, which can make it easier to miss. Don't worry! We're here to point out our favorite films that carry a heavy dose of real math information. Grab your popcorn and make your next movie night a math-themed experience!

The Classic: A Beautiful Mind

A Beautiful Mind is based loosely on the real life story of John Nash, a Nobel Prize winner. Russell Crowe plays John Nash, a mathematical genius that specialized in game theory, differential geometry, and differential equations. Game theory can be utilized in fields such as economics and political science. In fact, Nash won his Nobel Prize in economics. In a famous scene, the film dramatizes Nash's discovery of the Nash equilibrium, a term used in economics and game theory. The film is said to take artistic interpretation of Nash's real life, but the mathematics in the movie are based on real theorems and theories. The director of A Beautiful Mind enlisted the help of a mathematics consultant, Dave Bayer of Columbia University, to ensure the mathematics were correct throughout the film. The movie contains some math jokes and facts that may only be clear to those well versed in mathematics. For an example, at the end of the film, a student wants to show Nash a proof exploring the idea that "finite Galois extensions are the same as covering spaces", which is actually a true statement. Along with his prowess in mathematics, A Beautiful Mind also demonstrates Nash's story of mental illness and schizophrenia. His schizophrenia initially impacts his career, but he is able to recover and take his place as one of the leading mathematical and economic minds of his time.

Watch the dramatized discovery of Nash's Equilibrium here!

The Romantic Drama: Proof

Proof, starring Gwenyth Paltrow and Anthony Hopkins, ties in universal themes with the backdrop of mathematics. The movie tells the story of 2 mathematicians, father and daughter, fighting mental illness and attempting to prove various mathematical theorems. Before he dies, Robert (the father) makes note of the interesting characteristics of the number 1729. The daughter eventually takes the place of her ailing father and dedicates herself to mathematics, even though she lacks formal training. Critics explain that the film realistically expresses the field of mathematics. It shows the nuances of proving a theorem, as well as the work and studying that are involved. Paltrow's character, Catherine, describes how she feels when she is attempting to solve a difficult problem, comparing "elegant proofs" to music. Proof also makes reference to other real mathematicians that may or may not be known to the public, such as Sophie Germain and Carl Friedrich Gauss. The director of the film consulted heavily with Timothy Gowers, a Fields medalist from Cambridge University.

For the Sport's Enthusiast: Moneyball

Watch a mental math scene here

Moneyball is the perfect flick for sports enthusiasts and statisticians alike. Starring Brad Pitt, Moneyball takes a twist on the average sports film, incorporating mathematics into the strategy of baseball. The movie is adapted from a book of the same name, written by Michael Lewis. The plot is based on the 2002 Oakland A's and General Manager Billy Beane. Billy Beane took a different route when gathering players, in order to deal with the economic impositions placed on the team. He searched for undervalued players, and looked specifically at statistical analysis in order to determine who was worth the cost. Other 'risky' plays like stealing bases and bunting were thrown out the window under Beane's guidance. The heavy dose of mathematics in the movie is centered on the Pythagorean Expectation, which is used to calculate wins based on runs scored and allowed. The sabermetric approach to baseball is placed head to head against more traditional methods, and definitely makes for an entertaining film.

Pythagorean Expectation

The Teen Rom-Com: Mean Girls

This may be an outlier of the group, as the movie is geared towards teenage girls and features all the facets of your typical high-school movie. Mean Girls star Lindsey Lohan plays a homeschooled girl who tries her luck at navigating high school for the first time. She comes across some new friends, and they attempt to steer her in the right direction. Cady Herring, Lohan's character, has an aptitude for math and even joins the math club. Her math teacher, played by Tina Fey, is featured several times in the movie explaining various high-school math concepts. Late in the movie, Cady attends a mathlete competition and is faced with a limits problem. There are also scenes where she is being tutored in calculus, and the math errors showed are typical errors a high school student would make. This movie is perhaps the epitome of teen-movies circa 2004 but the mathematics represented, although correct, definitely correspond with the high school setting.

Click here to watch the scene!

The Story of An Underdog: Good Will Hunting

Known as a classic math movie, Good Will Hunting is a must-see. This underdog tale has a romantic twist and stars both Matt Damon and Robin Williams. Matt Damon's character, Will Hunting is a troubled young adult who's life path weaved in and out of foster homes and trouble with the law. While working as a janitor at MIT, he is able to solve two math problems that were created for graduate students. A professor at MIT took interest in Will, noticing his affinity towards mathematics and genius-level ability. Along his journey, Will faces his inner struggles with the help of a therapist (Robin Williams), and meets a romantic interest who helps to shape his life. The math problem that Will faces on the board is actually a real problem, although not as difficult as it is made out to be. The problem involves a feature of graph theory, homeomorphically irreducible trees. Pictured below is the problem that sent Will into the realm of academia. Interestingly enough, the math brains behind the movie actually appeared on screen as well. Patrick O'Donnell, who had a minor roll in the bar scene, actually ran the math department at University of Toronto at the time. O'Donnell and John Mighton, who plays the professor's assistant, chose the equations and theorems used in the movie. 

See the solution here!

Holiday Math: Dreidel Simulation

Happy Hanukkah, Center of Math fans! Since today's the first day (in Cambridge) of the holiday, I decided to take a mathematical/computational look at the dreidel game, one of the many traditions practiced on Hanukkah. Keep reading to learn how the game is played, and to see what I found out with my simulation of the game—the results may surprise you!