Fall 2016 quarter 
10/2/2016  We will start with a few beautiful warmup problems and proceed to learn some bare minimum of graph theory needed to fully understand the winning MathMovesU presentation Dan Tsan made last year. Handouts: handout We will begin by looking at Lineland with different shapes passing through this new and interesting world. This will help us next week when we take a look into Flatland. Handouts: Handout #1 Handouts: Handout Perhaps when you were bored in a math class you started looking at the patterns on the wallpaper around you.
Many types of wallpaper use a simple repeating pattern, but occasionally you might notice that the wallpaper has a more subtle kind of symmetry.
How many essentially different types of symmetry are there? As we will find out, the answer is seventeen.
Handouts:
We considered an area model for multiplication, which explains the concept of distribution, and we built a method for multiplying numbers. By the end of the session, most student were able to mentally multiply numbers between 10 and 20. When told that the method had no name, the students came up with names Multiplication by Distribution and Easier Multiplication. The session ended with the discovery of a pattern involving the squares of integers, also explained by the area model, which is where we will pick up next week. Handouts: Blank Worksheet  Answer Key 
10/9/2016  We will continue the study of the 10/2 handout. Handouts: handout Animations shown in class and extras for those interested in further investigation of the topic: Today we will mix techniques from physics, geometry, and algebra to solve classical geometry problems. Dr. Michael Hall is back with the full classification of wallpaper groups. We will discuss combinatorics this week, including the concepts of Stars and Bars and Young diagrams. Handouts: Handout  Solutions Handouts: Blank Worksheet (same as the previous week)  Answer Key (same as the previous week) 
10/16/2016  We will discuss planar graphs, Euler characteristic, and related topics.
We will investigate a game with dice to find the best strategy. Handouts: Handout #3  Solutions #3 We will continue studying Mass Point Geometry from the book A Decade of the Berkeley Math Circle. Our goal will be to prove that our methods are legitimate using classical Euclidean geometry. If you only know the sum of two numbers, and your friend only knows the product, neither of you are likely to figure out the original numbers. But, how much can you figure out if you just make statements about what you know? We solve problems on combinations with restrictions. Handouts: Handout  Solutions This week, we delve into hexaflexagons: flat objects that actually have three sides. Handouts: Blank Worksheet 
10/23/2016  Dan Tsan will give us a lecture based on his awardwinning MathMovesU presentation for the first hour. Then we will study planar graphs, Kuratowski Theorem and Euler characteristic of a graph.
Handouts: Handout #4  Solutions #4
We will discuss what we mean by information and what it means to transmit information. Of course, as you transmit information, errors can occur . Think of the children’s game “Broken Telephone”, where players take turns whispering a message made up by the first player to each other. The final message received by the last player could be very different from the original. While this is fun in a game, we usually try to avoid this as much as possible in real life. Can we create a code that has a builtin protection against transmission errors? What is the price that we have to pay for increasing accuracy? We will touch upon several ideas of Shannon’s Information Theory and work through several examples to find out.
Handouts: Handout  Solutions With Andrew out of town, Taylor Womack will be presenting a lesson on a new multiplication algorithm. Handouts: Answers  Blank Worksheet 
10/30/2016  The majority of the class have stopped working in the vicinity of Problem 9 from the 10/9 handout. We will resume from there, study the Euler characteristic of planar graphs and prove that the graphs K_3,3 and K_5 are not planar. The faster students who are finished or nearly finished with the handout will be given a bunch of MathOlympiadstyle problems to solve. Handouts: handout Handouts: Handout #5 Today we will learn about the permutations of the tiles of a Rubik's cube. We will follow the section on the Rubik's cube in our books. We go over problems of geometric probability and some problems that use coordinate geometry. Handouts: Handout  Solutions This week we covered various halloweentheremed math problems. Handouts: Blank Worksheet  Answers 
11/6/2016  We will finish studying the 10/30 handout. If time permits, we will use graph theory to solve the Instant Insanity puzzle. We will be exploring fake coins problems as well some additional puzzles Handouts: Handout  Solutions Today we will review the concept of mathematical induction from last year. We will follow the section in A Decade of the Berkeley Math Circle on this topic. Guest instructor Chris Ohrt leads a session on the geometry of origami! We will be looking at the prerequisites for understanding Burnside's lemma. This includes the definition of symmetry of an object and how colorings of an object can form equivalence classes. Handouts: Handout  Solutions This week we discuss a different, but somewhat similar algorithm to Egyptian Multiplication. Handouts: Blank Worksheet 
11/13/2016  We will attempt to finish the 10/30 handout. If/when finished, we will study the final handout of the Intro to Graphs minicourse, Introduction to Ramsey Theory. Handouts: handout Handouts: Handout Today we have a special guest speaker: Jeremy Brightbill, a PhD student at UCLA. He will give a talk on number fields. Following up on last week, we'll take a look at what sorts of things that Origami can do than regular rulerandcompass cannot. We will explore the orbit and stabilizer of a coloring of an object and how they can be related to the number of symmetries of an object. This is known as the orbitstabilizer lemma. Handouts: Handout  Solutions 
11/20/2016  We will discuss problem 9 of the 10/30 handout for warmup and then get back to studying graph theory.
Handouts: Handout  Solutions Today we will get back to our study of mathematical induction, and hopefully we'll get to start talking about strong induction. We will wrap up our study of symmetry by using the OrbitStabilizer lemma to obtain Burnside's lemma and look at the questions we can solve. Handouts: Handout  Solutions Handouts: Blank Worksheet 
12/4/2016  We will resume studying the 10/30 handout at problem 12. Once finished, we will begin studying the next, and final, handout of the minicourse.
