
LAMC Meetings Archive • Fall 2007–Summer 2020For the current schedule, visit the Circle Calendar 2007–2008  2008–2009  2009–2010  2010–2011  2011–2012  2012–2013  2013–2014  2014–2015  2015–2016  2016–2017  2017–2018  2018–2019  2019–2020   Fall 2019 quarter  10/6/2019  We start off the new year with some problems from the Moscow Math Olympiad.
Handouts: Handout We start off the new year with some problems from the Moscow Math Olympiad.
Handouts: Week1 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: Wallpaper Symmetries Introduction to Gaussian integers, as well as prime and irreducible Gaussian integers
Handouts: Gaussian Integers 1 Did you ever want to communicate with your friends in a secret way that only those who knew the secret could understand? A cipher is a tool for writing and reading secret messages. We will study three ciphers this time: the reverse cipher, Polybius cipher, and Casear cipher. You will find a template for making a Caesar cipher disk at the link below.
Handouts: Caesar cipher template Did you ever want to communicate with your friends in a secret way that only those who knew the secret could understand? A cipher is a tool for writing and reading secret messages. We will study three ciphers this time: the reverse cipher, Polybius cipher, and Casear cipher. You will find a template for making a Caesar cipher disk at the link below. Handouts: Caesar cipher template To start out the quarter, we will be developing logic skills by looking at problems with hats and doors. The goal of this handout is to learn how to make an assumption, test the assumption, and readjust the original assumption if necessary.
Handouts: handout  solutions To start out the quarter, we will be developing logic skills by looking at problems with hats and doors. The goal of this handout is to learn how to make an assumption, test the assumption, and readjust the original assumption if necessary.
Handouts: handout  solutions We will start the BNP curriculum.
We will solve a variety of fun problems.
Today we will be looking at how to optimally divide cakes among any amount of people by using an ancient Egyptian technique of breaking fractions into their unit parts but doing so in a specific way, which we will work with and then prove to be optimal.
Handouts: Egyptian Fractions 1 Today we will be looking at how to optimally divide cakes among any amount of people by using an ancient Egyptian technique of breaking fractions into their unit parts but doing so in a specific way, which we will work with and then prove to be optimal. Handouts: Egyptian Fractions 1 We will be starting off the new quarter with some algebra activities.
Handouts: Handout  Solutions We will be starting off the new quarter with some algebra activities.
Handouts: Handout  Solutions We will discuss a few properties concerning representations of nonnegative integers and rationals in a general base (such as divisibility criteria), and particularly in base 10. Then we will apply this knowledge to several Olympiadtype Number Theory problems.
Handouts: Handout  Solutions  10/9/2019  We will go through a sample AMC exam from 2019.
 10/13/2019  We start off probability by introducing some set theory notation and defining sample spaces. Then we apply those to classical probability questions.
Handouts: Handout  Homework We start off probability by introducing some set theory notation and defining sample spaces. Then we apply those to classical probability questions.
Handouts: Week2  Homework2 We will continue to answer some of the questions posed about wallpaper symmetries. For those that finish the worksheet on wallpaper symmetries, we started discussing the origins and early properties of the padic numbers.
Handouts: Wallpaper Symmetries  Absolute Values and pAdics Characterization of which positive integers are sums of squares
Handouts: Gaussian Integers 2 An anagram is a word or phrase formed by rearranging letters of another word or phrase. Most often, all the original letters are used once. For example, the words "silent" and "listen" form an anagram. This time, we will have anagrams for a warmup and then study two ciphers: pigpen cipher and rail fence cipher.
An anagram is a word or phrase formed by rearranging letters of another word or phrase. Most often, all the original letters are used once. For example, the words "silent" and "listen" form an anagram. This time, we will have anagrams for a warmup and then study two ciphers: pigpen cipher and rail fence cipher. An alternate method of multiplication based in binary numbers and the basic ideas of the distributive property.
Handouts: Handout  solutions An alternate method of multiplication based in binary numbers and the basic ideas of the distributive property.
Handouts: Handout  solutions We will continue the BNP curriculum.
We will solve a variety of fun problems.
This week we continue from last week and try to prove a mathematical algorithm, which is a very important activity in mathematics. Handouts: Egyptian fractions 2 This week we continue from last week and try to prove a mathematical algorithm, which is a very important activity in mathematics. Handouts: Egyptian fractions 2 We will discuss boardtiling problems is it possible to completely cover a region with a particular set of tiles without overlap?  and interesting mathematical problems that arise from these puzzles.
Handouts: Handout We will discuss boardtiling problems is it possible to completely cover a region with a particular set of tiles without overlap?  and interesting mathematical problems that arise from these puzzles.
Handouts: Handout We will solve a variety of geometry problems involving the computation of a length or an area, or using notions about areas to prove an identity. The problems range in difficulty from introductory to fairly challenging.
Handouts: Handout We will go over these fundamental concepts and do a range of problems.
Handouts: Handout  10/16/2019  We'll continue going through the solutions for the 2019 AMC exams
 10/20/2019  We continue our probability unit with a worksheet on Conditional Probability. This includes the Law of Total Probability and Bayes' Theorem.
Handouts: Handout  Homework We are continuing solving probability problems. The topics for this class are Conditional Probability, Law of Total Probability, Bayes' Rule. Handouts: Week3  Homework3 We will study the symmetries of frieze patterns, especially those of unimodular frieze patterns. We work toward an interesting result proved by Conway and Coxeter about polygonal structures in frieze patterns.
