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Archive for March, 2014


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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armlMetroplex Math Team students practice for ARML. We plan to practice Team, Individual, and Relay rounds and discuss solutions to some of these challenging problems. Join us!

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Cayley-graph-of-the-cyclic-group-Z-by-8Z-with-the-generators-S23Abstract Algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields.

In the following introduction to this topic, we will discuss Binary Operations, Groups, Subgroups, Cyclic Groups, Cayley Digraphs, and how they relate to each other. With a thorough understanding of these topics, students will have the basis to further examine the subject that is Abstract Algebra.

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