Posts Tagged ‘recursion’

While Metroplex Math Circle is on Thanksgiving break please enjoy this replay of the excellent talk given by Mathew Crawford in our last session:

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This weekend we have a very special guest lecturer.  Mathew Crawford, who will be well known to many MMC attendees as the author of the popular AOPS titles:  Introduction to Number Theory and  Intermediate Algebra.   Mr. Crawford will be bringing with him 50 copies of problem materials which will be available only to the first 50 families to join us.  If you are unfamiliar with Mathew Crawford and his extensive work, the following information comes from the AOPSWiki:

Mathew Crawford is the founder and CEO of MIST Academy, a school for gifted and talented students, headquartered in Birmingham, Alabama. Crawford won numerous national math championships as a student before attending Washington University in St. Louis on a Compton Fellowship where he studied mathematics and worked on the Human Genome Project at the Institute for Biomedical Computing. After spending several years on Wall Street and eventually running a finance operation from the basement of his apartment, Crawford founded his first education company in 2001, Universal Set Educational Resources, with childhood friend Cameron Matthews. In 2003, Crawford became the first employee of Art of Problem Solving where he helped to write and teach most of the online classes during the first three years of the AoPS online school.

His competition achievements include:

  • National MathCounts written test champion in 7th grade (perfect score of 46) and second place in 8th grade (score of 44).
  • Two-time perfect scorer on the AHSME.
  • Perfect score on the AIME as a freshman.
  • Three-time invitee to the Mathematical Olympiad Summer Program.
  • Member of a top 4 Putnam team.
  • Youngest winner of the National Mu Alpha Theta convention.
  • Only 5-time winner of the Alabama State Written Examination (Algebra II/Trig once, Comprehensive four times).
  • Twice among ARML high scorers (tie-breakers) and Zachary Sobol Award winner.

Crawford also writes competition problems and performs duties for many math competitions:

  • USAMTS problem writer and grader (2004-2006)
  • iTest head test writer (2007 and 2008)
  • Birmingham and Alabama MATHCOUNTS coordinator
  • Mu Alpha Theta test writer and proof reader
  • Co-coach of the Missouri ARML team (1996,1997)
  • Coach of the San Diego ARML team (2005,2006)
  • Coach of the Alabama ARML team (2008, 2010-present)
  • Headed up the grading of the Power Round for the Georgia ARML site (2010)
  • San Diego Math League test writer and problem writer for the San Diego Math Olympiad (2004-6)

His first book, Introduction to Number Theory was published by AoPS in June, 2006. He is also coauthor of the Intermediate Algebra text, which came out in April, 2008.

Crawford’s user page can be found here.

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Sequences of numbers are often defined using a recursive relation and initial conditions, for example, the sequence of Fibonacci numbers is defined with F_{n+2} = F_{n+1} + F_{n} and initial conditions F_{1} = 1, F_{2} = 1. In this lecture we will see (a) how to solve various types of recursions (b)How to determine various sequence properties of number sequences directly from the recursions, and (c) how this knowledge can come in handy in many competition problems as well as in the study of computer science.

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The 2010-2011 season of Metroplex Math Circle will have a strong start with our first lecture by Dr. Branislav Kisačanin.

Dr. Kisačanin is a computer scientist at Texas Instruments working in the field of computer vision. In his spare time he likes to use computer vision for fun project such as his Tetris-playing robot. He loved math and physics competitions and nowadays likes to challenge his wife, kids, and friends with math problems and puzzles. Dr.Kisačanin wrote a book about selected mathematical gems:Mathematical Problems and Proofs.

Dr. Kisačanin has chosen Number Sequences as his first topic.  These sequences are critical parts of the tool kit of any middle school problem solver but also offer unexplored mysteries for professional mathematicians.  Following are some of the topics Dr. Kisačanin may cover in the course of his talk:

Triangular numbers

  • Derive the formula and the relation to square numbers both algebraically and geometrically
  • Sum of n squares (Gregory’s vanishing triangle, induction)
  • Sum of n cubes (connection back to triangular numbers)
  • Excercise: Sum of n odd numbers (again alg + geom)
  • OPTIONAL: Sums of higher powers (telescoping)

Fibonacci numbers

Binomial coefficients

  • Coefficients in (a+b)^n
  • Combinations
  • Pascal’s Triangle
  • Excercise: each row sums to 2^n, finding triangular and Fibonacci numbers, …

Other sequences and open problems

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