Handouts: Triangulated Polygons and Frieze Patterns We will study the symmetries of frieze patterns, especially those of unimodular frieze patterns. We work toward an interesting result proved by Conway and Coxeter about polygonal structures in frieze patterns. Handouts: Frieze Patterns 1 We will finish the study of the rail fence cipher and take a quiz on ciphers. Remember, we do not quiz students, we quiz teachers. If you get a low grade, it means we didn't do a good job. Please study for the quiz  don't let us down!
We will finish the study of the rail fence cipher and take a quiz on ciphers. Remember, we do not quiz students, we quiz teachers. If you get a low grade, it means we didn't do a good job. Please study for the quiz  don't let us down! We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2
Handouts: handout  answers We will continue with different ways to multiplying 2 numbers. This week we will look at Russian Peasant Multiplication, which, surprisingly, has no relation to Russia or Peasants. However, this will be another good way to show the students how to write numbers as sum of powers of 2
Handouts: handout  answers We will continue the BNP curriculum.
We will solve a variety of fun problems.
This week we begin on a very very very important topic in mathematics; Modular Arithmetic. Although the importance of this topic is not fully realized until a course in Abstract Algebra, it is still a very important topic because it can instill an understanding that the mathematics or arithmetic that we see commonly is not the only way of operating and that mathematics can be far more general and far reaching. Handouts: Part 1 This week we begin on a very very very important topic in mathematics; Modular Arithmetic. Although the importance of this topic is not fully realized until a course in Abstract Algebra, it is still a very important topic because it can instill an understanding that the mathematics or arithmetic that we see commonly is not the only way of operating and that mathematics can be far more general and far reaching. Handouts: Part 1 We will continue to expand on last week's topic of tilings.
Handouts: Handout  Solutions We will continue to expand on last week's topic of tilings.
Handouts: Handout  Solutions We solve a couple of introductory problems to Graph Theory and several harder ones. The lesson spans over two weeks. Handouts: Handout Using unique factorization into primes and modular arithmetic to solve problems.
 10/23/2019  We will go through geometry problems with related to triangles and circles in the context of AMC
 10/27/2019  Our third week of probability includes a handout on random variables and expected value, as well as some more interesting conditional probability problems.
Handouts: Handout Our third week of probability includes a handout on random variables and expected value, as well as some more interesting conditional probability problems. Handouts: Week4 We continue our study of the symmetries of unimodular frieze patterns with some challenge problems. We introduce the result proved by Conway and Coxeter connecting polygonal triangulations to frieze patterns.
Handouts: Triangulated Polygons and Frieze Patterns II We continue our study of the symmetries of unimodular frieze patterns with some challenge problems. We introduce the result proved by Conway and Coxeter connecting polygonal triangulations to frieze patterns. Handouts: Frieze Patterns 2 We will take the first look at one of the most important concepts in mathematics, that of a function.
We will take the first look at one of the most important concepts in mathematics, that of a function. This packet introduces the topic of exponents, some of their properties, and size comparison! Handouts: answers pgs 13  handout This packet introduces the topic of exponents, some of their properties, and size comparison! Handouts: answers pgs 13  handout We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 12 contest.
A continuation of next week. Please use this week's sheet as a a set of problems that build stronger arithmetic skills and provide insight into other ways of "counting." Handouts: Part II A continuation of next week. Please use this week's sheet as a a set of problems that build stronger arithmetic skills and provide insight into other ways of "counting." Handouts: Part II We will be working with some algebra problems this week.
Handouts: Handout  Solutions We will be working with some algebra problems this week.
Handouts: Handout  Solutions We continue the lesson from last week with a few Olympiadtype problems in Graph Theory. Handouts: Handout Euler's totient function and more numbertheoretic functions Handouts: Handout  10/30/2019  We will go through AMC geometry problems with the emphasis on polygons.  11/3/2019  We will be working through a fun application of probability and graph theory to electrical circuits. No knowledge of physics is necessary, everything needed is contained in the handout. Handouts: Handout We will be working through a fun application of probability and graph theory to electrical circuits. No knowledge of physics is necessary, everything needed is contained in the handout. Handouts: ElectricalCircuits We work in groups to write rigorous proofs of statements from number theory, graph theory, combinatorics, set theory and other topics. Handouts: Formal Proofs Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs Handouts: Proofs 1 Students will learn the y = f(x) notation. Students will learn the y = f(x) notation. Continuing last week's handout! Handouts: handout  pg 4 answers Continuing last week's handout! Handouts: pg 4 answers  handout We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 12 contest.
This week we are working with some logic problems in the guise of "real world" sentences. The hope is that students can see that a basic logical structure is embedded in common speech, and that logical structure is the basis for mathematics and is taken to its most pure form and to its greatest lengths. Please use these worksheets as a warm up for the logical skills needed for induction next week. Handouts: Handout This week we are working with some logic problems in the guise of "real world" sentences. The hope is that students can see that a basic logical structure is embedded in common speech, and that logical structure is the basis for mathematics and is taken to its most pure form and to its greatest lengths. Please use these worksheets as a warm up for the logical skills needed for induction next week. Handouts: Handout Continuing our work in algebra last week, we will look at remainders and divisibility. Handouts: Solutions  Handout Continuing our work in algebra last week, we will look at remainders and divisibility. Handouts: Solutions  Handout We discuss various techniques for solving problems involving integer inequalities, specific to Olympiad Number Theory. Handouts: Handout NT problems including showing there are infinitely many 4k+1 primes. Handouts: Handout  11/6/2019  We will cover some of the applications of trigonometry in AMC.  11/10/2019  We dive further into the connection between random walks and electrical circuits, deriving series and parallel circuits. We then discuss electrical circuits on an integer lattice and its connection to Polya's problem. Handouts: Handout We dive further into the connection between random walks and electrical circuits, deriving series and parallel circuits. We then discuss electrical circuits on an integer lattice and its connection to Polya's problem. Handouts: Week6Handout We work in groups to write rigorous proofs of statements from number theory, graph theory, combinatorics, set theory and other topics. Handouts: Formal Proofs II Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs Handouts: Proofs 2 We will continue studying functions. We will continue studying functions. Review what we learned about exponents and continue to build how to compare numbers Handouts: handout  solutions I  solutions II Review what we learned about exponents and continue to build how to compare numbers Handouts: handout  solutions I  solutions II We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 12 contest.
