Los Angeles Math Circle

LAMC Meetings Archive // Fall 2007 - Spring 2017

For the current schedule, visit the Circle Calendar

Fall 2007 - Spring 2008Fall 2008 - Spring 2009Fall 2009 - Spring 2010Fall 2010 - Spring 2011Fall 2011 - Spring 2012Fall 2012 - Spring 2013Fall 2013 - Spring 2014Fall 2014 - Spring 2015Fall 2015 - Spring 2016Fall 2016 - Spring 2017
We will start with a few beautiful warm-up 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
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.

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
We will continue the study of the 10/2 handout.
Handouts: handout
Handouts: Blank Worksheet (same as the previous week) | Answer Key (same as the previous week)
Today we will mix techniques from physics, geometry, and algebra to solve classical geometry problems.
Animations shown in class and extras for those interested in further investigation of the topic:
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
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
We will discuss planar graphs, Euler characteristic, and related topics.
This week, we delve into hexaflexagons: flat objects that actually have three sides.
Handouts: Blank Worksheet
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.
Dan Tsan will give us a lecture based on his award-winning MathMovesU presentation for the first hour. Then we will study planar graphs, Kuratowski Theorem and Euler characteristic of a graph.
With Andrew out of town, Taylor Womack will be presenting a lesson on a new multiplication algorithm.
Handouts: Answers | Blank Worksheet

