Los Angeles Math Circle

# LAMC Meetings Archive // Fall 2007 - Spring 2019

## 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 2017Fall 2017 - Spring 2018Fall 2018 - Spring 2019
 10/7/2018 High School I: Gaussian Integers I We will introduce Gaussian integers in order to decide which integers can be written as a sum of two squares.Handouts: Gaussian Integers IHigh School II: Math on Chess Board (Avet Grigoryan) We will start the school year with an an overview of math problems and puzzles involving a chessboard (with and without chess pieces) and solving techniques including tiling, coloring and invariance principle. Handouts: Problems and Select SolutionsJunior Circle: Cryptarithms and Logic Handouts: HandoutEarly Elementary I: Split the difference (Oleg Gleizer) Handouts: Split the Difference handoutAdvanced: Warm-up problems We warm up for the new academic year by solving problems from some recent mathematical competitions.Handouts: Lesson HandoutEarly Elementary II: First Meeting! Ciphers Its our first meeting! We will be introducing ciphers to the class. Only complete pages 1-4Handouts: Ciphers Part 1Beginners: A survey of Olympiad problems For our first week of the year, we'll be working on a handout containing olympiad problems from a long time ago. The problems do not follow any particular theme other than general knowledge of algebra.Handouts: Handout 10/14/2018 High School I: Gaussian Integers II We will wrap up the discussion of Gaussian integers and prove which integers are the sum of two squares.Handouts: Gaussian Integers IIBeginners: Fractions and Decimals This week we'll develop a theory for converting fractions to decimals and vice versa. Everyone knows that lots of things can be represented by both a fraction or decimal, but most people are at least a little fuzzy on how the two are related. This week we'll cut through the fuzz and by the end you'll be a pro at fractions and decimals.Handouts: HandoutJunior Circle: Leap Frog A look at how even with the opportunity to team up against another player or all work together this game is still very tough (maybe too tough??).Handouts: Solutions | HandoutEarly Elementary II: Ciphers Part 2 We wi be continuing past page 4 on our Ciphers worksheet. The goal for this class is to work on our problem solving skills and to try again when our first assumption does not work. (Pages 5-8) (WE DID NOT DO PROBLEM #8)Handouts: Ciphers Part 2High School II: Symmetries and Groups (James Cameron) In this meeting we will explore the symmetries of different objects like squares, rectangles, and coins. We will see how these symmetries interact with each other to form a structure called a group.Handouts: Problems and Multiplication TablesAdvanced: Quadratic Equations I (Vieta's Theorem) We start our exploration of quadratic equations with Vieta's Theorem. As usual, we also do some geometry.Handouts: Lesson Handout | HomeworkEarly Elementary I: Sharing Problems and Birthday Cakes (Oleg Gleizer) We will first check homework focusing on harder problems. Then we will solve a warm-up problem, get back to sharing problems, and end up cutting birthday cakes of various shapes. Handouts: Sharing Problems handout 10/21/2018 High School I: Gini Index We introduce the Gini index, an economic metric to measure wealth inequality.Handouts: Gini IndexAdvanced: Functions and Quadratic Equations II We solve some problems on the properties of general functions, and learn to apply Vieta's theorem from last week. As usual, some geometry.Handouts: Lesson Handout | HomeworkEarly Elementary I: Roman Numerals I (Oleg Gleizer) We will begin to study Roman numerals. Handouts: Roman Numerals I handoutJunior Circle: Sets and Functions Part I Handouts: Handout | Later Pages of HandoutHigh School II: Symmetries and Groups II (James Cameron) Having experimented with the groups of symmetries of the rectangle+string model, we will define groups and explore associated concepts such as subgroups, group actions, isomorphisms, orbits, and stabilizers. Handouts: Problems Early Elementary II: Mayan Numbers Part 1 Today will have have an introduction to Mayan Numbers plus Ken-Ken warm up! We will also be collecting and correcting the Cipher worksheet so make sure your student brings it to class! Homework: Finish the Mayan Numbers worksheet.Handouts: Mayan Numbers Part 1 Beginners: Place-Value Numerals The students ventured away from our decimal system and explored binary, trinary, and hexadecimal base systems. These form the basis of communication between modern computers, and also allow for encryption of English messages much more simply than the decimal system would, so it is important that our students are familiar with them.Handouts: Handout 10/28/2018 High School I: Quadratic Reciprocity I We introduce Legendre symbols and quadratic reciprocity to study residues modulo primes. Handouts: Quadratic Reciprocity IJunior Circle: Sets and Functions Part II Handouts: HandoutHigh School II: Circle Inversions In this meeting, our goal is to construct a strange new geometry where straight lines are circles and triangles have angles that sum to less than 180. We start with circle inversions and then introduce the Poincaré disc model. Handouts: Handout | SolutionsBeginners: Place value systems continued This weekend we will be continuing our study of place-value systems that are not base 10. Last week was a nice gentle introduction to the topics that consisted of a lot of computation. This week