This week we are starting induction! If these problems are challenging (they most likely are) do not panic. These are meant to be attempted to the best of everyone's ability and asking questions is extremely encouraged. Handouts: Induction This week we are starting induction! If these problems are challenging (they most likely are) do not panic. These are meant to be attempted to the best of everyone's ability and asking questions is extremely encouraged. Handouts: Induction We will continue our work in algebra with GCDs. Handouts: Handout  Solutions We will continue our work in algebra with GCDs. Handouts: Handout  Solutions We continue the lecture from last time, with a few additional Olympiadtype Number Theory problems. Handouts: Handout The useful Lifting the Exponent Lemma, which states that for odd prime p and m,n such that p(mn), vp(m^pn^p) = vp(mn)+1. Handouts: Handout  Solutions  11/17/2019  We start a new unit: Complex Numbers! We will see that while complex numbers might seem unnatural, they are actually extremely useful and can simplify problems that have seemingly nothing to do with the square root of 1. Handouts: Handout We start a new unit: Complex Numbers! We will see that while complex numbers might seem unnatural, they are actually extremely useful and can simplify problems that have seemingly nothing to do with the square root of 1. Handouts: Week7Handout We work on Olympiad style problems about Combinatorics, Number Theory, Probability, Geometry, and basic Set and Function Theory. Handouts: Olympiad Problems Handouts: Olympiad Style Problems 1  Solutions We will review functions. Then students will take quiz 2. We will review functions. Then students will take quiz 2. A look at how we can organize multiple sets of objects/people/things using venn diagrams! Handouts: handout 1  handout 2  solutions 1  solutions 2 A look at how we can organize multiple sets of objects/people/things using venn diagrams! Handouts: handout 1  handout 2  solutions 1  solutions 2 We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 12 contest.
This week we are continuing on induction proofs. It is fully expected that these problems are challenging, but they are extremely important to attempt, not necessarily get correct. The point of these two weeks is to 1. prepare students for induction proofs (the basis of discrete math and many proofs about finite objects) and 2. to challenge them to see the power of mathematical logic to prove an infinite amount of statements in a finite time. Handouts: Replacements for 3 and 5  Main worksheet (ignore problems 3 and 5) This week we are continuing on induction proofs. It is fully expected that these problems are challenging, but they are extremely important to attempt, not necessarily get correct. The point of these two weeks is to 1. prepare students for induction proofs (the basis of discrete math and many proofs about finite objects) and 2. to challenge them to see the power of mathematical logic to prove an infinite amount of statements in a finite time. Handouts: Replacements for 3 and 5  Main hand out (ignore problem 3 and 5) The Euclidean Algorithm is a way to find the greatest common divisor of two numbers. Using what we've learned in algebra in the past weeks, we will investigate how this method works. Handouts: Handout  Solutions The Euclidean Algorithm is a way to find the greatest common divisor of two numbers. Using what we've learned in algebra in the past weeks, we will investigate how this method works. Handouts: Handout  Solutions We define convexity, convex hulls and triangulations, then solve a few Olympiadtype problems in Geometric Combinatorics. Handouts: Handout Techniques for solving equations with the constraint that the variables must take integer values. Handouts: Handout  11/20/2019  We will go through AMC problems covering several sections of number theory  11/23/2019  We define and discuss basic concepts from trigonometry, including the law of sines and the law of cosines. We then apply these notions to solving several Olympiadtype geometry problems. Handouts: Handout  11/24/2019  We will explore roots of unity and some geometric aspects of the complex numbers. Handouts: Notes  Handout We will explore roots of unity and some geometric aspects of the complex numbers. Handouts: Notes  ComplexNumbersII We do further Olympiad style problems about Combinatorics, Number Theory, Probability, Geometry, and basic Set and Function Theory. Handouts: Olympiad Problems Hints  Olympiad Problems II  Olympiad Problem Solutions Handouts: Olympiad Style Problems 2  Solutions  Hints This week we will be looking at what it means for shapes to be similar, as well as exploring how we can add on to a shape to create another that is similar to the original. Handouts: handout 1  handout 2  solutions 1  solutions 2 This week we will be looking at what it means for shapes to be similar, as well as exploring how we can add on to a shape to create another that is similar to the original. Handouts: handout 1  handout 2  solutions 1  solutions 2 We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 12 contest.
