Fall 2015 quarter  8/5/2015  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. Handouts: Combinatorics I  Combinatorics II  Combinatorics (Solutions)  10/4/2015  Working in teams, we will solve a variety of fun problems from a Russian olympiad for middle school students. Handouts: Handout For our first meeting, we will solve logic problems from a Russian math contest called the Math Festival. These problems require logical reasoning and will help the students exercise their minds. Handouts: Math Festival  Math Festival Solutions Welcome to Math Circle! We will be learning about ciphers today and writing some coded messages of our own. Handouts: Introduction to Ciphers Handout  Introduction to Ciphers Solutions 4PM6PM In IPAM What makes an election fair? Handouts: Week 1 Handout In our first meeting, we will solve some interesting problems from Moscow Math Olympiads, including problems in geometry, number theory, algebra, and combinatorics. Handouts: Problem set We will study Python for the first hour. We will solve problems preparing for AMC8 for the second hour. Handouts: handout
Handouts: Handout  Solutions  10/11/2015  The teams formed during the first session will present their solutions to last week's handout to the class. There is no new handout for this week. Handouts: Handout 1 Solutions We will explore fractions and decimals in depth this quarter. This week, we will prove the relationship between certain fractions and their terminating decimal expansions. Handouts: Fractions and Decimals 1  Fractions and Decimals 1 Solutions We will continue learning about different types of ciphers and how to use each of them. Handouts: More Fun With Ciphers Handout  More Fun With Ciphers Solutions 4PM6PM in IPAM
Is a fair voting system even possible? 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 integervalued polynomials. Handouts: Handout 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. Handouts: handout We will learn about infinity by examining Hilbert's Paradox. We will also review some of last week's problems from the Russian Olympiad. Handouts: Handout02  10/18/2015  Today we will learn about mathematical induction and use it to prove statements involving the natural numbers. Handouts: Handout  Some Solutions This week, we will delve further into fractions and decimals, and prove that every fraction has a repeating decimal expansion. Handouts: Fractions and Decimals 2  Fractions and Decimals 2 Solutions 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. Handouts: Math Kangaroo Practice Problems  Math Kangaroo Solutions  EE Homework Week 3 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. Handouts: Handout We will use recursive functions to draw fractals with the help of the Python's Turtle module. We will further study the properties of the fractals. Handouts: handout We will continue to explore infinity with Infinity Rockets in Infinite Space. Handouts: Handout  10/25/2015  This week, we will continue to use mathematical induction to prove statements about the natural numbers. Handouts: Handout This week, we will continue with fractions and decimals and move on to rational and irrational numbers. Handouts: Fractions and Decimals 3  Fractions and Decimals 3 Solutions We will have a spooky time solving Halloween problems! Handouts: Halloween Handout  Halloween Handout Solutions  Halloween Homework Solutions Handouts: Handout for this week and last We will continue our exploration of the stable marriage problem, including variants such as dishonest preference lists, incomplete preference lists, manytoone matching (the hospital/residents problem), and the stable roommates problem. Handouts: Handout 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.
Cryptarithms are mathematical puzzles in which digits are replaced by letters of the alphabet. We will learn to solve some of these. Handouts: Handout  Solutions  11/1/2015  Today we will solve a variety of problems that have shown up on the AMC 8 math competition over the past thirty years. 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". Handouts: Modular Arithmetic and Ciphers  Modular Arithmetic and Ciphers Solutions We will be working with cubes and their 2D representations. Handouts: Nets Handout  Nets Solutions  Nets Homework Solutions Handouts: Handout for this week 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. Handouts: Handout We will continue the study of the fractals from the 10/18 handout.
This week we will start working on Permutations! Handouts: Handout  Solutions  11/8/2015  Today, we will learn about the mathematical concept of invariant, and see what a powerful problem solving tool it is. Handouts: Handout We begin our exploration of means by looking at arithmetic means and what they represent. Handouts: Arithmetic Mean  Arithmetic Mean Solutions We will be continuing our topic with nets today, but this time, with pyramids instead of with cubes. Handouts: Pyramids and Nets Handout  Homework Solutions Fair We continue our study of queueing theory from last week. Handouts: Handout 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) Handouts: handout We will take another look at permutations to continue last week's work. Handouts: Handout  11/15/2015  Today we will work in teams to find invariants and use them to solve problems! Handouts: Handout  Handout Solutions We continue our exploration of means by looking at harmonic means and what they represent as well as comparing them to arithmetic means. Handouts: Arithmetic and Harmonic Means  Arithmetic and Harmonic Means Solutions Handouts: Splitting the Difference and Problem Solving Handout  Splitting the Difference and Problem Solving Solutions "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. Handouts: Handout The AMC8 competition takes place on Tuesday, Nov. 17th. Since most of our students participate, we will have a preparation session this time.
We will study Venn Diagrams this week. Handouts: Handout  11/22/2015  We finish our exploration of arithmetic and harmonic means and review the concepts we have learned this quarter. Handouts: Arithmetic and Harmonic Means (Continued)  Arithmetic and Harmonic Means (Continued) Solutions  Quarter Review  Quarter Review Solutions We will be working on logic problems, as well as word problems. Handouts: Flipping Triangles  Flipping Triangles Solutions 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 PonceletSteiner Theorem. Handouts: Handout We will study another fractal, called the Sierpinski triangle. If time remains, we will discuss dimensions of fractals.