we are going to use some of the intuition that we built up last week to solve some more theoretical questions, and come to some surprising conclusions!Handouts: HandoutAdvanced: Quadratic Equations III (Quadratic Inequalities) We continue exploring quadratic equations by completing the square and learning the formula for the roots. We also start working towards quadratic inequalities, in particular seeing when a quadratic function is positive or negative.Handouts: Lesson Handout | HomeworkEarly Elementary I: Roman numerals II (Oleg Gleizer) We will continue studying the first handout on Roman numerals. Once finished, we will switch to the second handout. Handouts: Roman Numerlas II handoutEarly Elementary II: Mayan Numbers Part 2 We will continue with our lesson on Mayan Numbers and turn in the Mayan Number Part 1 worksheet. For the Mayan Numbers Part 2 worksheet, we did not do number 7.Handouts: Mayan Numbers Part 2 11/4/2018 High School I: Quadratic Reciprocity II We explore applications of quadratic reciprocity. Handouts: Quadratic Reciprocity IIHigh School II: Hyperbolic Geometry I We will continue our exploration of the Poincaré disc and prove facts about hyperbolic lines and shapes. Handouts: Solutions | Handout Early Elementary I: Roman Numerals II, continued (Oleg Gleizer) We will continue to study the Roman Numerals II handout. Handouts: Quiz 2Advanced: Quadratic equations IV We take a step back from learning fundamental facts about quadratic equations, and practive what we have already learnt on some problems. Geometry also makes an appearance.Handouts: Lesson Handout | HomeworkBeginners: Problem Solving Last week we finished talking about place value systems, and this week we'll be letting out hair down and doing some good, old fashioned, problem solving. There is no specific theme for this week but we will be using some of the things that we developed in the past couple of weeks. Handouts: HandoutJunior Circle: Sets and Functions Part III Our final look at sets and functions. In our final meeting on this topic we will discuss bijections betwen infinite sets and show how that can help determine equivalence.Handouts: HandoutEarly Elementary II: Logic Gates Part 1 We wi be learning about the different types of Logic Gates and how to use them. We will also be correcting Mayan Numbers Part 2. Homework: pages 1-17Handouts: Logic Gates Full 11/11/2018 High School I: Tropical Geometry I Instead of our typical definition of addition and multiplication, tropical arithmetic looks at minimum and addition operations. We will graph and find roots of tropical polynomials.Handouts: Tropical Geometry I and IIEarly Elementary II: We will have class! This is Veteran's Day weekend (a three day weekend) but we will be having class. Today we will be finishing the Logic Gates worksheet! **IF YOUR CHILD MISSED THIS CLASS YOU ARE EXCUSED AS THE AIR QUALITY AND EVACUATIONS LED TO STUDENT ABSENSES.Handouts: Logic Gates FullBeginners: Permutations This weekend we are going to start a multi-week study of the topic of permutations. We are going to start our study by defining what a mathematical permutation is, learn how mathematicians notate permutations and prove some elementary results.Handouts: HandoutJunior Circle: Parity and Puzzles Today, we will look at how certain problems can be resolved by viewing them in the context of parity.Handouts: HandoutAdvanced: Quadratic equations V (Parabolas) We introduce parabolas -- graphs of a quadratic equations, and solve some problems regarding them.Handouts: Lesson Handout | HomeworkHigh School II: Hyperbolic Geometry II We will keep working in the Poincare disc and discover phenomena peculiar to hyperbolic geometry such as AAA congruence, Lobachevskii's Theorem, and Schweikart's constant.Handouts: Section 2 11/18/2018 High School I: Tropical Geometry II We will continue our study of tropical arithmetic by proving a version of the Fundamental Theorem of Algebra for tropical quadratic polynomials.Handouts: Tropical Geometry I and IIHigh School II: Hyperbolic Geometry III We look at more peculiar hyperbolic facts, like the hyperbolic Pythagorean theorem and the angle of parallelism. Handouts: SolutionsEarly Elementary I: Roman numerals IIS (Oleg Gleizer) We will discuss the ambiguity of Roman numerlas, will learn how to convert decimals to Roman numerals, and will learn writing dates as Roman numerals. Handouts: handoutAdvanced: Quadratic equations VI We use the theoretical properties of quadratic equations established during the last few weeks to solve some concrete problems. Also: more geometry.Beginners: Permutations continued This weekend we will be continuing our study of permutations. Now that we have a basic understanding conceptualization of permutations and have some basic notation down, we are going to apply that notation to better understand the 15 puzzle. By the end of the class we won't have solved the puzzle, but we will be a lot closer. Handouts: HandoutEarly Elementary II: Solids and Projections Part 1 Today we will be manipulating shapes in our heads and drawing their projections on our papers. This is Part 1. Homework will be announced in class. Junior Circle: Practicing with Percentages We will look at several problems dealing with percentage decreases, increases, and changes in general.Handouts: Handout 11/25/2018 Early Elementary II: NO CLASS - THANKSGIVING WEEKEND No class this weekend because everyone is on Thanksgiving Break!Beginners: No