This week we are working on finding the perimeter of some interesting shapes. These problems are designed to prepare students to quickly recognize the shapes they are looking at and how they can break them into easily solvable pieces. Handouts: Worksheet This week we are working on finding the perimeter of some interesting shapes. These problems are designed to prepare students to quickly recognize the shapes they are looking at and how they can break them into easily solvable pieces. Handouts: Perimeter Some more fun with algebra, primes, and GCDs! Handouts: Handout  Solutions Some more fun with algebra, primes, and GCDs! Handouts: Handout  Solutions Handouts: Handout  12/1/2019  First, students will learn basic facts about polygons. Then they will proceed to build some solids out of cubes and to draw their 2D projections. First, students will learn basic facts about polygons. Then they will proceed to build some solids out of cubes and to draw their 2D projections.  12/8/2019  For the last meeting of the quarter we will split up into groups and compete! We're playing a traditional endof quarter game. Handouts: Problems with Solutions A 2 hour long math competition against the other high school group! Handouts: Problems with Solutions In this class, students will draw 2D projections of more complicated solids. In this class, students will draw 2D projections of more complicated solids. For the last meeting of 2019, we will be playing a Math Review Game! For the last meeting of 2019, we will be playing a math review game! We will continue the BNP curriculum.
We will solve problems from the Math Kangaroo grades 12 contest.
This week we are playing a Math Circle classic! The rules are attached. Hope everyone has a great time :). Handouts: How to Play This week we are playing a Math Circle classic! The rules are attached. Hope everyone has a great time :). Handouts: How to Play We will finish out the quarter with some friendly competition! We will finish out the quarter with some friendly competition! Review of NT, review of counting and intro to combinatorial identities. Handouts: Handout  Winter 2020 quarter  1/12/2020  We start off the new (calendar) year with something you probably haven't seen before: tropical geometry. Throw out everything you know about addition and multiplication; we define new operations and explore polynomials under the new rules. Handouts: Handout We start off the new (calendar) year with something you probably haven't seen before: tropical polynomials. Throw out everything you know about addition and multiplication; we define new operations and explore polynomials under the new rules. Handouts: Tropical Polynomials  Solutions We motivate the study of metrics by introducing the taxicab metric (the distance traveled by a taxi in a city with a grid layout). How would you define a circle, line segment between points, or a parabola with a new notion of distance? Handouts: Taxicab Geometry  Taxicab Geometry Challenge Problems We introduce a new notion of distance  the taxicab metric. We investigate how geometry behaves with this new distance, and try to find similarities and differences between this and the Euclidean metric. Handouts: Metrics 1 Students were asked to study the 8th packet, More Solids, during the Winter break. Since the packet is rather challenging for the age, we will through it class. We started a warmup discussing auction theory. We will begin our exploration of game theory by starting with the example of subtraction games of varying subtraction sets. Handouts: handout  solutions We started a warmup discussing auction theory. We will begin our exploration of game theory by starting with the example of subtraction games of varying subtraction sets. Handouts: handout  solutions This week we will be looking at geometrical numbers and the successive difference found in the sequence of these numbers. Interestingly, these types of numbers have been studied for thousands of years ever sense the Ancient Greeks. These sorts of problems are very useful to build strategies for induction proofs and pattern spotting. Handouts: Worksheet  Solutions This week we will be looking at geometrical numbers and the successive difference found in the sequence of these numbers. Interestingly, these types of numbers have been studied for thousands of years ever sense the Ancient Greeks. These sorts of problems are very useful to build strategies for induction proofs and pattern spotting. Handouts: Worksheet  Solutions Welcome back to Math Circle! We will kick off the quarter with some applications of graphs and geometry. Handouts: Handout  Solutions  Homework  Homework Solutions Welcome back to Math Circle! We will kick off the quarter with some applications of graphs and geometry. Handouts: Handout  Solutions We state and prove Ceva's theorem, then solve a few Olympiadtype problems. Handouts: Handout Handouts: Handout  1/15/2020  We will go through AMC problems related to statistics and probability theory  1/17/2020  Handouts: Handout  1/19/2020  Handouts: Tropical Orthogonal Representations Using the taxicab metric as a guide, we define the general notion of metric and give numerous examples. Handouts: Metrics We continue our study of metrics, this time considering more abstract examples. We discuss sequences, and we learn that with regards to sequences, the taxicab metric and the Euclidean metric are equivalent. Handouts: Metrics 2 Students will solve problems from the 2014 Canadian Math Kangaroo test. Students will solve problems from the 2014 Canadian Math Kangaroo test. This week we started off class discussing "I cut you choose." We then continued will be continuing our discussion of subtraction games, in particular the Game 21. With 21 sticks, each player can either take away 1,2, or 3 sticks each turn, what is the winning strategy if you don't want to take the last stick? If you want to take the last one? How are these two related? This week we started off class discussing "I cut you choose." We then continued will be continuing our discussion of subtraction games, in particular the Game 21. With 21 sticks, each player can either take away 1,2, or 3 sticks each turn, what is the winning strategy if you don't want to take the last stick? If you want to take the last one? How are these two related? Handouts: Homework Problems This week we will be working on some very interesting logical problems. These sorts of problems, although not generalization to higher mathematical theory, are very useful in practicing mathematical logic and intuition. Handouts: Worksheet  Solutions This week we will be working on some very interesting logical problems. These sorts of problems, although not generalization to higher mathematical theory, are very useful in practicing mathematical logic and intuition. Handouts: Worksheet  Solutions More work with graphs and geometry. Handouts: Handout  Solutions More work with graphs and geometry. Handouts: Handout  Solutions The students took a 2hour long practice exam for BAMOtype competitions. There were 3 problems (in Geometry, Combinatorics and Number Theory). Handouts: Handout Handouts: Handout  1/22/2020  AMC problems related to high school level functions theory  1/26/2020  In the first part of a two (or three) part series on cardinality, we deal with bijections and give a proper definition for the cardinality of a set. Handouts: Handout  Homework In the first part of a 3part series on cardinality, we deal with bijections and give a proper definition for the cardinality of a set. Handouts: Infinity1  Homework We can extend the integers by including square roots of integers like 1 or 3. Can we predict which prime integers remain prime in the extension? Handouts: Primes in Extensions of the Integers I We can extend the integers by including square roots of integers like 1 or 3. Can we predict which prime integers remain prime in the extension? Handouts: Prime Splitting 1 First, we will go through a few problems from the 9th packet. Then students will take a quiz on polygons and solids. Then we will finish discussing the packet. First, we will go through a few problems from the 9th packet. Then students will take a quiz on polygons and solids. Then we will finish discussing the packet. This week, we will be preparing for the Math Kangaroo contest by working out various types of problems! Handouts: solutions  handouts This week, we will be preparing for the Math Kangaroo contest by working out various types of problems! Handouts: handouts  solutions More work with graph theory and geometry proofs. Handouts: Handout  Solutions More work with graph theory and geometry proofs. Handouts: Handout  Solutions Today we discussed the solutions for Practice Exam 1, and then solved three other problems. Handouts: Handout Handouts: Handout  Solutions  1/29/2020  We will go through an actual AMC test, and discuss some of the solutions  2/2/2020  We continue with part 2 of our Infinity sequence. This time we define cardinality through injectivity and bijectivity. Handouts: Homework  Handout In the second part of the sequence, we explore more bijections and prove several surprising facts about the cardinalities about the naturals, integers, and the reals. Handouts: Infinity 2  Homework Happy Groundhog Day! With the help of famous results about quadratic reciprocity we will classify primes in most quadratic extensions of the integers. Handouts: Primes in Extensions of the Integers II With the help of famous results about quadratic reciprocity we will classify primes in most quadratic extensions of the integers. Handouts: Primes Splitting 2 We will solve logic problems from the Knights and Liars packet. (Oleg Gleizer) We will solve logic problems from the Knights and Liars packet. (Oleg Gleizer) Intro: How many days are in a year? How are leap years counted? Why do we include every 400 years, but not 100, 200, or 300? Hint: a year is technically about 365.25 days  11 minutes. In this handout we examine how to systematically perform calculations to find the day of the week (Sunday, Monday, Tuesday, etc.) a particular date is, e.g., your birthday Handouts: handout  solutions Intro: How many days are in a year? How are leap years counted? Why do we include every 400 years, but not 100, 200, or 300? Hint: a year is technically about 365.25 days  11 minutes. In this handout we examine how to systematically perform calculations to find the day of the week (Sunday, Monday, Tuesday, etc.) a particular date is, e.g., your birthday Handouts: handout  solutions This week we will be playing a game with the Math Kangaroo Practice tests. The Math Kangaroo is a math test for young mathematicians and is usually held in the middle of the year. The game we are playing this week will encourage team work, communication, and dieligent justification of answers. Handouts: Test 2013  Test 1998 This week we will be playing a game with the Math Kangaroo Practice tests. The Math Kangaroo is a math test for young mathematicians and is usually held in the middle of the year. The game we are playing this week will encourage team work, communication, and dieligent justification of answers. Handouts: Test 2013  Test 1998 More graphs and geometry. Handouts: Handout  Solutions  Homework  Homework Solutions More graphs and geometry. Handouts: Handout  Solutions The students took an Olympiadstyle practice exam with 3 problems (in Geometry, Algebra, and Combinatorics). Handouts: Handout Handouts: Solutions  Handout  2/3/2020  This week we start on the topic of Graph Theory! This topic considers a particular object of mathematics, the Graph, which is defined by its vertices and the edges connecting them. We will be considering the degree of the vertices as well a particular type of Graph, the Bipartite Graph. Handouts: Graph Theory Sheet  Solutions  2/9/2020  Handouts: Handout This week we are concluding our conversation about infinity with some more challenge problems on constructing bijections. We also look into some mathematical paradoxes related to infinity and not only. Handouts: Infinity3 We will introduce the generating function, a creative, combinatorial tool that can simply solve many interesting problems. By the end, we will have used generating functions to study the Fibonacci sequence, dice games, and ways to pay the unlucky cashier with coins. Handouts: Generating Functions  Generating Functions Solutions We will introduce the generating function, a creative, combinatorial tool that can simply solve many interesting problems. By the end, we will have used generating functions to study the Fibonacci sequence, dice games, and ways to pay the unlucky cashier with coins. Handouts: Generating Functions Students will learn the notion of a statement, simple and composite, and how the truth function takes value on statements. Students will learn the notion of a statement, simple and composite, and how the truth function takes value on statements. WarmUp: What is the trick to square numbers ending with 5 and why does it work? More math kangaroo problems! Handouts: handouts  solutions WarmUp: What is the trick to square numbers ending with 5 and why does