Handouts: Handout  11/29/2015  Please note that we will not have class on 11/29 due to the Thanksgiving holiday.  12/6/2015  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. Handouts: Math Dominoes Rules  Math Dominoes Questions  Math Dominoes Solutions 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!
We will be playing a endofquarter review game. Handouts: Game Questions  Winter 2016 quarter  1/10/2016  Today we'll learn tricks for instantly performing calculations in our heads. Handouts: Handout  Problems 1  Problems 2  Problems 3  Answers For the first session of 2016, we will discuss some geometry and go over problems from Math Kangaroo contests. Handouts: Handout  Solutions 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. Handouts: Math Kangaroo Practice  Math Kangaroo Practice Solutions  Homework #1 Handouts: Cryptography We will study the Gini index (or Gini coefficient), a statistic commonly used in economics to describe income inequality or wealth inequality. Handouts: Handout We will use selfsimilarity to figure out dimensions of various geometric figures from a square, cube, and tesseract to the Koch curve.
Handouts: handout We will study probability in the first half and end the session with the Monty Hall Problem. Handouts: Handout  Solutions  1/17/2016  Today we will remember what greatest common divisors are from elementary school and learn a powerful way of computing them. Handouts: Handout  Solutions This week, we will discuss shadow geometry and similarity in triangles. We will also practice some hard Math Kangaroo problems. Handouts: Handout  Solutions Today we will be solving problems related to the balance scale. Handouts: Introduction to Balance Scale  Introduction to Balance Scale Solutions  Advanced Balance Scales  Advanced Balance Scales Solutions  Homework #2 Handouts: Handout 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. Handouts: Handout We will finish our study of fractal dimensions. If time permits, we will begin the new topic, Going Back and Forth between Rational and Decimal Representations of Fractions. Handouts: handout We will finish up the Monty Hall Problem discussed last week and move on to Fibonacci numbers. Handouts: Handout  1/24/2016  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. Handouts: Handout Our main goal for this section is to learn how to determine whether or not a solution exists for the 15 Puzzle. We begin doing this by learning about permutations this week. Handouts: Handout  Solutions We will be starting our binary unit by weighing with powers of 2. Handouts: Binary Part 1  Binary Part 1 Solutions  Homework #3 Handouts: Handout We introduce the principles of special relativity, Lorentz transformations, spacetime diagrams, and spacetime intervals, and we contrast special relativity with Galilean relativity. Handouts: Handout
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 coprime integers) and the set of real numbers having the terminating (finite) or infinite recurring form.
Handouts: Handout  1/31/2016  Today we will explore the concept of a bijection between two sets and see how it can make the notion of "counting" infinite sets rigorous. Handouts: Handout 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 continue learning about permutations. Handouts: Handout  Solutions We will continue our unit of binary numbers. Handouts: Warm Up  Binary Part 2  Binary Part 2 Solutions  Homework #4 We continue our introduction to special relativity, focusing on the spacetime interval, proper time, time dilation, and the twin paradox. Handouts: Handout
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.
Handouts: handout Handouts: Handout  2/7/2016  Today we will see that there are actually more than one kinds of infinity. In particular, we will learn that the infinity of real numbers is larger than the infinity of natural numbers. Handouts: Handout
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.
Handouts: Handout  Solutions We will finish our unit on binary numbers. Handouts: Binary Part 3  Binary Part 3 Solutions  Homework #5 Handouts: Handout In the first of two sessions on platonic solids and their symmetries, we give a gentle introduction to the platonic solids and Euler's formula. Handouts: Handout We will continue our study of the 1/7 handout. Once finished, we will switch to the new one.
Handouts: Handout  2/14/2016  Today we will learn how to work with quadratic equations and derive the quadratic formula. Handouts: Handout There will be no class this week. See you on the 21st!  2/21/2016  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. Handouts: Handout  Solutions We will be working with estimation this week! Please bring rulers to class. Handouts: Warm Up  Estimation Handout  Estimation Solutions  Homework #6 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. Handouts: Cube and Tetrahedron Model Template  Dodecahedron Model Template  Icosahedron Model Template  Octahedron Model Template  MAIN HANDOUT We will continue our studies of the 1/17 and 1/31 habdouts.
Handouts: Handout  2/28/2016  Today we will learn about graphs, prove some of their most important properties, and use them to solve problems. Handouts: Handout This week, we will discuss prime numbers, Euclid's lemma, the proof of irrationality using Euclid's lemma, and the Goldbach Conjecture. Handouts: Handout  Solutions We will be exploring how to break up numbers into parts. Handouts: Young's Diagrams Handout  Warm Up  Young's Diagrams Solutions  Homework #7 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. Handouts: Handout We will resume the minicourse at Problem 17 of the 1/17 handout.