Class this Week! Have a happy Thanksgiving!High School II: No class - Thanksgiving Break 12/2/2018 High School I: Golomb Ruler Given a ruler, how many inch markings can you remove and still measure each increment between 1 and 12 inches? Is there some way to construct a 12-inch ruler such that each increment from 1 to 12 can be measured in a unique way?Handouts: Golomb RulerEarly Elementary I: Cut the log (Oleg Gleizer) We will do some log cutting as a first step in learning topology. Handouts: handoutBeginners: Permutations Finale Today we are going to finish up our study of permutations and finally resolve the case of the 15 puzzle. After that is done, we will take a look back at what we have done, and take note of some interesting results that we have proved along the way. Handouts: HandoutHigh School II: Electrical Circuits and Random Walks We will have a short review quiz. Then, Aaron Anderson will talk about how electrical circuits correspond to random walks on the vertices of graphs. Handouts: HandoutJunior Circle: Review of Fall Quarter Material We will review several of the most difficult topics from this quarter.Advanced: Individual Problem Solving We see what we have learned about geometry and quadratic equations with an individual problem-solving session.Handouts: Handout 12/9/2018 High School I: Presentations High School II: Electric Circuits and Random Walks II We will continue talking about the correspondence between voltage, resistance, and current in circuits with random walks on graphs. Handouts: Challenge ProblemsJunior Circle: Math Dominoes A fun competition during class to determine which team of students can solve questions from this topics this quarter and beyond the quickest and most accurately.Beginners: Permutations post Finale and Math Relay This weekend we are going to have our final class for the quarter. For the first hour we are going to take stock of everything that we have proved so far about the 15 puzzle, and the second half will be a class-wide relay with prizes!Handouts: HandoutEarly Elementary I: Cut the bagel. (Oleg Gleizer) We will cut some bagels as a step in learning topology. Handouts: handoutAdvanced: Math Dominoes Fun and games! 1/13/2019 Junior Circle: Special Lesson by Nitya (Futoshiki Squares) Handouts: HandoutHigh School II: Math Relay We will have a competition to solve problems for prizes!High School I: Continued Fractions I We will introduce continued fractions and learn how to calculate them. We will also investigate the relationship between the irrationality of a number and properties of its continued fraction expansion.Handouts: Continued Fractions IBeginners: Geometry I This weekend we are going to start our study of geometry starting waaaay back at the start of Greek mathematics. This weekend we'll be (re)learning how to use a compass and straight edge. As such, please remember to being a compass and straightedge with you to class today! I hope that you are all as excited to resume the LAMC as I am. Handouts: HandoutEarly Elementary I: Squares on a chessboard (Oleg Gleizer) The utlimate goal of this lesson is to count all the squares on the chessboard, 1 by 1, 2 by 2, 3 by 3, and all the way to 8 by 8. Handouts: handoutAdvanced: Weighings, Logic and Geometric Constructions We start a few new topics for the quarter -- weighings, logic and ruler & compass constructions.Handouts: Lesson Handout | HomeworkEarly Elementary II: Quilt Mending! First Class of the quarter and we will be doing a brand new topic!Handouts: Quilt Mending | Quilt Mending Solutions 1/20/2019 High School I: Continued Fractions II We will continue our study of continued fractions with an imporant application in number theory: Given an irrational number, how efficiently can it be approximated by rational numbers? Continued fraction expansions play an important role in solving this problem.Handouts: Continued Fractions IIEarly Elementary I: Games on the chessboard (Oleg Gleizer) We will figure out ways to always win in a bunch of fun chessboard games. Handouts: handoutBeginners: Geometry II Today we are going to continue our studies of Geometry, and learn more about what you can do using a compass and ruler, and finally talk about geometry as you have seen it in school. Today will be a nice mix of hands on drawing/calculation with the compass and ruler, as well as a bit of proving using claim / reason charts.Handouts: HandoutHigh School II: Algorithms We will start a lesson studying algorithms: what are they, and how do they work?Handouts: Early Elementary II: Math Kanagroo Practice Part 1 We will be practicing Math Kanagroo questions. We will focus on reading the questions fully, strategies find the answer, and practicing to show our work. Handouts: Math Kangaroo Practice 1Junior Circle: Pseudo Paradoxes A look at a series of proven statements. But something about their conclusions seem a bit odd...Handouts: HandoutAdvanced: Weighings, Logic and Geometric Constructions II We continue with the topics of weighings, logic and straightedge & compass constructions.Handouts: Lesson Handout | Homework 1/27/2019 High School II: Algorithms II We will learn about more aspects of algorithms, such as efficiency and computational complexity. Handouts: Challenge ProblemsAdvanced: Weighings, Logic and Geometric Constructions III More straightedge and compass constructions, along with harder weighings and logic problems.Handouts: Lesson Handout | HomeworkBeginners: Geometry III Today will