Tag Archives: open problem

We Use Math, Integermania, and Best-of-Seven

Welcome to this week’s Math Munch!

astronaut“When will I use math?” Have you ever asked this question? Well, then you are in for a treat, because the good people of We Use Math have some answers for you! This site was created by the Math Department at Brigham Young University to help share information about career paths that are opened up by studying mathematics. Here’s their introductory video:

The We Use Math site shares write-ups about a wide range of career opportunities that involve doing mathematics. I was glad to learn more about less-familiar mathy careers like technical writing and cost estimation. Also, my brother has studied some operations management in college, so it was great to read the overview of that line of work. In addition, the We Use Math site has pages about recent math discoveries and about unsolved math problems. Check them out!

Next up is one of my long-time favorite websites: Integermania!

Perhaps you’ve heard of the four 4’s problem before. Using four 4’s and some arithmetic operations, can you make the numbers from 1 to 20? Or even higher? Some numbers are easy to make, like 16. It’s 4+4+4+4. Some are sneakier, like 1. One way it can be created is (4+4)/(4+4). But what about 7? Or 19? This is a very common type of problem in mathematics—which math objects of a certain type can be built with limited tools?

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Steven J. Wilson

Integermania is a website where people from around the world have submitted number creations made of four small numbers and operations. It’s run by Steven J. Wilson, a math professor at Johnson County Community College in Kansas. (Steven has even more great math resources at his website Milefoot.com)

There are many challenges at Integermania: four 4’s, the first four prime numbers, the first four odds, and even the digits of Ramanujan’s famous taxicab number (1729).

Here are some number creations made of the first four prime numbers. Can you make some of your own?

Here are some number creations made of the first four prime numbers.
Can you make some of your own?

One of my favorite aspects of Integermania is the way it rates number creations by “exquisiteness level“. If a number creation is made using only simple operations—like addition or multiplication—then it’s regarded as more exquisite than if it uses operations like square roots or percentages. I also love how Integermania provides an opportunity for anyone to make their mark in the big world of mathematical research—it’s like scrawling a mathematical “I wuz here!” After years of visiting the site, I just submitted for the first time some number creations of my own. I’ll let you know how it goes, and I’d love to hear about it if you decide to submit, too.

Here are recaps of all the World Series since 1903 from MLB.com

Here are recaps of all the World Series since 1903 from MLB.com

Now coming to the plate: my final link of the week! Monday was the first day of the new Major League Baseball season. I want to share with you a New York Times article from last December. It’s called Keeping Score: Over in Four About a Fifth of the Time. The article digs into the outcomes of all of the World Series championships—not so much who won as how they won. It takes four victories to win a seven-game series, and there are 35 different ways that a best-of-seven series can play out, put in terms of wins and losses for the overall winner. For instance, a clean sweep would go WWWW, while another sequence would be WWLLWW. The article examines which of these win-loss sequences have been the most common in the World Series.

(Can you figure out why there are 35 possible win-loss sequences in a seven-game series? What about for a best-of-five series? And what if we tried to model the outcome of a series by assuming each team has a fixed chance of winning each game?)

worldseriesstats

A clip of the stats that are displayed in the Times article. Click through to see it all.

I was curious to know if the same results held true in other competitions. Are certain win-loss sequences rare across different sports? Are “sweeps” the most common outcome? After sifting through Wikipedia for a while, I was able to compile the statistics about win-loss sequences for hockey’s Stanley Cup Finals. This has been a best-of-seven series since 1939, and it has been played 73 times since then. (It didn’t happen in 2005 because of a lockout.) You can see the results of my research in this document. Two takeaways: sweeps are also the most common result in hockey, but baseball more frequently requires the full seven games to determine a winner.

It could be a fun project to look at other best-of-seven series, like the MLB’s League Championship Series or basketball’s NBA Finals. If you pull that data together, let us know in the comments!

Batter up, and bon appetit!

