Los Angeles Math Circle

# LAMC Meetings Archive // Fall 2007 - Spring 2017

## For the current schedule, visit the Circle Calendar

Fall 2007 - Spring 2008Fall 2008 - Spring 2009Fall 2009 - Spring 2010Fall 2010 - Spring 2011Fall 2011 - Spring 2012Fall 2012 - Spring 2013Fall 2013 - Spring 2014Fall 2014 - Spring 2015Fall 2015 - Spring 2016Fall 2016 - Spring 2017
 8/5/2015 Beginners: Combinatorics 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 Advanced: Problems from Russian Olympiads (Noah Olander) Working in teams, we will solve a variety of fun problems from a Russian olympiad for middle school students. Handouts: HandoutJunior Circle: Problems from Russian Olympiads Handouts: Handout | SolutionsHigh School II: Problems from Moscow Math Olympiads 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 setIntermediate: Intro to Python, part 1, and AMC8 preparation. (Oleg Gleizer) We will study Python for the first hour. We will solve problems preparing for AMC8 for the second hour. Handouts: handoutBeginners: Math Festival 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 SolutionsEarly Elementary: Introduction to Ciphers 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 SolutionsHigh School I: Voting Theory 1 4PM-6PM In IPAM What makes an election fair?Handouts: Week 1 Handout 10/11/2015 High School I: Voting Theory 2 4PM-6PM in IPAM Is a fair voting system even possible?Intermediate: Intro to Python, part 2, and preparation for AMC 8. (Oleg Gleizer) 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: handoutHigh School II: Integer-Valued Polynomials 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.Handouts: HandoutBeginners: Fractions and Decimals I 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 SolutionsAdvanced: Problems from Russian Olympiads continued (Noah Olander) 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 SolutionsEarly Elementary: More Fun with Ciphers! 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 SolutionsJunior Circle: Hotel Infinity 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 Beginners: Fractions and Decimals II 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 SolutionsHigh School II: The Stable Marriage Problem, part 1 (Will Rosenbaum) 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: HandoutIntermediate: Recursive functions and fractals. (Oleg Gleizer) 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: handoutJunior Circle: Infinity: Part 2 We will continue to explore infinity with Infinity Rockets in Infinite Space. Handouts: HandoutEarly Elementary: Math Kangaroo Practice 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 3Advanced: Induction I (Michael Puthawala) Today we will learn about mathematical induction and use it to prove statements involving the natural numbers.Handouts: Handout | Some SolutionsHigh School I: Cardinality 1: Countable Sets 10/25/2015 Early Elementary: Halloween Fun! We will have a spooky time solving Halloween problems! Handouts: Halloween Handout | Halloween Handout Solutions | Halloween Homework SolutionsHigh School I: Cardinality 2: Uncountability Handouts: Handout for this week and lastBeginners: Fractions and Decimals III 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 SolutionsHigh School II: The Stable Marriage Problem, part 2 (Will Rosenbaum) 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.Handouts: HandoutIntermediate: Nature of randomness. (Mark Ponomarenko) Recursive functions and fractals, part 2. (Oleg Gleizer) 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. Advanced: Induction II (Michael Puthawala) This week, we will continue to use mathematical induction to prove statements about the natural numbers.Handouts: HandoutJunior Circle: Cryptarithms 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 High School I: Ordinal Infinities Handouts: Handout for this weekAdvanced: AMC 8 Prep (Noah Olander) Today we will solve a variety of problems that have shown up on the AMC 8 math competition over the past thirty years.Junior Circle: Permutations: Part 1 This week we will start working on Permutations! Handouts: Handout | SolutionsEarly Elementary: 3D Shapes and Nets We will be working with cubes and their 2D representations. Handouts: Nets Handout | Nets Solutions | Nets Homework SolutionsHigh School II: Queueing Theory, part 1 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: HandoutBeginners: Modular Arithmetic and Ciphers 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 SolutionsIntermediate: Recursive functions and fractals. (Oleg Gleizer) We will continue the study of the fractals from the 10/18 handout. 