be out third and likely final Geometry session of the quarter. During the first week we got some practice using the straightedge and compass, during the second we had a gentle introduction to two column proofs, and for this last week we'll be solving problems using some of what we have learned. Not all of the problems look like they are 'classic' straightedge and compass problems, but we will find that using just those two implements, you can do more than you might think. For this final week please bring your straightedge and compass with you to class. Handouts: HandoutJunior Circle: Pseudo Paradoxes II We will continue with last week's handout.High School I: Mathematics of musical scales (Olga Radko In this power-point presentation, we will address the following questions: Why do some musical intervals sound pleasant, while others do not? Why do we have exactly 12 notes in an octave of a piano? Why aren't distances between frets on a flute or a guitar equal to each other? The answers, surprisingly, involve deep mathematical analysis involving continued fractions, the problem of doubling the cube, and rational approximations. Early Elementary II: Babushka Squares Students will be doing an activity and worksheet that involve manipulating shapes to fit them all in the smallest box possible. If your child missed this week, problem #2 sadly cannot be completed without the manipulatives we used in class. Homework is the beginning of problem #3. All students have to do is create all many 7 square shapes as they can.Handouts: Babushka SquaresEarly Elementary I: One more chessboard game, then the abacus. (Oleg Gleizer) After a warm-up, students will figure out a winnign strategy for a fun chessboard game, called Move a Rook into the Corner. If time remains, students will start learning how to use an ancient computer, called the abacus. This will allow students to better understand the working and advantages of the decimal place-value numeral system currently in use by humanity. Handouts: handout 2/3/2019 Junior Circle: Parity and Proof Writing We will look at several parity related problems and then continue onto two different important topics related to proofs.Handouts: HandoutHigh School I: Limits of Sequences We will introduce the formal defnition of a limit of a sequence and develop basic properties.Handouts: Limits of SequencesHigh School II: Cantor Set We're going to look at the famous Cantor set, its construction, and some of its oddities.Handouts: HandoutEarly Elementary II: Math Kangaroo Practice 2 We will be continuing to practice Math Kangaroo. HOMEWORK: 3rd and 4th grade practice test.. The MK Practice 2 is not homework, please don't have your student complete this at home.Handouts: MK Practice 2 | Practice TestEarly Elementary I: The abacus. (Oleg Gleizer) We will use the abacus to study decimal place-value numerals. Beginners: Vectors Today we will be working with vectors, and connecting them to the previous work that we have been doing on geometric constructions. Vectors are in some ways just like the counting numbers, and in other ways are very geometric, unlike the counting numbers. This dual nature of vectors makes them both interesting and useful. We will start to uncover this duality today!Handouts: Handout | Auxiliary HandoutAdvanced: Invariants and Geometric Constructions We focus more on using invariants in combinatorial problems about processes, and also continue with straightedge & compass constructions.Handouts: Lesson Handout | Homework 2/10/2019 Junior Circle: IFF and Parity Handouts: HandoutHigh School II: Cantor Set II We will continue studying the Cantor set, invesitgating properties such as its cardinality and "dimension." Once we develop some notions of dimension, as a bonus we will also look at other fractal sets and their dimensions.Handouts: Handout | Bonus ProblemsBeginners: Math Kangaroo This week we are going to do something completely different from what we have been working on the past month. We are making a hard right turn from geometry and instead we will be focusing on problem solving with an emphasis on solving problems for the upcoming math kangaroo competition. You certainly don't have to aim to take the exam to enjoy this weekend's class, indeed When I was still in school, I enjoyed these contests no so much because I was competitive, but more because the questions themselves were often quite beautiful. You do not need anything special for this week's lesson!High School I: Sequences II We will continue our practice with formally proving limits of sequences and we will prove some additional properties of sequence limits.Handouts: Limits of Sequences IIEarly Elementary II: Final Math Kangaroo Review We will be going over the practice test from last week and possibly doing some extra practice problems while discussing strategies for taking the Math Kangaroo exam.Advanced: Invariants and Geometric Constructions II We continus with our topics of invariants and straightedge & compass constructions.Handouts: Lesson Handout | HomeworkEarly Elementary I: The abacus and extra problems. (Oleg Gleizer) We will attenpt to finish the abacus handout. If time remains, we will solve extra problems from the next handout. Handouts: extra handout 2/17/2019 Early Elementary II: NO CLASS: PRESIDENT'S DAY WEEKEND No class this week because of the holiday/three day weekendBeginners: No Class Today On account of the holiday, there will be no Math Circle this weekend. See you