it work? More math kangaroo problems! Handouts: handout  solutions This week we start on the topic of Graph Theory! This topic considers a particular object of mathematics, the Graph, which is defined by its vertices and the edges connecting them. We will be considering the degree of the vertices as well a particular type of Graph, the Bipartite Graph. Handouts: Solutions  Graph Theory Sheet We will be finishing up our unit on graphs and geometry this week! Handouts: Homework  Handout  Solutions  Homework Solutions We will be finishing up our unit on graphs and geometry this week! Handouts: Handout  Solutions Today we went through the solutions for the practice exam, and we solved a few extra problems in class. Next time there will be no class (on President's Day). Handouts: Handout Handouts: Handout  2/16/2020  Please use the worksheet from last week. This week we completed the rest of the Graph Theory packet, going over topics such as the coloring of a graph, which is the least number of colors required to fully "color" a graph. An interesting theory associated with this is that if one considers ANY map that separates the area into counties, states, cities, etc. then one can color that map with four colors such that no two adjacent counties, states, cities, etc. have the same color. This theorem was proved using computers! Handouts: Graph Theory Sheet  Solutions Please use the worksheet from last week. Handouts: Graph Theory Sheet  Solutions Handouts: Handout  2/23/2020  We start off our two week unit on Point Mass Geometry. This is a topic that lets you get the nice results of planar geometry without the tedious work! Handouts: Handout  Solutions We will learn a new method of solving geometry problems with assigning masses to points and using intuition from physics. Handouts: Handout  Solutions In the first week of a three week sequence, we introduce basic probability concepts such as conditional probability, Bayes' Rule, and tower property Handouts: Measure and Probability I Handouts: Probability 1 We will continue studying the Mathematical Logic minicourse. We will continue studying the Mathematical Logic minicourse. WarmUp: Multiply twodigit and threedigit numbers using the crisscross method! It really reduces the amount of work required! Today we will be introducing the topic of Polyhedras and learning the terminology. Handouts: handout  solutions WarmUp: Multiply twodigit and threedigit numbers using the crisscross method! It really reduces the amount of work required! Today we will be introducing the topic of Polyhedras and learning the terminology. Handouts: handout  solutions This week we will be solving the instant insanity puzzle. This puzzle is a mathematicians favorite as it is difficult to solve by “brute force,” but permits a very simple and elegant mathematical solution. We will begin by trying brute force tactics and when that fails us we will find a way to use graph theory to solve the puzzle very simply. here is the amazon link to the puzzle: Winning Moves Games Instant Insanity https://www.amazon.com/dp/B004KCN6EQ/ref=cm_sw_r_cp_api_i_QwSuEb6C13BXG This week we will be solving the instant insanity puzzle. This puzzle is a mathematicians favorite as it is difficult to solve by “brute force,” but permits a very simple and elegant mathematical solution. We will begin by trying brute force tactics and when that fails us we will find a way to use graph theory to solve the puzzle very simply. here is the amazon link to the puzzle: Winning Moves Games Instant Insanity https://www.amazon.com/dp/B004KCN6EQ/ref=cm_sw_r_cp_api_i_QwSuEb6C13BXG We will begin a new unit on game theory. Handouts: Handout  Homework  Homework Solutions We will begin a new unit on game theory. Handouts: Handout We solve five final practice problems for the Bay Area Mathematical Olympiad. Handouts: Handout Handouts: Handout  3/1/2020  Handouts: Handout  Solutions Handouts: Handout We will continue our study of probability by introducing random variables and distributions. Handouts: Measure and Probability II Handouts: Probability 2 We will fist check some problems from the packet 12 homework. Then we will begin studying logic gates. We will fist check some problems from the packet 12 homework. Then we will begin studying logic gates. WarmUp: If I give you two numbers, like 998 and 992, and I ask you to multiply them together using conventional math techniques, you end up writing a lot of numbers to generate the answer. But notice that 998 is just 2 shy of 1000, and 992 is just 8 shy of 1000. If you multiply 2 times 8, you get 16. And if you take 8 away from 998, or you take 2 away from 992, you get 990. And guess what? The correct answer is 990016. Similarly, if I ask you to multiply 990 times 991, you could work it out ... or you could recognize that 990 is 10 below 1000, 991 is 9 below, the product of 9 and 10 is 90, and 990 minus 9 is 981, and 991 minus 10 is also 981. The answer: 981090. The insight: if you rewrite 998 as (10002) and 992 as (10008), multiply the two we get 1000*1000  2*1000  8*1000 + 8*2. Hence we get (100028)*1000 + 16 = 990,016. We will be continuing our topic on Polyhedra from last week! Handouts: handout  solutions WarmUp: If I give you two numbers, like 998 and 992, and I ask you to multiply them together using conventional math techniques, you end up writing a lot of numbers to generate the answer. But notice that 998 is just 2 shy of 1000, and 992 is just 8 shy of 1000. If you multiply 2 times 8, you get 16. And if you take 8 away from 998, or you take 2 away from 992, you get 990. And guess what? The correct answer is 990016. Similarly, if I ask you to multiply 990 times 991, you could work it out ... or you could recognize that 990 is 10 below 1000, 991 is 9 below, the product of 9 and 10 is 90, and 990 minus 9 is 981, and 991 minus 10 is also 981. The answer: 981090. The insight: if you rewrite 998 as (10002) and 992 as (10008), multiply the two we get 1000*1000  2*1000  8*1000 + 8*2. Hence we get (100028)*1000 + 16 = 990,016. We will be continuing our topic on Polyhedra from last week! Handouts: handout  solutions This week we will be working with probability! We will be learning how to calculate the expected value of a given game. This sort of "average" thinking is used all the time by everyone unconsciously and explicitly. Handouts: Probability sheet This week we will be working with probability! We will be learning how to calculate the expected value of a given game. This sort of "average" thinking is used all the time by everyone unconsciously and