11/8/2015 Early Elementary: Nets, continued -- Pyramids We will be continuing our topic with nets today, but this time, with pyramids instead of with cubes. Handouts: Pyramids and Nets Handout | Homework SolutionsIntermediate: The area of Koch snowflake. 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: handoutHigh School I: Combinatorial Game Theory FairJunior Circle: Permutations: Part II We will take another look at permutations to continue last week's work. Handouts: HandoutHigh School II: Queueing Theory, part 2 We continue our study of queueing theory from last week.Handouts: HandoutBeginners: Arithmetic Mean We begin our exploration of means by looking at arithmetic means and what they represent.Handouts: Arithmetic Mean | Arithmetic Mean SolutionsAdvanced: Invariants I (Noah Olander) Today, we will learn about the mathematical concept of invariant, and see what a powerful problem solving tool it is.Handouts: Handout 11/15/2015 High School II: Nonclassical Constructions - Marked Ruler (Christopher Ohrt) "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: HandoutIntermediate: AMC8 preparation session. (Oleg Gleizer) The AMC8 competition takes place on Tuesday, Nov. 17th. Since most of our students participate, we will have a preparation session this time. Junior Circle: Venn Diagrams We will study Venn Diagrams this week. Handouts: HandoutHigh School I: Combinatorial Game Theory 2 Beginners: Arithmetic and Harmonic Means 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 SolutionsEarly Elementary: Splitting the Difference and Problem Solving Handouts: Splitting the Difference and Problem Solving Handout | Splitting the Difference and Problem Solving SolutionsAdvanced: Invariants II (Noah Olander) Today we will work in teams to find invariants and use them to solve problems!Handouts: Handout | Handout Solutions 11/22/2015 High School I: What Is Dimension? Intermediate: Back to fractals. (Oleg Gleizer) We will study another fractal, called the Sierpinski triangle. If time remains, we will discuss dimensions of fractals. High School II: Nonclassical Constructions - Poncelet Steiner (Christopher Ohrt) 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.Handouts: HandoutEarly Elementary: Flipping and Counting Triangles We will be working on logic problems, as well as word problems. Handouts: Flipping Triangles | Flipping Triangles SolutionsBeginners: Arithmetic and Harmonic Means Continued and Quarter Review 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 SolutionsJunior Circle: Transformations Via Permutations Handouts: Handout 11/29/2015 Early Elementary: NO CLASS -- Thanksgiving Break Please note that we will not have class on 11/29 due to the Thanksgiving holiday. 12/6/2015 Intermediate: Proof Bee. (James Newton) 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! Early Elementary: Math Dominoes! 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 SolutionsJunior Circle: Math Dominoes We will be playing a end-of-quarter review game. Handouts: Game QuestionsHigh School II: Math Relays 1/10/2016 Intermediate: Fractal dimensions. (Oleg Gleizer) p { margin-bottom: 0.1in; line-height: 120%; We will use self-similarity to figure out dimensions of various geometric figures from a square, cube, and tesseract to the Koch curve.Handouts: handoutEarly Elementary: Math Kangaroo Practice 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 #1High School I: Cryptography Part 1 Handouts: CryptographyHigh School II: Gini index We will study the Gini index (or Gini coefficient), a statistic commonly used in economics to describe income inequality or wealth inequality.Handouts: HandoutBeginners: Geometry I For the first session of 2016, we will discuss some geometry and go over problems from Math Kangaroo contests.Handouts: Handout | SolutionsJunior Circle: Probability (Rong Huang) We will study probability in the first half and end the session with the Monty Hall Problem. Handouts: Handout | SolutionsAdvanced: Tricks for Mental Math (Ethan Waldman) Today we'll learn tricks for instantly performing calculations in our heads.Handouts: Handout | Problems 1 | Problems 2 | Problems 3 | Answers 1/17/2016 Early Elementary: Meet the Balance Scale 