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Archive for the ‘math circle’ Category


220px-Full_Adder.svgAddition is the most elementary arithmetic operation required of computers. Their invention and development has almost entirely been to do arithmetic extremely fast.  I will present the simplest binary adder circuits (ripple carry adder), show some surprising results about carry propagation, develop carry look-ahead circuits for log-time addition, and then talk about redundant number systems and constant time addition and, consequently, log-time multiplication.
I will also present residue number systems that enable constant time addition and multiplication.
It’s all integer math, modulo arithmetic, and some simple logic circuits.

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Sieve_of_Eratosthenes_animation
Why do prime numbers fascinate us?
Why prime numbers are still an active area of research?
What advances have mathematicians made in this field this year?
You will find answers to these questions and many more if you come to the lecture.
See you there!

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Please join us this Saturday as Adrian Andreescu leads the group through a selection of some of the most interesting AMC 8 problems.

cof-1stplace

And while you attend, if you meet the requirements below you can sign up for the AMC 8 according to the following instructions:

Hello All,

Hooray, it’s getting close to AMC 8 time again!  I’m handling signups differently this year.  First, here are the particulars for the test:

  • When:  November 19th at 5pm SHARP
  • Where:  The Davis Library
  • Requirements:  Must be in 8th grade or below and current school does not offer the test.
  • Cost:  $10 (The Davis library charges $50 for the program room, so I’ve had to increase the cost to cover all related fees)
  • What to bring:  number 2 pencils, protractor, compass (scratch paper will be provided).  NO calculators

If you meet the above requirements, then registration for the test will happen in person on November 2nd.  Here are the particulars to sign up:

If you have any questions, feel free to email me.

Best,

Kathy Cordeiro

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Lissajous_animation
Let’s talk about the sine and cosine functions. One does not need to use very much information about these commonly seen functions in order to understand a large number of curves which can be drawn by graphing sine and cosine in Cartesian and polar coordinates. We will see sine curves, sums of sine curves, Lissajous figures, cycloids, hypocycloids, epicyclodes, and, of course, many rows of roses.

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Titu PortraitThis Saturday, Dr. Andreescu will be answering the eternal question:  “Why Math on a Saturday Afternoon?”

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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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364px-nine-point_circle_svgThis Saturday, April 20th Dr. Kisačanin will return for another of his fantastic lectures.  Triangles factor into almost every math contest in addition to being endlessly fascinating objects in themselves.  Here is Dr. Kisačanin’s description of the session with links to resources:

In this talk about geometry of triangles we will see two different proofs of Stewart’s theorem, derive formulas for important cevians, and solve several interesting geometric problems.

We will also look at other important points in triangles (Fermat point, centers of excircles, …) and look at the Euler line, the nine-point circle, and related problems.

http://en.wikipedia.org/wiki/Stewart%27s_theorem
http://en.wikipedia.org/wiki/Fermat_point
http://en.wikipedia.org/wiki/Excircle
http://en.wikipedia.org/wiki/Euler_line
http://en.wikipedia.org/wiki/Nine_point_circle

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stanford-paul-2009-08For 2ⁿ – 1 to be prime we also need n itself to be prime, but that is not sufficient. For example, 2¹¹ – 1 is composite even though 11 is prime.  However, if you look at tables of Mersenne primes it is interesting to note that if you start with 2 and use that to make a new number 2ⁿ – 1 with n = 2 you get 3, then recycling the 3 you get 7, use n = 7 and you get 127, another prime! How long could this go on?

Let f(n) = 2ⁿ – 1.  The iterations you get, starting from 2, are f⁰(2) = 2, f¹(2) = 3, f²(2) = 7, f³(2) = 127, f⁴(2) = 1701411834604692317316873037158884105727.

It turns out these are all prime!  But what of the next one???  Well, it may be a very long time before any of us know. The largest value of n for which 2ⁿ – 1 is known to be prime is n = 57885161, and that after a concerted effort using volunteers from around the globe.  Not much chance of answering this one in our lifetimes, unless some really new idea arrives.Before you make a hasty conjecture (as has already been done), a cautionary piece of history is in order.  If you define a new function to iterate you get some other interesting numbers.  Let g(n) = n² – 2n + 2. Then the iterates are g⁰(3) = 3, g¹(3) = 5, g²(3) = 17, g³(3) = 257, g⁴(3) = 65537, and all of these are prime!  So, with forgivable excitement, the conjecture was made that all of these will be prime, especially as the next one, g⁵(2) = 4294967297, was much too large at the time for mere mortals to conceive of factoring with their bare hands.

None, that is, until Euler combined his genius with an impish disbelief in Fermat’s conjecture to discover that g⁵(2) = 4294967297 = 641 * 6700417.  And since then we have found many more composite Fermat numbers, and no further Fermat primes, leading to the complementary conjecture that all the rest are composite!  It seems that we never learn to be humble around these things…

It takes a larger number to be “forever beyond reach” these days.  Rather than the now puny 4294967297 we cower before f⁵(2) = 2¹⁷⁰¹⁴¹¹⁸³⁴⁶⁰⁴⁶⁹²³¹⁷³¹⁶⁸⁷³⁰³⁷¹⁵⁸⁸⁸⁴⁴¹⁰⁵⁷²⁷ – 1, and who can blame us?

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Titu Portrait With the AMC tests behind us and the AIME to come, don’t miss this opportunity to hear Dr. Andreescu talk about some of his favorite problems!

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image001In a follow up lecture on mathematical functions, we will explore more stories and problems related to polynomials, trigonometric functions, and functional equations. Furthermore, we will dive deeper into the original historical context of functions – curves such as cycloids, cardioids, catenaries, circles, ellipses, hyperbolas, parabolas. Finally, we will try to understand why exponential and trigonometric functions turn up in solutions of so many fundamental problems in math, physics, and engineering. Come and learn with us!

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