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Posts Tagged ‘combinatorics’


PascalsTriangleCoefficientThe first hour of this talk will be targeted primarily at the younger part of our audience (roughly in grades 4-7) – we will explain the fundamental rules of counting that can be used to solve even very hard problems: rule of sum (addition principle), rule of product (multiplication principle), pigeonhole principle, and the inclusion-exclusion principle. We will use them to solve a number of interesting problems from various competitions. For those students already familiar with these concepts, we will have a set of problems to keep them busy during the first hour.

The second hour will be targeted at the more experienced members of the Math Circle community (roughly grades 8-12). First, we will discuss several techniques used to prove combinatorial identities (combinatorial arguments, algebraic manipulations, and the method of generating functions). We will prove several important combinatorial identities using all of these techniques, illustrating the diversity of approaches found in combinatorics. After that we will look at applications of combinatorics in number theory, geometry, and graph theory, illustrating them with more interesting and challenging problems. While the younger part of our audience may not be able to follow everything during the second hour, it will be a great exposure to advanced mathematical topics, to show them they have a lot more to learn.

Throughout the talk we will highlight the mathematicians who developed this beautiful mathematical field, from its beginnings in gambling to modern applications in medicine, science, and technology.

PascalTriangleAnimated2

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220px-TernaryTreesWe are pleased to announce that the first topic of our 2013-2014 Metroplex Math Circle will be Pólya-Burnside Enumeration in Combinatorics, presented by our own Adithya Ganesh on September 14, 2013.

Burnside’s lemma from group theory has a broad scope of application in combinatorial enumeration problems.  Pólya’s enumeration theorem, which generalizes Burnside’s lemma using generating functions, provides a remarkable framework to easily solve counting problems in which we want to regard two entities as equivalent under some symmetry.

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Ivan Matic, assistant director of the Berkeley Math Circle and a former IMO medalist will teach our students a variety of pricinples related to Combinatorial Games.

Winning strategy for a particular game is a procedure that will ensure a victory no matter how the opponent is playing.   We will discuss some of the two player games that have winning strategies, and try to recognize the patterns for finding the strategies.   After that we will talk about multi-player games and probabilistic games.

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180px-Four_Colour_Map_Example.svgDon’t let the title mislead you, coloring isn’t kid’s stuff.  Coloring of graphs is a rich mathematical area of research which young mathematicians will encounter frequently.

Mr. Maier is going to talk about coloring arguments and some of their applications to combinatorial problems, especially problems from the theory of plane tilings, but also from knot theory, probability, and some popular puzzles.  Is it possible to cover an eight-by-eight inch chessboard with two-by-one inch dominos? Is it possible to cover the remainder with dominos? If so, how, and if not, why not?

If you’d like to learn more about coloring the following links may be useful:

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discrete mathA good friend of the Metroplex Math Circle, Dr. Arthur Benjamin, has just released a new lecture course through the Teaching Company titled “Discrete Mathematics.” We have our pre-ordered copy and its seems to have the unique combination of humor and depth that we know from Dr. Benjamin’s excellent “mathemagic” presentations.

For any students just starting with Math Circles, they will benefit greatly from becoming familiar with the topics on these DVDs: number theory,  combinatorics and graph theory.

Here is the description of the course from the Teaching Company:

Welcome to Discrete Mathematics, a subject that is off the beaten track that most of us followed in school but that has vital applications in computer science, cryptography, engineering, and problem solving of all types.

Most of the mathematics taught after elementary school is aimed at preparing students for one subject—calculus, which is the mathematics of how things grow and change continuously, like waves in the water or clouds in the sky. Discrete mathematics, on the other hand, deals with quantities that can be broken into neat little pieces, like pixels on a computer screen, the letters or numbers in a password, or directions on how to drive from one place to another.

While continuous mathematics resembles an old-fashioned analog clock, whose second hand sweeps continuously across a dial, discrete mathematics is like a digital watch, whose numbers proceed one second at a time. As a result, discrete mathematics achieves fascinating mathematical results using relatively simple means, such as counting.

Explore this modern realm of digital math in Discrete Mathematics, 24 mind-expanding lectures by veteran Teaching Company Professor Arthur T. Benjamin, an award-winning educator and mathemagician who has designed a course that is mathematically rigorous and yet entertaining and accessible to anyone with a basic knowledge of high school algebra.

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Alan Davis recap


In his lectures about combinatorics, Mr. Davis touched upon various topics. He gave students problems about counting the ways to rearrange and pick items, using candy as a visual to help the students understand. He also talked about the various strategies used to solve these problems. These strategies included using “bars and stars” to represent the problem, and using the Inclusion-Exclusion Principle. Lastly, he demonstrated the numerous applications of the famous Pascal’s Triangle.

Sample Problems:

Say there are 5 Jolly Ranchers and 3 Starbursts. How many ways are there to pick a Jolly Rancher or a Starburst? How many are there to pick a Jolly Rancher and a Starburst?

How many different ways are to rearrange the letters in the word ALABAMA?

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A
How many paths are there from point A to point B only going up and right?

If there are 3 people and 5 different candies, how many ways are there to distribute the candies if each person has to get at least one candy?

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combinatorics-for-undergradAlan Davis will continue his previous talk on Combinatorics with more challenging problems and concepts.  Those who missed his first talk are encouraged to attend promptly for a quick review at 2:00.