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 #2Beginners: Geometry II This week, we will discuss shadow geometry and similarity in triangles. We will also practice some hard Math Kangaroo problems.Handouts: Handout | SolutionsHigh School II: Circumcenter of mass (Emmanuel Tsukerman) 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: HandoutHigh School I: Crytography 2: Public Key Encryption Handouts: HandoutIntermediate: Fractal dimensions and more. (Oleg Gleizer) 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: handoutAdvanced: Divisibility and the Division Algorithm (Noah Olander) Today we will remember what greatest common divisors are from elementary school and learn a powerful way of computing them.Handouts: Handout | SolutionsJunior Circle: Fibonacci Numbers We will finish up the Monty Hall Problem discussed last week and move on to Fibonacci numbers.Handouts: Handout 1/24/2016 High School I: Quaternions: Algebraic Origins Handouts: HandoutAdvanced: Cool Results on Primes (Noah Olander) 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: HandoutEarly Elementary: Weighing in Powers of 2 -- Introduction to Binary We will be starting our binary unit by weighing with powers of 2.Handouts: Binary Part 1 | Binary Part 1 Solutions | Homework #3Intermediate: Rational and decimal representations of fractions. (Oleg Gleizer) P { margin-bottom: 0.08in; } 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. Junior Circle: Scaling Areas and Volumes (Andrew George) Handouts: HandoutHigh School II: Special relativity, part 1 (Jared Claypoole, Julio Parra, and Andrew Yuan) We introduce the principles of special relativity, Lorentz transformations, spacetime diagrams, and spacetime intervals, and we contrast special relativity with Galilean relativity.Handouts: HandoutBeginners: Cracking the 15 Puzzle - Part 1 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 1/31/2016 High School I: Quaternions: Geometric Applications High School II: Special relativity, part 2 We continue our introduction to special relativity, focusing on the spacetime interval, proper time, time dilation, and the twin paradox.Handouts: HandoutIntermediate: Geometric sequences, series, their limits, and applications. (Oleg Gleizer) P { margin-bottom: 0.08in; }A:link { } 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: handoutBeginners: Cracking the 15 Puzzle - Part 2 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 | SolutionsEarly Elementary: Binary Part 2 We will continue our unit of binary numbers.Handouts: Warm Up | Binary Part 2 | Binary Part 2 Solutions | Homework #4Junior Circle: Math Kangaroo Practice Handouts: HandoutAdvanced: Infinity I (Michael Puthawala) 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 2/7/2016 High School I: The Gini Index Handouts: HandoutAdvanced: How many infinities are there? (Michael Puthawala) 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: HandoutEarly Elementary: Binary Part 3 We will finish our unit on binary numbers. Handouts: Binary Part 3 | Binary Part 3 Solutions | Homework #5High School II: Platonic Solids, part 1 (Christopher Ohrt) 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: HandoutIntermediate: Geometric sequnces, series, and their application to converting fractions to the decimal form. (Oleg Gleizer) We will continue our study of the 1/7 handout. Once finished, we will switch to the new one. Junior Circle: Rates and Distances: Part 1 Handouts: HandoutBeginners: Cracking the 15 Puzzle - Part 3 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 2/14/2016 Early Elementary: NO CLASS -- President's Day Weekend There will be no class this week. See you on the 21st!Advanced: Quadratics (Noah Olander) Today we will learn how to work with quadratic equations and derive the quadratic formula.Handouts: Handout 2/21/2016 High School II: Platonic Solids, part 2 (Christopher Ohrt) 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 HANDOUTEarly Elementary: Estimation We will be working with estimation this week! Please bring rulers to class. Handouts: Warm Up | Estimation Handout | Estimation Solutions | Homework #6Intermediate: Sequences, limits, and fractions. (Oleg Gleizer) We will continue our studies of the 1/17 and 1/31 habdouts. Junior Circle: Rates and Distances: Part 2 Handouts: HandoutHigh School I: Ramsey Theory 1 Beginners: Cracking the 15 Puzzle - Part 4 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 