Dana Randall Gives the 2009 Arnold Ross Lecture, Followed By Who Wants to Be a Mathematician
The Arnold Ross Lecture and Who Wants to Be a Mathematician traveled to the National Science Center's Fort Discovery in Augusta, Georgia on October 29 for a morning of mathematics.
"Thanks so much for all the work you guys did in bringing this lecture to Augusta. All the students had a great time."
"I thought that the lecture was good. It is a great way to show what mathematicians do. Thanks for a great program and a stimulating lecture. The students certainly benefited from it. "
"Thank you for bringing the math game show to Augusta. Andrew Ding is a deserving winner, and his good work and good fortune have created a lot of enthusiasm here at Augusta Prep. I know that you and the sponsors take pride in increasing students' appreciation and enjoyment of mathematics. "
First, Dana Randall, professor in the Schools of Mathematics and Computer Science at Georgia Tech gave an introduction to tilings, discussing many aspects of the topic. Then Andrew Ding of Augusta Preparatory Day School won US$3000 and a TI-Nspire playing Who Wants to Be a Mathematician with seven other high school students from Georgia and South Carolina. Below is a summary of the events.
Domino Tilings of the Chessboard: An Introduction to Sampling and Counting, Dana Randall, Georgia Institute of Technology
Rob Dennis (above left), president and CEO of the National Science Center, welcomed the crowd of nearly 150 students and teachers to Fort Discovery and introduced AMS Secretary Bob Daverman (above right), University of Tennessee. Daverman told the audience about Arnold Ross and about Randall, who earned her undergraduate degree in math from Harvard University and her Ph.D. in computer science from the University of California, Berkeley.
Randall first talked about her experience in high school at summer math camps and recommended that students look into attending such camps.
She introduced tilings by asking how many ways there are to build a 2 x n wall with 1 x 2 bricks? After calculating the number for some small n, Randall explained that the number of ways is a Fibonacci number. The brick problem is one example of a domino tiling (covering a region with unmarked 2 x 1 dominoes without overlap). Tiling is an appealing subject in its own right, but why should students care about it? Randall not only answered that question (the short answer: "Because they're cool"), but also answered
- Where tilings exist?
- How many tilings exist for a region?
- What do tilings look like?
- When do algorithms involved in tilings stop?
For an n x n chessboard, n must be even in order for a tiling to exist, and if some squares of the board are removed to form a related region, those regions can only be tiled with dominoes if an equal number of black squares and white squares are removed. This latter condition is not sufficient, however. Randall concluded the "Where" part of the lecture by noting that an algorithm by William Thurston can tell when a region is tilable and if so, find a tiling.
She then showed different regions, some of which can be thought of in three dimensions, and showed ways of counting the number of tilings. To count the number of ways to tile a chessboard, Randall set up a correspondence between special non-intersecting paths from one edge to the opposite edge and used determinants (which eliminated some double counting) to get the number of tilings.
One nice result that also related to random tilings involved tiling Aztec diamonds. For a diamond with 2n rows, the number of 1 x 2 domino tilings is 2n(n+1)/2. Even though the picture below may not look random, Randall said that coloring Aztec diamonds at random often leads to concentrations of colors at the vertices of the diamond.
Image courtesy of Dana Randall.
Tilings are interesting to specialists in other fields besides mathematics. There are applications in chemistry--to the strength of bonds in randomly generated hydrocarbons--and in physics--to problems in thermodynamics.
What a typical tiling looks like is helpful when analyzing random tilings. Randall talked about Markov chains and related algorithms that generate random tilings to card shuffling. The algorithm she described was rapid, taking polynomial time rather than exponential, so that the answer to When? is "not long."
At the end of her lecture, Randall stated that domino tilings involve algebra, combinatorics, geometry, probability, algorithmic theory, and relate to chemistry, biology, physics, and nanotechnology.
Who Wants to Be a Mathematician
Who Wants to Be a Mathematician followed the lecture and a nice refreshment break. The contestants are listed below, and pictured with Dana Randall and AMS Public Awareness Officer and host of Who Wants to Be a Mathematician, Mike Breen.
Front (left to right):
Back (left to right):
As mentioned above, Andrew Ding was the big winner, taking home $3000 from the AMS and a TI-Nspire graphing calculator from Texas Instruments. Both games were pretty close--each having many lead changes. In game one, Elijah led at the halfway point, then Andrew Byrd led for two questions. Taylor took the lead at question seven and held on for the victory. This earned him $500 and a TI-Nspire. Taylor had an enthusiastic rooting section, as did Brooke in game two, which made up part of the big audience.
In game two, Joe led at the halfway point and after question five. Andrew Ding and Emily were tied after question six, then Andrew pulled away to win. Joe gave good explanations for his answers, as did Andrew, and used game one contestant Elijah for his Help.
Taylor and Andrew squared off for the chance at the Bonus Question, worth $2000.
The Square-Off question took a while, but Andrew signaled in first and answered correctly. Taylor said that he was closing in on the answer when Andrew signaled in, but wasn't quite ready to make his choice. He congratulated Andrew and stayed on stage for the Bonus question. The audience, which had been cheering at all the right times during the games, was very quiet during the three minutes that Andrew was allotted for the question.
Andrew did some figuring and made his choice with plenty of time to spare. Then members of the audience got a chance to express their preferences and their most popular choice turned out to be Andrew's selection as well. He gave a nice explanation about why he chose the answer that he did and earned the $2000 for the bonus, in addition to $500 and a TI-Nspire from his game one victory and $500 from winning the Square-Off Round.
Here are all the prizes and money won that day.
- TI-Nspire graphing calculator from Texas Instruments and $3000 from the AMS: Andrew Ding
- TI-Nspire graphing calculator from Texas Instruments and $500 from the AMS: Taylor Harvey
- Maple 13 from Maplesoft: Andrew Byrd and Emily Herndon
- Calculus by Anton, Bivens and Davis from John Wiley and Sons: Elijah Coleman and Joe Shepherd
- What's Happening in the Mathematical Sciences from the AMS: Kristan Shuford and Brooke Hargrove
The AMS thanks sponsors Texas Instruments, Maplesoft, and John Wiley and Sons for their continued generous support of Who Wants to Be a Mathematician. Thanks also to the National Science Center for hosting the event and to the Augusta area teachers and students in Georgia and South Carolina who attended.
Read the Augusta Chronicle article about the lecture and game: "Game equals fun, money for math students," by Preston Sparks.
Photographs by Who Wants to Be a Mathematician judge and co-creator Bill Butterworth (DePaul University Department of Mathematical Sciences), Robin Aguiar (Meetings and Professional Services), and by Who Wants to Be a Mathematician host and AMS Public Awareness Officer Mike Breen.