next week! 2/24/2019 Junior Circle: Math Kangaroo Review Handouts: HandoutHigh School I: Integer-Valued Polynomials We characterize all polynomials that have integer outputs for integer inputs.Handouts: Integer-Valued PolynomialsHigh School II: The Gini Index (Dan Hoff) We’ll explore a measure of economic inequality known as the Gini Index. In particular, we’ll learn what it is, how to calculate it, and what some of its strengths and limitations are. Handouts: Handout | SolutionsEarly Elementary I: Balance scale. (Oleg Gleizer) Students will first take a quiz on the abacus. Then the class will learn how to use a balance scale for weighing objects and solving math problems. Handouts: handoutBeginners: Combinatorics I Today we are going to talk about a subject of math that really counts, combinatorics. Combinatorics is known as the math of counting, however the counting itself is usually not the point. The point is the clever arguments that allow the counting to be done at all. Combinatorics is a mainstay of mathematical puzzles and competitions alike, as it is an extremely rich field of math which is still elementary.Handouts: Handout 2 | Handout 1 | Handout 2 Solutions | Handout 1 SolutionsAdvanced: Invariants and Geometric Constructions III We continue with invariants, semi-invariants and geometric constructions.Handouts: Lesson Handout | HomeworkEarly Elementary II: Finding Fake Coins Today we will find fake coins in different scenarios using a balance scale. Handouts: Fake Coins 3/3/2019 Junior Circle: Planes, Tetraheda, and Cross Sections Handouts: HandoutEarly Elementary I: Math Kangaroo preparation. (Oleg Gleizer) We will study in preparation for the Math Kangaroo competition, up-coming on 3/21. Beginners: Combinatorics II This week we will be continuing what we stared last week and talk more about combinatorics. We will be starting with a brief review of what we spoke about last week, before moving onto completely new problems.Handouts: HandoutHigh School II: Weighings, information exchange, and other abstractions (Sherry Gong) In this lesson we will do combinatorial weighing and probability problems, with some related problems about information exchange.High School I: Mass Point Geometry The center of mass of a system of finitely many point masses is relatively easy to calculate. We will explore certain planar geometric problems that can be easily solved when we assign masses to relevant points.Handouts: Mass Point GeometryAdvanced: Invariants and the Intercept Theorem We continue our route to triangle siilarity by learning the intercept theorem.Handouts: Lesson Handout | Homework | ReadingEarly Elementary II: Evens and Odds Today we will be doing a worksheet that allows the students to discover the patterns of even and odd numbers. The warm-up will be Ken-Ken. We will be collecting the Fake Coins worksheet from last week.Handouts: Evens and Odds 3/10/2019 Junior Circle: Pi Day Based Problems Handouts: HandoutEarly Elementary II: Pie Day! Today we celebrate Pie Day as it is coming up next week! We will be collecting last week's worksheet and correcting a couple of problems for credit.Handouts: EE Pi Day WorksheetHigh School I: Pi Day (almost) Pi originated as the ratio of a circle's circumference to its diameter in the plane. We will see how this differs for circles on a sphere.Handouts: Pi DayEarly Elementary I: Balance scale and binary numbers. (Oleg Gleizer) We will first use a balance scale to introduce binary numbers. Then, we will study their properties. Handouts: handoutHigh School II: Pi and Probability To celebrate Pi day, we'll look at some probability questions involving pi. For example: suppose you have equally spaced lines and you drop a toothpick. What is the probability that the tootpick crosses a line? Beginners: Pi day! Today we'll be taking a break from our normally scheduled content to talk about everyone favorite geometric constant, pi! Pi is of course one of the most well known mathematical constants and has been studies from ancient Greece until now. We'll do a couple problems about Pi and compute a whole bunch of things geometrically. Handouts: HandoutAdvanced: Similar Triangles Our multi-week journey towards similar triangles culminates with the proof of the two main theorems about them.Handouts: Lesson Handout | Homework 3/17/2019 Junior Circle: Pi Day Part II (Area) Handouts: HandoutEarly Elementary II: Creative Student Worksheet Today we will be completing a worksheet that consists of word problems and math questions created by the Early Elementary II students! This is the last class before a 2 week break!Handouts: Creative Student WorksheetBeginners: Math Relay For our last meeting of the quarter, we will be having another math relay! As we did last quarter, we'll be splitting people up into teams an seeing which team can get through the most problems.High School I: Presentations High School II: More Pi Problems We will continue the worksheet on problem related to pi. We will find the probability that two randomly selected integers are coprime and calculate some continued fractions.Handouts: Handout | Quiz RubricEarly Elementary I: Balance scale and binary numbers, continued. (Oleg Gleizer) We will continue the study of binary numbers using a balance scale. 3/24/2019 Early Elementary II: NO CLASS We will not have class today. Remember to re-register for Spring Quarter! 3/31/2019 Early Elementary II: NO CLASS We will not have class today. Remember to re-register for Spring Quarter! 