2/28/2016 Intermediate: Geometric sequences, series, and fractions, continued. (Oleg Gleizer) We will resume the mini-course at Problem 17 of the 1/17 handout. High School II: Homotopy Theory (Sanath Devalapurkar) 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: HandoutEarly Elementary: Young's Diagrams We will be exploring how to break up numbers into parts. Handouts: Young's Diagrams Handout | Warm Up | Young's Diagrams Solutions | Homework #7Junior Circle: Graphing with Rates and Distances: Part 3 Handouts: HandoutBeginners: Euclid and Prime Numbers I This week, we will discuss prime numbers, Euclid's lemma, the proof of irrationality using Euclid's lemma, and the Goldbach Conjecture.Handouts: Handout | SolutionsHigh School I: Ramsey Theory 2 Advanced: Graph Theory I (Michael Puthawala) Today we will learn about graphs, prove some of their most important properties, and use them to solve problems.Handouts: Handout 3/6/2016 Beginners: Euclid and Prime Numbers II 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 | SolutionsHigh School I: Infinite Ramsey Theory Advanced: Graph Theory II (Michael Puthawala) Today we will continue our study of graph theory and use it to solve real life problems.High School II: Complex Numbers and Geometry We explore applications of complex numbers in plane geometry.Handouts: HandoutJunior Circle: Math Kangaroo Practice (Part II) Handouts: Handout (updated)Intermediate: Geometric sequences. (Oleg Gleizer) 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. Early Elementary: Math Kangaroo Practice 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 3/13/2016 Early Elementary: Math Dominoes! Handouts: Math Dominoes Rules | Math Dominoes Questions | Math Dominoes SolutionsIntermediate: Resolving Zeno's paradox. (Oleg Gleizer) 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. High School II: Math Relays Advanced: Math Relays (Noah Olander) We will end the quarter with a competition, with teams racing each other to solve the most problems!Handouts: ProblemsJunior Circle: Tournament We will play a final review tournament. Beginners: Practice Math Kangaroo We finished the remaining packet on prime numbers, and the students took a Math Kangaroo test for practice. 4/3/2016 Junior Circle: Welcome back! Modular Arithmetic. (Andrew George) 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.Early Elementary: Fun and Games on a Chessboard -- Counting Squares We will be using a chessboard today to have some fun with math!Handouts: Fun on a Chessboard | Fun on a Chessboard SolutionsHigh School II: Game Theory, part 1 (Brent Woodhouse) We introduce game theory and winning strategies, with examples such as Nim and Chomp.Handouts: HandoutAdvanced: Proof by Contradiction (Noah Olander) 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: HandoutIntermediate: Mathematical Induction and Peano Axioms. (Oleg Gleizer) 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. Handouts: handoutBeginners: Math Games For the first meeting of spring quarter, we will take a look at mathematical games and strategies!Handouts: Handout | SolutionsHigh School I: Bounded Sets 4/10/2016 High School II: Game Theory Part 2 (Brent Woodhouse) We explore games with payoff matrices and mixed strategies.Handouts: HandoutIntermediate: Peano axioms and properties of addition. (Oleg Gleizer) We will continue studying the 4/3 handout. Advanced: Computability (Michael Puthawala) We will discuss some applications of math to computer science.Junior Circle: One More Time Around The (Mod) Clock. Modular Arithmetic Part II. (Andrew George) 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.Early Elementary: Fun and Games on a Chessboard -- Part Two We will continue our topic on chessboards. Handouts: Chessboard II | SolutionsHigh School I: Non-Euclidean Geometry Beginners: Successive Differences This week, we examine how sequences can be defined by the differences between each element. (Pages 1-8)Handouts: Handout | Solutions 4/17/2016 High School I: Hyperbolic Geometry Intermediate: Peano axioms and commutativity of addition. (Oleg Gleizer) ```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. ```Early Elementary: Roman Numerals We will learn about Roman Numerals today!Handouts: Roman Numerals | SolutionsHigh School II: Finite Automata This session introduces automata theory, a branch of the theory of computation, with deterministic and nondeterministic