******

UPDATE (4/4/13): My first set of five number creations was accepted and are now posted on the Ramanujan challenge page. Here are the three small ones! Can you find a more exquisite way of writing 47 than I did?

myintegermania

Marjorie Rice, Inspired by Math, and Subways

Welcome to this week’s Math Munch!

A few weeks ago, I learned about an amazing woman named Marjorie Rice.  Marjorie is a mathematician – but with a very unusual background.

mrice_picMarjorie had no mathematical education beyond high school.  But, Marjorie was always interested in math.  When her children were all in school, Marjorie began to read about and work on math problems for fun.  Her son had a subscription to Scientific American, and Marjorie enjoyed reading articles by Martin Gardner (of hexaflexagon fame).  One day in 1975, she read an article that Martin Gardner wrote about a new discovery about pentagon tessellations.  Before several years earlier, mathematicians had believed that there were only five different types of pentagons that could tessellate – or cover the entire plane without leaving any gaps.   But, in 1968, three more were discovered, and, in 1975, a fourth was found – which Martin Gardner reported on in his article.

Marjorie's first type of pentagonWhen she read about this, Marjorie became curious about whether she could find her own new type of pentagon that could tile the plane.  So, she got to work.  She came up with her own notation for the relationships between the angles in her pentagons.  Her new notation helped her to see things in ways that professional mathematicians had overlooked.  And, eventually… she found one!  Marjorie wrote to Martin Gardner to tell him about her discovery.  By 1977, Marjorie had discovered three more types of pentagons that tile the plane and her new friend, the mathematician Doris Schattschneider, had published an article about Marjorie’s work  in Mathematics Magazine.

type11There are now fourteen different types of pentagons known to tile the plane… but are there more?  No one knows for sure.  Whether or not there are more types of pentagons that tile the plane is what mathematicians call an open problem.  Maybe you can find a new one – or prove that one can’t be found!

Marjorie has a website called Intriguing Tessellations on which she’s written about her work and posted some of her tessellation artwork.  Here is one of her pentagon tilings transformed into a tessellation of fish.

fishgrid fishsm

By the way, it was Marjorie’s birthday a few weeks ago.  She just turned 90 years old.  Happy Birthday, Marjorie!

wild about math logoNext up, I just ran across a great blog called Wild About Math!  This blog is written by Sol Lederman, who used to work with computers and LOVES math.  My favorite part about this blog is a series of interviews that Sol calls, “Inspired by Math.”  Sol has interviewed about 23 different mathematicians, including Steven Strogatz (who has written two series of columns for the New York Times about mathematics) and Seth Kaplan and Deno Johnson, the producer and writer/director of the Flatland movies.  You can listen to Sol’s podcasts of these interviews by visiting his blog or iTunes.  They’re free – and very interesting!

subway map 2Finally, what New York City resident or visitor isn’t fascinated by the subway system? And what New York City resident or visitor doesn’t spend a good amount of time thinking about the fastest way to get from point A to point B?  Do you stay on the same train for as long as possible and walk a bit?  Or do you transfer, and hope that you don’t miss your train?

chris and matt

Chris and Matt, on the subway.

Well, in 2009, two mathematicians from New York – Chris Solarz and Matt Ferrisi – used a type of mathematics called graph theory to plan out the fastest route to travel the entire New York City subway system, stopping at every station.  They did the whole trip in less than 24 hours, setting a world record!  Graph theory is the branch of mathematics that studies the connections between points or places.  In their planning, Chris and Matt used graph theory to find a route that had the most continuous travel, minimizing transfers, distance, and back-tracking.  You can listen to their fascinating story in an interview with Chris and Matt done by the American Mathematical Society here.

If you’re interested in how graph theory can be used to improve the efficiency of a subway system, check out this article about the Berlin subway system (the U-bahn).  Students and professors from the Technical University Berlin used graph theory to create a schedule that minimized transfer time between trains.  If only someone would do this in New York…

Bon appetit!