Mr. Davis will focus on the essence of the inclusion-exclusion principle with some interesting problems.

For the curious students who are eager to sharpen their problem-solving skills in Combinatorics, Mr. Davis recommends these two collections that Titu Andreescu and Zuming Feng collaborated on:

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We are very fortunate to have Alan Davis talk about combinatorics on February 21, 2009. His talk about combinatorics will include the following topics: Counting permutations and combinations, counting with repetition, binomial coefficients, Pascal’s triangle, Pascal’s identity, Vandermonde’s identity, and the inclusion-exclusion principle.
Combinatorics is related to many other areas of mathematics such as algebra, probability, and geometry, and also computer science and statistics. More information about this interesting branch of mathematics can be found here.

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Our November 19, 2008 Special Presentation will be by Dr. Arthur Benjamin, who in addition to being a sought after speaker is also a successful author.  Many people are familiar with Dr. Benjamin from the course that he authored for the Teaching Company entitled Joy of Mathematics. Following is an excerpt from the course description:

Fun with Numbers

Here is an example:

Think of a number between 1 and 10. Triple it. Add 6. Then triple again. Now take your answer, probably a two-digit number, and add the digits of your answer. If you still have a two-digit number, add those digits again. You should now be thinking of the magical number 9. The reason this works is based on algebra and the fact that the digits of any multiple of 9 must sum to a multiple of 9.

This is one of the many wonders of modular arithmetic, sometimes called clock arithmetic, where numbers wrap around in a circle. A useful application of this field is casting out nines, a simple and ancient technique for checking the answers to arithmetical problems.

Modular arithmetic also provides a very handy method for mentally computing the day of the week for any date in history.

This connection between entertaining number tricks and the deeper properties of mathematics reflects Dr. Benjamin’s specialty, which is combinatorics, the branch of mathematics that deals with the subtleties of counting. Some examples: How many different six-symbol license plates are possible? And for the book collector, how many ways are there of arranging 10 books on a shelf? (Would you believe more than 3 million?) These simple questions introduce concepts such as the factorial function.

Drawing on his dual fascination with combinatorics and games, Dr. Benjamin used his analytical skill to win first place in the American Backgammon Tour in 1997.

In addition to his scholarly articles, Dr. Benjamin has also written books that are highly accessible to the public.  One such book is Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks co-written with Michael Shermer with a foreword by Bill Nye the Science Guy.  Praise for this book includes:

secretsofmentalmath-bookcover“A great introduction to the wonder of numbers, from two superb teachers.”
— Brian Greene, author of The Elegant Universe

“A magical mystery tour of mental mathematics! Fascinating and fun.”
— Joseph Gallian, president of the Mathematical Association of America

“The clearest, simplest, most entertaining, and best book yet on the art of calculating in your head.”
— Martin Gardner, author of Mathematical Magic Show and hundreds of Mathematical Games columns for Scientific American.

“This book can teach you mental math skills that will surprise you and your friends. Better, you will have fun and have valuable practical tools inside your head.”
— Dr. Edward O. Thorp, mathematician and author of Beat the Dealer and “Beat the Market”

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In addition to being the subject of books like Count Down, the Director of Metroplex Math Circle, Dr. Titu Andreescu is also the author of multiple books on problem solving. These books draw on his many years of experience as the director of AMC, coach of the US International Math Olympiad team and author of many contest problems.

To help the Metroplex Math Circle community we have created an Amazon List with some of Dr. Andreescu’s currently available books. In addition to Dr. Andreescu’s books for experienced problem solvers we have also included some books and resources on the list for students just starting into problem solving.

Not only does Metroplex Math Circle benefit from Dr. Andreescu himself, but many of his co-authors are also friends of MMC and frequent lecturers.

Following are the author descriptions from the book 104 Number Theory Problems: From the Training of the USA IMO Team:

About the Authors

Titu Andreescu received his Ph.D. from the West University of Timisoara, Romania. The topic of his dissertation was “Research on Diophantine Analysis and Applications.” Professor Andreescu currently teaches at The University of Texas at Dallas. He is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002), director of the Mathematical Olympiad Summer Program (1995–2002), and leader of the USA IMO Team (1995–2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world’s most prestigious mathematics competition. Titu co-founded in 2006 and continues as director of the AwesomeMath Summer Program (AMSP). He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a “Certificate of Appreciation” from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titu’s contributions to numerous textbooks and problem books are recognized worldwide.

Dorin Andrica received his Ph.D. in 1992 from “Babes-Bolyai” University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at “Babes-Bolyai” since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. He is an invited lecturer at university conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called “Andrica’s Conjecture.” He has been a regular faculty member at the Canada–USA Mathcamps between 2001–2005 and at the AwesomeMath Summer Program (AMSP) since 2006.

Zuming Feng received his Ph.D. from Johns Hopkins University with emphasis on Algebraic Number Theory and Elliptic Curves. He teaches at Phillips Exeter Academy. Zuming also served as a coach of the USA IMO team (1997-2006), was the deputy leader of the USA IMO Team (2000-2002), and an assistant director of the USA Mathematical Olympiad Summer Program (1999-2002). He has been a member of the USA Mathematical Olympiad Committee since 1999, and has been the leader of the USA IMO team and the academic director of the USA Mathematical Olympiad Summer Program since 2003. Zuming is also co-founder and academic director of the AwesomeMath Summer Program (AMSP) since 2006. He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1996 and 2002.

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