explicitly. Handouts: Probability sheet More game theory! We will introduce winning and losing positions. Handouts: Handout  Solutions  Homework More game theory! We will introduce winning and losing positions. Handouts: Handout  Solutions We introduce and study the basic properties of complex numbers, with the goal of using them to solve geometry problems. Handouts: Handout Handouts: Handout  3/4/2020  We will go through some AIME problems from past exams  3/5/2020  We continue our discussion of complex numbers and start applying them to Euclidean geometry problems. Handouts: Handout  3/8/2020  We continue working on the second handout on point mass geometry. This gets into negative masses and barycentric coordinates. Handouts: Handout  Solutions Handouts: Handout We will continue our study of random variables by introducing the Bernoulli, geometric, and binomial distributions. As a final surprising example, we construct a famous example of a nonmeasurable set. Handouts: Measure and Probability I, II, and III We will finish studying logic gates and will study double negation. We will finish studying logic gates and will study double negation. This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces! Handouts: handout This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces! Handouts: handout This week we played the famous Math Dominoes Game with an added CHALLENGE ROUND. I hope everyone has a great time! Continuing our unit on game theory. Handouts: Handout  Solutions Continuing our unit on game theory. Handouts: Handout  Solutions Handouts:  3/15/2020  For our final (virtual) meeting of the quarter, we will be hosting a Kahoot competition over the Zoom application. Please make sure to download the app at https://zoom.us/download and join the meeting from 46 pm at the link sent via email. See you there! We will prepare for the upcoming Math Kangaroo competition. We will prepare for the upcoming Math Kangaroo competition. This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces! Handouts: handout This week we will be continuing our topic of Polyhedras, but this time developing a more rigorous relationship between edges, vertices, and faces! Handouts: handout Handouts:  Spring 2020 quarter  4/5/2020  Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs Handouts: Handout We'll go over various proof techniques and practice writing rigorous proofs. Handouts: FormalProofs We review the material from last quarter. Handouts: Time Travel Travel back in time and revisit some of the old topics from Winter 2020 Handouts: Time Travel We will discuss some problems from packet 14. Then students will take a quiz. We will discuss some problems from packet 14. Then students will take a quiz. This helps students see what ageappropriate math problems look like in a different country! Handouts: handout  solutions This helps students see what ageappropriate math problems look like in a different country! Handouts: handout  solutions This week we solved a mathematical trick using equivalence relations! Equivalence relations are extremely important in mathematics and allow mathematicians to formally categorize "objects" in a particular "Space." Handouts: Handout  Solutions This week we solved a mathematical trick using equivalence relations! Equivalence relations are extremely important in mathematics and allow mathematicians to formally categorize "objects" in a particular "Space." Handouts: Handout  Solutions Welcome back to (virtual) Math Circle! We're kicking off the quarter with some proofs by induction. Handouts: Handout  Homework Handouts: Lecture Notes  Handout  4/12/2020  Introduction to formal proof writing. Sets, functions, basic combinatorics and number theory proofs Handouts: Handout We'll continue working on the handout on writing formal and rigorous proofs. Handouts: FormalProofs2 We use our knowledge of metrics to study and prove theorems about contraction maps. Handouts: Fixed Points We study fixed points, and a few theorems that let us find them. Handouts: Fixed Points We will meet inhabitants of a 1dimesional world, called Lineland. We will meet inhabitants of a 1dimesional world, called Lineland. This week, we will be introducing the distance formula! Handouts: handout  solutions This week, we will be introducing the distance formula! Handouts: handout  solutions We will continue working on proofs by induction. Handouts: Handout  Homework We solve a few more difficult geometry problems using complex numbers. Handouts: Handout We prove that for all n there exists a prime number between n and 2*n. Handouts: Handout  Lecture Notes  4/13/2020  This week we worked on some intuitive mathematics! By that I mean mathematics that can be reliably answered by the physical intuition we have all built up over our lives. This "weighty" way of thinking that we are all seemingly borne with was first formalized mathematically in Ancient Greece and this week we get to explore some of its implications. Handouts: Worksheet  Solutions  4/17/2020   4/19/2020  Handouts: Handout We are finishing the handout on writing formal and rigorous proofs. Sperner's Lemma is an interesting result about the coloring of special graphs. We connect the result with a famous result, the Brouwer fixedpoint theorem. Handouts: Sperner's Lemma We discuss some interesting graph theory, including the Euler characteristic, and proving Brouwer's Fixed Point Theorem with Sperner's Lemma. Handouts: Graphs 1 We will continue to study packet 15 starting from the problem 15.10. We will continue to study packet 15 starting from the problem 15.10. We will be continuing the idea of rates from last week, but this time applying to trade! What is the best way to optimize the outcome of both parties in trade? When is trade not a winwin situation? Handouts: handout We will be continuing the idea of rates from last week, but this time applying to trade! What is the best way to optimize the outcome of both parties in trade? When is trade not a winwin situation? Handouts: handout This week we will be taking on some challenging problems from the Russian math Olympiad! These are questions designed to test some of the smartest young mathematicians. I hope everyone is challenged and excited! Handouts: Handout This week we will be taking on some challenging problems from the Russian math Olympiad! These are questions designed to test some of the smartest young mathematicians. I hope everyone is challenged and excited! Proofs by induction in algebra, and some geometry. Handouts: Handout  Homework  Solutions We introduce basic notions of