4/7/2019 High School I: Groups of Symmetries We will discuss some surprising symmetries with a visual aid. This week will lead to an in-depth exposition of abstract group theory.High School II: Continued Fractions We will continue studying continued fractions in more detail, looking at special properties and details of convergence.Handouts: Beginners: How big are big numbers really? We've all seem some pretty big numbers in our day. Sure maybe you've seen a 81, a 104 or maybe even (if you are very worldly) 3841. But, jut how big are big numbers really? How big is a number like 52!, the number of ways to arrange the number of cards in a 52 card deck? What about the number of times you would have to flip 200 coins before you got all heads? Can you even say which one is larger? Today we'll answer this question and more, by introducing the logarithm, a function that is extremely useful for making sense of the super massive.Handouts: HandoutEarly Elementary I: Decimal and binary numbers. (Oleg Gleizer) We will reviev binary numbers and then take a quiz. Then we will start figuring out what is in common between decimal and binary numbers. Handouts: handoutJunior Circle: Insect World (I) Handouts: HandoutAdvanced: Combinations We start a new topic of combinations and Pascal's triangle. Also a problem on triangle similarity.Handouts: Lesson Handout | HomeworkEarly Elementary II: Continuation of Student-Made Worksheet Welcome back! Please bring your worksheets from the last time we had class. We will be continuing to go through the problems at all the students created! Homework is to finish at least 25 problems on the worksheet. Reminder to re-register your student for Spring Quarter!Handouts: Creative Student Worksheet 4/14/2019 Junior Circle: Insect World (II) High School II: Continued Fractions (Continued) We will continue studying the properties of continued fractions, in particularly proving convergence and the 'best approximation law.' Advanced: Combinations II (Pascal's Triangle) We continue studying combinations, introducing the binomial formula and Pascal's triangle.Handouts: Lesson Handout | HomeworkBeginners: Goats Maybe on the whole you felt like last week was entirely too much. Maybe you thought that the numbers that we spoke about last time were too large, logarithms were too confusing and you are ready to take a mathematical break and return to a more pastoral existence. Good news! This next week we are talking about goats. That's right goats, everyone's favorite ornery, stubborn, ravenous livestock. We will find that making sure that goats have enough to eat is more mathematical then you might have thought. This just goes to show that you can try and leave math, but math will always find you!Handouts: HandoutEarly Elementary I: Decimal and binary numbers, continued. (Oleg Gleizer) We will continue comapring binary and decimal numbers. We will see that binary and decimal numbers are place-value while roman numbers are not. High School I: Introduction to Groups I We introduce the definition and basic properties of a group in the context of the symmetries of a square index card in the plane.Handouts: Introduction to Groups IEarly Elementary II: Cubes and Nets Part 1 We will be doing a worksheet that introduces nets of cubes to the studentsHandouts: Cubes and Nets Part 1 4/21/2019 Junior Circle: Insect World (III) Handouts: HandoutEarly Elementary I: Odd and even numers (Oleg Gleizer) We will study odd and even numbers in the decimal and binary form. Handouts: handoutBeginners: Knights, Knaves and Knormals. Logic puzzles are a mainstay of recreational mathematics, and today we'll be solving problems involving people that always tell the truth (knights), always lie (knaves), and sometimes tell the truth and other times lie (knormals). Solving these problems can be challenging, but we'll learn how to approach them is a systematic way so that you can always find the answer. Although these problems seem like all fun and games, they actually have some connections to mathematical logic; the most fundamental branch of modern math.Handouts: HandoutHigh School II: Expected number of hacky sacks and temper tantrums (Paul Zeitz) Watching a math circle for 2nd and 3rd graders leads to nontrivial questions about probability and expected value. We will discover interesting and non-intuitive phenomena about randomness. Handouts: HandoutAdvanced: Combinations III We continue with the problems from last week on pascal's triangle and combinations.Handouts: Lesson Handout (Anton's group only) | HomeworkHigh School I: Introduction to Groups II We continue our discussion of groups with a focus on group morphisms and quotient groups.Handouts: Introduction to Groups IIEarly Elementary II: NO CLASS Class canceled due to a lot of students being on Spring Break vacation or celebrating the holiday. 