This week, we explored combinatorics, the study of the method of counting. Specifically, we went over the multiplication principle, the addition principle, multiple independent events, permutations and combinations.
Which polynomials take integer values p(x) at all integer points x? (It's not just the ones that have integer coefficients!) We'll introduce the finite difference operator and apply properties of it to arrive at a simple but surprising characterization of integer-valued polynomials.
During the first hour, we will learn some simplest forms of input and output and examine a very efficient division algorithm. During the second hour, we will be training for the upcoming AMC8 competition.
We will be working on some practice problems for the Math Kangaroo this week. The Math Kangaroo is an annual mathematics competition that takes place in March. If your child is interested in competing, please register at their website (www.mathkangaroo.org) as soon as possible, as registration fills up quickly.
In this meeting we will explore the Stable Marriage Problem, a classical problem in economics initially studied by David Gale and UCLA professor Lloyd Shapley. The pioneering work of Gale and Shapley has inspired hundreds of research articles and several books. We will give a gentle introduction to the Stable Marriage Problem and its applications to college admissions.
We will continue our exploration of the stable marriage problem, including variants such as dishonest preference lists, incomplete preference lists, many-to-one matching (the hospital/residents problem), and the stable roommates problem.
For the first hour, Mark Ponomarenko will present his winning project for the last year MathMOvesU competition, Dice, Coin Flips, Quantum Mechanics, and Randomness. We will get back to studying recursive functions and fractals during the second hour.
We explore the use of modular arithmetic in modern day cryptography. We do this by first exploring the Caesar cipher in the context of modular arithmetic and develop a better cipher called "Simplified RSA".
Queueing theory applies mathematical models for waiting lines, with applications in the design of telephone systems, computer networks, hospital emergency departments, and more. In a queueing system, customers arrive and are served by servers, and the arrival times of customers and the service times for customers may be random. We study one model of queues (the "M/M/1/K" model) and how customer arrival rate, service rate, and system capacity affect properties of the queue.
We will figure out the area of Koch snowflake. The area is finite, but the perimeter has infinite length. This way, Koch snowflake provides an example of a curve of infinite lenght bounding a finite area. (Oleg Gleizer)
"Classical" constructions in geometry in the ancient Greek tradition only allow the use of a straightedge (with no markings on it) and a compass. What constructions can be achieved with different restrictions? In this session, we explore constructions that make use of a marked ruler.
First proven by Steiner in 1833, every geometric construction with a compass and straightedge can be accomplished using a straightedge alone, as long as a single circle and its center are given. In this session, we will find the constructions that establish the Poncelet-Steiner Theorem.
This week, we will be playing a game called math dominoes. The students will work in pairs and compete against their classmates, and problems will mostly be based off of what we learned this quarter. In order to facilitate the process, please go over the rules with your child. The domino scoring system could be confusing at first, so please make sure your child knows how the system works prior to class on Sunday.
The students, split into pairs, will be competing in proving various mathematical statements, from fractals to geometry to pigeonhole principle. The winner of each pair will progress to the next round. At the end, there will be only one!
Welcome back! Math Kangaroo is in a couple of months, so we are doing practice for the competition. Please note that if your child wants to compete, registration is through the Math Kangaroo website, NOT through Math Circle.
We will define and study a variant of the center of mass of a polygon, called the circumcenter of mass. The circumcenter of mass is defined by triangulating the polygon, finding the circumcenter of each triangle, and taking the weighted average of those circumcenters, where each circumcenter is weighted by the area of its triangle. Analogues of the Archimedes Lemma and the Euler line result.
Today we will use the division algorithm we learned last week as our main tool in proving that square roots of prime numbers are irrational, that there are infinitely many prime numbers, and that prime factorization of integers is unique.
We will resume our study of fractions from Problem 8 of the 1/17 handout. We will learn geometric sequences and use them as a tool to find rational representations of real numbers having an infinite recurring part in the decimal form. We will further construct a bijection between the set of rational numbers (p/q, p and q co-prime integers) and the set of real numbers having the terminating (finite) or infinite recurring form.
Next time we will resume by discussing Problem 12 from the 1/17 handout at the board. We will proceed to study geometric sequnces, series, and their limits. We will use those as tools for converting real numbers having an infinite recurring decimal part to the rational form.
Our main goal for this section is to learn how to determine whether or not a solution exists for the 15 Puzzle. This week, we start tying together and applying what we have learned about permutations and taxicab geometry.
Our main goal for this section is to learn how to determine whether or not a solution exists for the 15 Puzzle. This week, we tie everything together by proving that configurations of the 15 puzzle with opposite parities cannot be solved, and also introduce some logic to show why this is not sufficient.
We investigate the rotational symmetries of the platonic solids. ***For this session, please bring scissors and tape*** for making paper models of the solids. Alternatively, you can make the models at home (see templates below - credit goes to mathsisfun.com) and bring them to the session.
A homotopy is a continuous deformation with bending, stretching, and squishing, but not tearing or gluing. We introduce the basic ideas of homotopy theory: homotopy equivalence and the fundamental group of a space.
We will finally finish studying the 1/31 handout. We will solve a few cool problems on geometric sequences and series. In particular, we will resolve the famous Zeno's paradox about Achilles and a tortoise. If time permits, we will start studying the book Algebra by I. Gelfand and A. Shen.
We will resume studying of the 1/31 handout from Problem 13. We will further use geometric series to resolve the most famous of Zeno's paradoxes, the one about Achilles and a tortoise. If time permits, we will start learning from the Algebra book by Gelfand and Shen.
At the beginning of this class we will (hopefully) finish the 1/31 handout, solving Problems 18 - 23. Then we will start a new topic, Mathematical Induction and Peano Axioms. The goal of the new mini-course is to show that a + b = b + a for any two non-negative integers a and b. To prove this seemingly obvious statement, we will need to teach an Artificial Intelligence (AI) some elementary arithmetic, proving that 1 + 1 = 2 as well as associativity and commutativity of addition along the way.
Two thirds of the class have stopped working around Problem 12 of the 4/3 handout. We will resume at Problem 12 next time. A third of the class has finished the handout. They will be given Olympiad-style problems.
We will go over the proof of commutativity of addition of non-negative integers one more time. Then we will proceed to solve problems from the next handout. If time permits, we will also discuss the solution of the functional equation xf(x+xy) = xf(x) + f(x^2)f(y). The problem was brought about by Matthew Roth - thanks, Matt!
Today we will be reviewing what we have learned this school year! This is to help with Math Dominoes in the final class, which will cover what we have studied this year. (Please note that not all the concepts are covered in this review for the sake of time.)