probability spaces, outcomes, events and independence, and solve a few related competitionstyle problems. Handouts: Handout Handouts: Handout  4/26/2020  We take a look at the HMGMAMQM inequalities and their applications. Handouts: Handout We take a look at the HMGMAMQM inequalities and their applications. Handouts: Inequalities We continue working on last week's worksheet, with a few extra problems on Brouwer's Fixed Point Theorem, Euler characteristic, and crossing numbers. Handouts: More Graphs We continue working on last week's worksheet, with a few extra problems on Brouwer's Fixed Point Theorem, Euler characteristic, and crossing numbers. Handouts: More Graphs We will prepare for the upcoming online MK competition. We will prepare for the upcoming online MK competition. This week, we will continue our discussion on trade, learning about opportunity costs as a way of determining when trade is a winwin situation and when it is not, as well as finding a rate that both traders will be happy with Handouts: handout This week, we will continue our discussion on trade, learning about opportunity costs as a way of determining when trade is a winwin situation and when it is not, as well as finding a rate that both traders will be happy with Handouts: handout This week we will reviewing the solutions to the Olympiad as well as starting a section on decimals and fractions. Handouts: Handout Handouts: Handout We continue with induction! Handouts: Handout  Solutions  Homework We introduce random variables, expectations, variance, independence and functions of random variables, then apply these notions to Olympiadtype problems. Handouts: Handout Handouts: Handout  5/3/2020  We continue with more applications of our HMGMAMQM inequalities. Handouts: Handout  Solutions We continue the handout on inequalities. Handouts: Inequalities2 We introduce braids, knots, and links. Handouts: Braids and Knots We study braids and begin to study knots. Handouts: Knots Week 1 The class will be a mixture of the Flatland handout and MK preparation. The class will be a mixture of the Flatland handout and MK preparation. A fun game played on a square grid... a twist on Latin squares! Handouts: handout  solutions A fun game played on a square grid... a twist on Latin squares! Handouts: handout  solutions Handouts: Handout Handouts: Worksheet This week we will introduce our unit on bipartite graphs. Handouts: Handout  Solutions  Homework We introduce the basic notions of convergence for sequences and series, and solve a few related problems. Handouts: Handout Handouts: Handout  5/10/2020  Handouts: Handout We introduce Reidemeister moves and the idea of a knot invariant. We work up to an interesting knot invariant that has the structure of a quandle. Handouts: Knot Invariants and Quandles We study a knot invariant, with the structure of a quandle. Handouts: Knots Week 2 Students will review the Lineland and Flatland material and take a quiz. Students will review the Lineland and Flatland material and take a quiz. This week we will introduce the idea of Prisoner's Dilemma! Handouts: teaching notes This week we will introduce the idea of Prisoner's Dilemma! Handouts: teaching notes Handouts: Handout Handouts: Worksheet More work with bipartite graphs and geometry. Handouts: Handout  Solutions  Homework We continue talking about sequences and series, and solve a few harder related problems. Handouts: Handout Definitions, examples, subgroups and isomorphisms, and some philosophy Handouts: Handout  5/17/2020  We introduce our most powerful knot invariants yet: the normalized Kauffman bracket and the Jones polynomial. Handouts: Kauffman Bracket and Jones Polynomial We study a more difficult knot invariant  the Jones polynomial. Handouts: Knots Week 3 Students will learn to solve problems working backward. Students will learn to solve problems working backward. This week, we will talk about a cool application of combinations and permutations! Handouts: homework  teaching notes This week, we will talk about a cool application of combinations and permutations! Handouts: teaching notes  homework Handouts: Handout Today we pivot from fractions and decimals and focus on percents, a related and equally important topic. Handouts: Handout This week, we focus on geometry. Handouts: Homework  Handout  Solutions We introduce continuous and differentiable functions, and study a few of their basic properties. Handouts: Handout Lagrange's theorem, quotients, maps between groups  5/24/2020   5/31/2020  We will continue studying backward reasoning. We are finishing the Inversion handout. Handouts: Problems Handouts: Game Theory We will continue studying backward reasoning. We will continue studying backward reasoning. We will be going over the hw from last time, and working through a new application! Handouts: handout  solution sketches We will be going over the hw from last time, and working through a new application! Handouts: handout  solution sketches Handouts: Handout Time to put your Math Circle knowledge to the test! We will be revisiting some of the topics we've covered this past year. Handouts: Handout  Solutions We finalize our discussion of differentiability with more rigor, and apply it to related Olympiadtype problems. Handouts: Handout Group actions and Burnside's Lemma  6/7/2020  We finish off the school year with a Kahoot individual competition emphasizing problems and techniques that we have done this year. Handouts: Problems Handouts: Problems We will help a mouse and two ants to travel through and around a cube. The journeys reveal a wealth of ideas and serve as an intro to studying nets of cubes. We will help a mouse and two ants to travel through and around a cube. The journeys reveal a wealth of ideas and serve as an intro to studying nets of cubes. We will be reviewing the concepts we have learned for the quarter Handouts: questions We will be reviewing the concepts we have learned for the quarter Handouts: questions Finishing out the quarter with some friendly competition. We introduce generating functions and apply them to combinatorial sequences such as binomial coefficients, the Fibonacci numbers and partitions. Handouts: Handout  Summer 2020 quarter  6/27/2020  Here are the two handouts for meeting 1 on 6/28. Handouts:   6/28/2020  We will begin to study Roman numerals. We will begin to study Roman numerals. We will study the Polybius and Caesar ciphers. We will study the Polybius and Caesar ciphers. We'll start off this summer with a lecture on basic testtaking techniques and an AMC 8 diagnostic test. 