4/28/2019 Junior Circle: Insect World (IV) Handouts: HandoutHigh School II: Wallpaper Patterns What kinds of patterns can be used as wallpaper. What are their groups of symmetries, and how can we classify them? How many are there? We will attempt to answer some of these questions and learn how to use Thurston's "orbifold notation" for wallpaper patterns.Handouts: HandoutAdvanced: Combinations IV (Stars & Bars) This week we introduce the binomial formula, and the "stars & bars" counting strategy.Handouts: Lesson Handout | HomeworkEarly Elementary I: Odd and even numbers, continued. (Oleg Gleizer) We will continue to study odd and even numbers in the decimal and binary form. If time remanis, we will do a bit of magic related to binary numbers. Handouts: solutions and teacher's notesBeginners: Divisibility `This weekend we'll be talking about divisibility. Just about everyone knows what it means to divide two integers, but this week we be doing almost no dividing. Instead we will be a lot more concerned with the question of when two one integer evenly divides (i.e. has no remainder after division) another. Divisibility might not sound like a terribly deep or nuanced topic, but it is, in fact, more nuanced and developed than you would believe. Further, it is a common entry into talking about abstract algebra, one of the largest branches of modern pure math.` Handouts: HandoutHigh School I: Electrical Circuits and Random Walks We discuss random walks on graphs as they relate to basic electrical circuit diagrams.Handouts: Electrical Circuits and Random WalksEarly Elementary II: Cubes and Nets Part 2 We will be continuing the topic of nets of cubes. In this worksheet we will explore different routes across vertcies and edges of the cube. Some questions on this worksheet are repeated from Part 1, so we crossed them out. We will also cut out nets of solids and see if they create cubes.Handouts: Cubes and Nets Part 2 | Nets of Cubes 5/5/2019 Early Elementary I: Odd and even numbers III. (Oleg Gleizer) We will continue studying the Odd and Even Numbers packet. High School II: Wallpaper Patterns II After practicing finding the signatures of many different wallpaper patterns, we will move on to classifying all the distinct wallpaper signatures using the "Signature Cost Theorem." As we will see, there are less than 20!Handouts: Handout | SolutionsBeginners: The Chinese remainder theorem This week we are going to continue our discussion of divisibility. Last week we introduced the general idea and solved some interesting problems, but this week we are going to try and turn our attention to proving the useful Chinese Remainder Theorem. This theorem gives the conditions under which you can find a solution to systems of simultaneous contingencies. Although this might sound very complicated, we'll find that it isn't so bad.Handouts: HandoutAdvanced: Combinatorics V (Stars & Bars) We continue with the more advanced verison of "starts & bars", plus various combinatorics problems.Handouts: Lesson Handout | HomeworkJunior Circle: Olympiad and Riddles Handouts: HandoutHigh School I: Electrical Circuits and Random Walks II We define effective resistance and introduce Polya's random walk problem. Handouts: Electrical Circuits and Random Walks IIEarly Elementary II: Venn Diagrams A look at how we can organize multiple sets of objects/people/things using our knowledge of venn diagramsHandouts: Venn Diagrams 5/12/2019 Advanced: Combinatorics VI (Young Tableaux) We introduce a new method to encode partitions of positive integers -- Young tableaux, and use them to prove some beautiful bijections on partitions.Handouts: Lesson Handout | HomeworkEarly Elementary I: Binary magic. (Oleg Gleizer) We will staudy a magic trick based on binary numbers. Handouts: handoutHigh School II: Integer Valued Polynomials In this lesson we will study polynomials with the property that every integer input gives an integer output. Handouts: HandoutBeginners: Modular Arithmetic and Ciphers Over the past two weeks we have been studying divisibility and the Chinese remainder theorem, but up until not we have been looking at them as purely mathematical problems with no regard towards any applications. This weekend we'll be talking about one very important of molecularity, cryptography. Cryptography is the study of how we can conceal information in such a way so that it can be perfectly uncovered by the right person, and look like gibberish to everyone else. Handouts: HandoutHigh School I: The Cantor Set We introduce a few definitions of the Cantor set and relate it to cardinality and a measure on the real line.Handouts: The Cantor SetJunior Circle: Isoperimetric Puzzle Handouts: HandoutEarly Elementary II: Taxicab Geometry Part I We will learn how to plot points on a grid and learn how to calculate distance between pairs of pointsHandouts: Taxicab Geometry 1 5/19/2019 High School II: Bipartite Graphs (Harris Khan) In this lesson we'll solve the "stable matching problem." Imagine two tennis clubs A and B competing in a tournament. Each player has a preference for which person they want to play from the other team. Can we find a pairing that is stable, i.e. where there is no pairing such that both players prefer to play someone else? High School I: Fractals How do we assign a dimension to a set? We define Minkowski, packing, and covering dimensions and relate these to the Cantor set.Handouts: FractalsJunior Circle: Towers of Hanoi Handouts: HandoutEarly Elementary I: Mobius strip We will study the most famous one-sided two-dimensional surface, the Mobius strip, by comparing it to a two-dimensional cylinder. The class includes quite a bit of cutting and gluing. Since many of students do not yet have the necessary hand-eye coordination, the class is taught in the PARENT-AND-ME format. Handouts: handoutBeginners: Problem Solving and Take Away Games Over this weekend and the next we'll be covering both take away games and problem solving. One class will do problem solving first and then take away games, but both classes will do both over the next