Handouts: Handout  3/6/2016  Today we will continue our study of graph theory and use it to solve real life problems. This week, we will continue with prime numbers, learning about the existence of infinitely many prime numbers, prime number theory, and twin primes. Handouts: Handout  Solutions Since the Math Kangaroo competition is very soon (March 17, 2016), we will be doing some more practice today. There is no homework this week  just finish the handout at home. Handouts: Math Kangaroo Practice Problems  Solutions We explore applications of complex numbers in plane geometry. Handouts: Handout 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.
Handouts: Handout (updated)  3/13/2016  We will end the quarter with a competition, with teams racing each other to solve the most problems! Handouts: Problems We finished the remaining packet on prime numbers, and the students took a Math Kangaroo test for practice. Handouts: Math Dominoes Rules  Math Dominoes Questions  Math Dominoes Solutions 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.
We will play a final review tournament.
 Spring 2016 quarter  4/3/2016  We will see that sometimes it's easier to prove a mathematical statement by showing that it is impossible that the statement is false. Handouts: Handout For the first meeting of spring quarter, we will take a look at mathematical games and strategies! Handouts: Handout  Solutions We will be using a chessboard today to have some fun with math! Handouts: Fun on a Chessboard  Fun on a Chessboard Solutions We introduce game theory and winning strategies, with examples such as Nim and Chomp. Handouts: Handout 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 minicourse is to show that a + b = b + a for any two nonnegative 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. Handouts: handout On our first day back, we will be looking into modular arithmetic. Files will be uploaded the day after the session. Handouts: Blank copy of worksheet.  Answer key.  4/10/2016  We will discuss some applications of math to computer science. This week, we examine how sequences can be defined by the differences between each element. (Pages 18) Handouts: Handout  Solutions We will continue our topic on chessboards. Handouts: Chessboard II  Solutions We explore games with payoff matrices and mixed strategies. Handouts: Handout We will continue studying the 4/3 handout.
We finish up modular arithmetic this week by moving past simple calculation and onto some interesting applications and characteristics of problems involving modular arithmetic. Handouts: Blank copy of worksheet.  Answer key.  4/17/2016  Today we will learn about the complex numbers and use them to solve problems in geometry and algebra. We start this week by finishing up the handout from last week. We then start on an introduction to graphs by looking at common problems involving handshaking and graph traversals. Handouts: Differences (cont'd)  Graph Theory  Handshaking and Chasing Kids  Solutions We will learn about Roman Numerals today! Handouts: Roman Numerals  Solutions This session introduces automata theory, a branch of the theory of computation, with deterministic and nondeterministic finite automata and regular languages. Handouts: Handout (revised) 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 Olympiadstyle problems.
 4/24/2016  We will continue to study complex numbers, and will see how they can be used to solve some geometry problems. We continue to study graphs by completing the proofs from last week and also looking at applications of what we have learned about graphs. Handouts: Handout  Solutions We will be working on solids and their projections today. Please bring cubes (any sort of blocks will work) to class. Handouts: Projections  Solutions Algorithms (in pseudocode) and analysis of their time complexity using bigO notation. Handouts: Handout We will continue our study of the 4/3 handout. Once finished, we will solve some hard Olympiadstyle problems.  5/1/2016  This week, we will finish up our study of graphs by looking at graph isomorphisms and seeing more applications of graph theory. Handouts: Handout  Solutions We will continue working with projections today. Please bring blocks if you have them. Handouts: Projections II  Solutions Handouts: Handout We will go over the proof of commutativity of addition of nonnegative 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! Handouts: handout This week we talk about strategy of a particular two player game. Handouts: Blank copy of worksheet. This week we have a second week of our surprisingly popular "Take Away Games" worksheet! This time around: the game of Nim. Handouts: Blank copy of worksheet.  Answer key.  5/8/2016  Today we will learn about making and breaking codes! Handouts: Handout  Solutions We discuss the exterior angle property in triangles and the angle sum property of polygons. Handouts: Handout  Solutions We will be working on our logic today! Handouts: Island of Knights and Liars  Solutions  Homework We continue continued fractions and show that in a certain sense they give the best rational approximations to irrational numbers. Handouts: Handout We will get back to studying the Algebra book by Gelfand and Shen. Handouts: handout  5/15/2016  Today we will learn what an algorithm is and see why they are useful. Handouts: Handout We continue discussing angles in a polygon, and then move to several visual proofs for Pythagoras Theorem. Handouts: Handout  Solutions Handouts: Warm Up Solutions  Perimeter and Area  Perimeter and Area Solutions
Handouts: Handout  5/22/2016  We will continue our study of algorithms from last week. Today we will finally get to see some applications of complex numbers to geometry! Handouts: Handout Handouts: Handout  Solutions 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.) Handouts: Handout  Solutions
Handouts: Handout We will continue studying the Algebra book by Gelfand and Shen based on the 5/15 handout.
 5/29/2016  Happy Memorial Day!
 6/5/2016  Thanks for your hard work all year! We will close out the year with math relays. The reenrollment test will only cover the topics we went over this quarter: games, successive differences, graph theory, geometry, combinatorics. Attendance is mandatory. Handouts: Math Dominoes Rules  Math Dominoes Questions  Math Dominoes Answer Key
 6/6/2016  