We will finish out the quarter as usual with a mathematical relay. We will be doing math dominoes this week on the topics we have learned this quarter. 
Winter 2017 quarter 
1/15/2017  We will resume at Problem 14 of the 10/30/2016 handout and proceed to Ramsey theory if time permits.
In our first meeting of this new year, we will be looking at how two dimensional shapes can be altered to create other two dimensional shapes with as few cuts as possible. Handouts: Handout  Solutions We will start the year by studying finite state automata. Handouts: Handout For our first meeting of the quarter, we'll use some topology to try to understand the answer to the question: if you sew a bunch of pants together so that there are no holes, then cut them back apart so that there are no pockets, how many pants do you get? Handouts: Handout  Solutions In this week, we cover a two dimensional game of Leap Frog and what victory positions are possible. Handouts: Blank worksheet  Answers 
1/22/2017  We will finish the proof of Theoprem 1 from the 10/30 handout. Then we will begin and, hopefully, finish the last handout of the Intro to Graphs course, the one on Ramsey Theory.
Handouts: Handout  Solutions Today we will study vectors. Handouts: Handout Chris Ohrt joins us to talk about generating functions, a fascinating confluence of sequences, calculus, and combinatorics. This week, we discuss several logic problems on parity in class and learn how to write proper proofs. Handouts: Handout  Solutions We explore what happens when we change the rules of movement. No longer is the shortest distance between two points a straight line! Handouts: Blank Worksheet  Answer key 
1/29/2017  We study a bit of Ramsey theory during the first hour of the classes. Handouts: Handout  Solutions Today we will learn the basics of group theory by discussing symmetry and playing sodoku. Handouts: Handout We continue our study of generating functions, this time using them to prove some fascinating facts about sequences. We discuss fun problems that use the concept of divisibility. Handouts: Handout  Solutions We now dive deep into taxicab gemoetry after a thorough review of coordinates and a short introduction last week. Handouts: Blank Worksheet  Answer Sheet 
2/5/2017  The students will be given a twohour test that covers the Intro to Graphs course we just have finished. Preparing for the test is a good way to review the course. The tests' results will give the students and the instructors the muchneeded feedback. The top five performers will get great math books as prizes! Handouts: test Handouts: Handout  Solutions Today we will continue our study of group theory and work with groups of symmetries of objects (including the Rubik's cube). Handouts: Handout We build on our study of generating functions, learning to understand sequences generated by recurrence relations. Handouts: Handout  Solutions This week we do our first round of practice problems for Math Kangaroo. Handouts: Blank Worksheet 
2/12/2017  We will study the theory of quadratic equations and solve a few surprisingly hard problems on the topic. Handouts: handout Handouts: Handout Today we will review complex numbers, this time using our knowledge of vectors as a tool. Handouts: Handout We begin our study of theoretical computer science looking at the basics of computational complexity and the runtime of sorting algorithms. Handouts: Handout  Solutions 
2/26/2017  We will continue stidying the 2/12 handout.
Handouts: Handout  Solutions We will continue our study of complex numbers. Handouts: Handout We learn about Turing Machines, the theoretical universal computer. We learn about expectations of random events and discuss problems related to lotteries, roulette and Monopoly. Handouts: Handout  Solutions
Handouts: Blank Worksheet  Answer Key 
3/5/2017  We will resume studying the 2/17 handout at Problem 13. Handouts: Handout This week we will study combinatorics from our Berkeley Math Circle books. Nondeterminacy; learn the statement of one of the great open problems, P vs. NP We discuss the relationship between probability, binomial coefficients and combinatorics. Handouts: Handout  Solutions

3/12/2017  Last time, many of our students felt uncomfortable with the weighted sums in the formula defining a convex function. To alleviate the feeling, we will take a second look at the topic we studied in April 2013, Barycentric Coordinates. Then we will get back to Problem 17 of the 2/17 handout. Handouts: handout Handouts: Handout Today we will solve some problems from classical Euclidean geometry. How many guards does it take to watch an art gallery? It depends on the shape of the gallery! Suppose we break a stick into three pieces randomly. What are the chances the resulting pieces will form a triangle? Handouts: Handout  Solutions 
3/19/2017  We will start with refreshing problem 18 from the 2/12 handout.
http://www.math.ucla.edu/~radko/circles/lib/data/Handout12721283.pdf
We will then use the derived formula as a tool to solve Problems 19 and 20. Then we will finish the handout and proceed to the next one.
We'll finish out the quarter with some fun math relays. Handouts: Problems  Solutions 
Spring 2017 quarter 
4/9/2017  We will see how the Vieta formulas for quadratic equations enable one to solve qubic equations as well. Handouts: handout Handouts: Handout Today we will study Propositional Logic. Handouts: Handout Welcome back for the spring quarter! We're starting off with a class about polyhedra, and their curvature. 
4/16/2017  We will review the derivation of the Cardano formula, then learn long division of polynomials, and then solve some cubic equations.
Today we will start a several week unit on graph theory, using the Instant Insanity puzzle as our motivation. This week, we'll be studying the theory of vector fields on surfaces! This week, we willl continue learning about functions and discuss onetoone and onto functions. In the second hour, we will learn how to write good proofs for harder problems. Handouts: Handout 
4/23/2017  We will continue the study of the Cardano formula, based on the handout posted on 4/9.
Today we will continue our study of graph theory. 