two weeks. For problem solving we will be diverging from our usual program and will be tackling some old math competition problems. These problems won't be emphasizing any particular mathematical principle and instead we'll be trying to discover how to approach these problems, and especially how you can make progress when you don't see the answer. For take away games we will have a friend of the math circle and current UCLA math PhD student Jeremy Brightbill give a special presentation on take away games. These mathematical games are typically played between two people where the goal is to take pieces in such a way that either you or the other person has to take the last piece. Over the course of the lesson we'll be playing a lot of games, so this is one lesson that you definitely don't want to miss! Handouts: Take Away Games Handout | Problem SolvingAdvanced: Combinatorics VII (Young Tableaux) We continue exploring Young tableaux and the partition function.Handouts: HomeworkEarly Elementary II: Taxicab Geometry Part II We will continue to learn about coordinate points and distance using the Taxicab worksheetHandouts: Taxicab Geometry 2 5/26/2019 Junior Circle: No Meeting (Memorial Day Weekend) Beginners: No Meeting this week In observation of Memorial day we will not be having Math Circle this weekend.Early Elementary II: NO CLASS - HOLIDAY WEEKEND We have no class today! Homework from last week is to finish as much of the Taxicab II Worksheet as possible. Please have your students come to class with questions about the worksheet. 6/2/2019 High School II: Oblivious Transfer & Private Secure Computation (Amit Sahai) Suppose an online bookstore has N books B1, ..., BN, and you want to buy a book, but you don't want the bookstore to know which book you're buying. In other words, you want be able to choose an integer i such that 0 < i < N+1, and you want to figure out a way that you can learn Bi, and yet the bookstore learns nothing about the integer i. This is called an Oblivious Transfer (OT). We will use modular arithmetic to construct OT, and see how to use OT to solve an even more general cryptographic problem called Private Secure Computation.Junior Circle: Triangular Numbers Handouts: HandoutEarly Elementary I: How to fight a dragon. (Oleg Gleizer) Students will learn how to use math to fight dragons. Handouts: handoutEarly Elementary II: High School I: Oblivious Transfer & Private Secure Computation (Amit Sahai) Suppose an online bookstore has N books B1, ..., BN, and you want to buy a book, but you don't want the bookstore to know which book you're buying. In other words, you want be able to choose an integer i such that 0 < i < N+1, and you want to figure out a way that you can learn Bi, and yet the bookstore learns nothing about the integer i. This is called an Oblivious Transfer (OT). We will use modular arithmetic to construct OT, and see how to use OT to solve an even more general cryptographic problem called Private Secure Computation.Beginners: Take Away Games and Problem Solving Today we'll be doing the complement of what we did tow weeks ago. You will have either a guest lecture or a lession on problem solving, depending on what you did not do last week. Advanced: Miscellaneous We finish out the year with problems on various topics.Handouts: Lesson Handout | Homework 6/9/2019 High School II: Math Relays For our last meeting we will have math relays! Come prepared to solve problems and win prizes.Early Elementary II: LAST CLASS BEFORE SUMMER Advanced: Math Dominoes We play math dominoes to celebrate the last class of the year.Early Elementary I: Adding and subtracting binary numbers. (Oleg Gleizer) Students will add and subtract binary numbers using long addition and subtraction parallel to the binary abacus. Handouts: handoutHigh School I: Competition Problems We will work in teams to solve problems as a fun end to the quarter. 6/16/2019 Beginners: Competition For our last LAMC meeting of the year, we will be doing a math relay type competition. The emphasis will be on working together as a team to solve increasingly difficult problems correctly and quickly. We will have problems that touch on topics that we have covered in this past year as well as problems taken from old math competitions. Bring your A game and prepare to send off this year in style! 6/18/2019 LAMC Chess Program: Opening Principles and Strategies Discussion of basic opening principles -- development, central control, king safety, etc. Handout for Group 1 is included. Handouts: Group 1 Handout 6/23/2019 Early Elementary: The abacus. We will study the abacus, a computer that was in use from the time of Babylon until the end of the 20th century. Students need to bring their own abaci to the class. Handouts: handout 6/25/2019 LAMC Chess Program: Basic Checkmates & Pawn Endings We continue our exploration of the three stages in a chess game by studying basic checkmates (group 1) and pawn endings (group 2). Handouts: Group 1 Handout | Piece Values (Group 1) | Chess Notation Guide | Group 2 Handout 7/2/2019 LAMC Chess Program: Tactical Motifs & Combinations Group 1 introduces two tactical motifs: forks and pins; Group 2 solves puzzles on a variety of tactical topics. Handouts: Handout 7/9/2019 LAMC Chess Program: More Tactics & Annotation Contest Group 1 continues exploring tactics (more pins and skewers), Group 2 has an annotation contest and solves puzzles. Handouts: Annotation Contest (Group 2) | Group 1 Handout