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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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