finite automata and regular languages.Handouts: Handout (revised)Advanced: Complex Numbers Today we will learn about the complex numbers and use them to solve problems in geometry and algebra.Beginners: Graph Theory - Handshaking and Chasing Kids 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 4/24/2016 High School II: Algorithms and Complexity (Iris Cong) Algorithms (in pseudo-code) and analysis of their time complexity using big-O notation.Handouts: HandoutIntermediate: Peano axioms and commutativity of addition, continued. (Oleg Gleizer) We will continue our study of the 4/3 handout. Once finished, we will solve some hard Olympiad-style problems. Advanced: Complex Numbers II We will continue to study complex numbers, and will see how they can be used to solve some geometry problems.High School I: Game Theory Early Elementary: Projections We will be working on solids and their projections today. Please bring cubes (any sort of blocks will work) to class.Handouts: Projections | SolutionsBeginners: Graph Theory II 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 5/1/2016 High School I: Topological Surfaces Junior Circle: Take Away Games (Florence Liu) This week we talk about strategy of a particular two player game.Handouts: Blank copy of worksheet.Junior Circle: Take Away Games II: Nim (Andrew George) 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.Early Elementary: Projections II We will continue working with projections today. Please bring blocks if you have them.Handouts: Projections II | SolutionsIntermediate: Problem solving session. (Oleg Gleizer) `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!`Handouts: handoutBeginners: Graph Theory III This week, we will finish up our study of graphs by looking at graph isomorphisms and seeing more applications of graph theory.Handouts: Handout | SolutionsHigh School II: Continued Fractions, part 1 (Christopher Ohrt) Handouts: Handout 5/8/2016 High School II: Continued Fractions, part 2 (Christopher Ohrt) We continue continued fractions and show that in a certain sense they give the best rational approximations to irrational numbers.Handouts: HandoutIntermediate: Back to the book. (Oleg Gleizer) We will get back to studying the Algebra book by Gelfand and Shen. Handouts: handoutHigh School I: Graph Algorithms 1 Early Elementary: Island of Knights and Liars and other Logic Puzzles We will be working on our logic today!Handouts: Island of Knights and Liars | Solutions | HomeworkBeginners: Geometry I We discuss the exterior angle property in triangles and the angle sum property of polygons.Handouts: Handout | SolutionsAdvanced: Cryptography Today we will learn about making and breaking codes!Handouts: Handout | Solutions 5/15/2016 High School I: Graph Algorithms 2 Early Elementary: Perimeter and Area Solutions Handouts: Warm Up Solutions | Perimeter and Area | Perimeter and Area SolutionsBeginners: Geometry II We continue discussing angles in a polygon, and then move to several visual proofs for Pythagoras Theorem.Handouts: Handout | SolutionsHigh School II: Tropical Polynomials (Bryant Matthews) Handouts: HandoutAdvanced: Algorithms I Today we will learn what an algorithm is and see why they are useful. Handouts: Handout 5/22/2016 High School I: Random Paths Early Elementary: Review 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 | SolutionsHigh School II: Tropical Orthogonal Representations (Bryant Matthews) Handouts: HandoutBeginners: Combinatorics on a Chessboard Handouts: Handout | SolutionsAdvanced: Algorithms II We will continue our study of algorithms from last week.Advanced: Complex Numbers III Today we will finally get to see some applications of complex numbers to geometry!Handouts: HandoutIntermediate: More of the book. (Oleg Gleizer) We will continue studying the Algebra book by Gelfand and Shen based on the 5/15 handout. 5/29/2016 Early Elementary: NO CLASS -- Memorial Day Beginners: No Meeting Happy Memorial Day! 6/5/2016 Early Elementary: Math Dominoes Handouts: Math Dominoes Rules | Math Dominoes Questions | Math Dominoes Answer KeyBeginners: Test The re-enrollment test will only cover the topics we went over this quarter: games, successive differences, graph theory, geometry, combinatorics. Attendance is mandatory.Advanced: Math Relays Thanks for your hard work all year! We will close out the year with math relays.High School II: Math Relays 6/6/2016 High School I: Math Relays