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 built-in 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 #4 | Solutions #4
Handouts: Handout | Solutions
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 Math-Olympiad-style problems to solve.
Handouts: handout
Handouts: Handout #5
We go over problems of geometric probability and some problems that use coordinate geometry.
Handouts: Handout | Solutions
This week we covered various halloween-theremed math problems.
Handouts: Blank Worksheet | Answers
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 will finish studying the 10/30 handout. If time permits, we will use graph theory to solve the Instant Insanity puzzle.
Guest instructor Chris Ohrt leads a session on the geometry of origami!
We will be exploring fake coins problems as well some additional puzzles
Handouts: Handout | Solutions
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
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.
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 orbit-stabilizer lemma.
Handouts: Handout | Solutions
Today we have a special guest speaker: Jeremy Brightbill, a PhD student at UCLA. He will give a talk on number fields.
We will attempt to finish the 10/30 handout. If/when finished, we will study the final handout of the Intro to Graphs mini-course, Introduction to Ramsey Theory.
Handouts: handout
Following up on last week, we'll take a look at what sorts of things that Origami can do than regular ruler-and-compass cannot.
Handouts: Handout
We will wrap up our study of symmetry by using the Orbit-Stabilizer lemma to obtain Burnside's lemma and look at the questions we can solve.
Handouts: Handout | Solutions
Handouts: Handout | Solutions
We will discuss problem 9 of the 10/30 handout for warm-up and then get back to studying graph theory.
Today we will get back to our study of mathematical induction, and hopefully we'll get to start talking about strong induction.
Handouts: Blank Worksheet
We will be doing math dominoes this week on the topics we have learned this quarter.
We will finish out the quarter as usual with a mathematical relay.
We will resume studying the 10/30 handout at problem 12. Once finished, we will begin studying the next, and final, handout of the mini-course.
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 resume at Problem 14 of the 10/30/2016 handout and proceed to Ramsey theory if time permits.
We will start the year by studying finite state automata.
Handouts: Handout
In this week, we cover a two dimensional game of Leap Frog and what victory positions are possible.
Handouts: Blank worksheet | Answers
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
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
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.
Today we will study vectors.
Handouts: Handout | Solutions
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 study a bit of Ramsey theory during the first hour of the classes.
We now dive deep into taxicab gemoetry after a thorough review of coordinates and a short introduction last week.
Handouts: Blank Worksheet | Answer Sheet
Today we will learn the basics of group theory by discussing symmetry and playing sodoku.
Handouts: Handout
Handouts: Handout | Solutions
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
Today we will continue our study of group theory and work with groups of symmetries of objects (including the Rubik's cube).
Handouts: Handout
Handouts: Handout | Solutions
The students will be given a two-hour 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 much-needed feedback. The top five performers will get great math books as prizes!
Handouts: test
Handouts: Handout | Solutions
This week we do our first round of practice problems for Math Kangaroo.
Handouts: Blank Worksheet
We build on our study of generating functions, learning to understand sequences generated by recurrence relations.
We will study the theory of quadratic equations and solve a few surprisingly hard problems on the topic.
Handouts: handout
Handouts: Handout
We begin our study of theoretical computer science looking at the basics of computational complexity and the runtime of sorting algorithms.
Today we will review complex numbers, this time using our knowledge of vectors as a tool.
Handouts: Handout
Handouts: Handout | Solutions
We will continue stidying the 2/12 handout.
We will continue our study of complex numbers.
Handouts: Handout | Solutions
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
This week we will study combinatorics from our Berkeley Math Circle books.
Handouts: Handout
We discuss the relationship between probability, binomial coefficients and combinatorics.
Handouts: Handout | Solutions
We will resume studying the 2/17 handout at Problem 13.
Non-determinacy; learn the statement of one of the great open problems, P vs. NP
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
Suppose we break a stick into three pieces randomly. What are the chances the resulting pieces will form a triangle?
Handouts: Handout | Solutions
How many guards does it take to watch an art gallery? It depends on the shape of the gallery!
Today we will solve some problems from classical Euclidean geometry.
Handouts: Handout
Handouts: Problem Solving
We will start with refreshing problem 18 from the 2/12 handout. http://www.math.ucla.edu/~radko/circles/lib/data/Handout-1272-1283.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
We went over the Math Kangaroo test.
Handouts: Math Kangaroo 2017
We will see how the Vieta formulas for quadratic equations enable one to solve qubic equations as well.
Handouts: handout
Welcome back for the spring quarter! We're starting off with a class about polyhedra, and their curvature.
Handouts: Handout | Solutions
Today we will study Propositional Logic.
Handouts: Handout
We practice our proof-writing techniques
Handouts: Handout
We started the quarter off checking what level of mathematics every student is at to better calibrate the rest of the quarter.
Handouts: Diagnostic Packet
We will review the derivation of the Cardano formula, then learn long division of polynomials, and then solve some cubic equations.The students finished with the current handout will be given a set of very hard geometry problems.
Handouts: geometry handout
This week, we willl continue learning about functions and discuss one-to-one and onto functions. In the second hour, we will learn how to write good proofs for harder problems.
Handouts: Handout
This week, we'll be studying the theory of vector fields on surfaces!
Handouts: Handout | Solutions
Today we will start a several week unit on graph theory, using the Instant Insanity puzzle as our motivation.
We study parity of numbers (odd or evenness) through chess pieces, checker boards, and geometric construction.
Handouts: Blank Worksheet | Solutions
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.
How do we determine the winner of an election? There are lots of answers, and there are certain properties we'd want those answers to satisfy. We'll prove that with just two of the most straightforward such properties, the only "fair" electoral system isn't very fair at all!
We define what it means for a set to be infinite.
Handouts: Handout
We delve deeper into parity with word problems that are quite tricky if you do not view them through a parity focused lens.
Handouts: Blank Worksheet | Solutions
Professor Sami Assaf from USC joins us to talk about two magic tricks that use nothing but math!
We will see how symmetries of an equilateral triangle act on roots of cubic equations. 
Handouts: handout
Handouts: Handout | Solutions
We discuss the cardinalities of countable and uncountable infinite sets and prove that the set of real numbers is bigger than the set of natural numbers.
Handouts: Handout
We use hinged Reflect-It mirrors to study the geometry of light.
Handouts: Blank Worksheet | Solutions
Professor Sami Assaf is back with two more mathematical magic tricks, including a surprising result about shuffling a deck!
Handouts: Handout | Solutions
We finish dicussing the handout from last week.
We continue our study of proper angles that make good kaleidoscopes, and then use our Relfect it Mirrors to discover the axes of symmetry that can be found in road signs while doing everyday driving!
Handouts: Blank Worksheet (same as previous week) | Blank Worksheet (Road Sign Symmetry) | Solutions (Kaleidoscopes) | Solutions (Road Sign Symmetry)
This week, we'll be meeting the hyperreal numbers, a formal way to do math with infinitely big and small quantities.
We will be revisiting the algebra curriculum of a function and introduce the idea of a function between general sets. In addition, we will define the notions of one-to-one and onto for a function. Then, we will look at a particular type of functions between n ordered objects known as permutations. We will then look at some enumeration results on permutations including the hat matching Combinatorics problem.

Handouts: handout
Handouts: Handout
We began our journey into understanding functions and how they connect the elements between sets. We took a very abstract approach to understanding functions, and did not just restrict our sets to those of numbers.
Handouts: Blank Worksheet | Solutions
We continue our exploration of the Hyperreals, using them to do parts of calculus with ease!
We'll continue our study of graph theory. This time we'll learn about the Euler characteristic of planar graphs and show that some graphs are not planar.
Handouts: handout
Handouts: Handout
We extend our study of functions to understand the size of infinite sets, such as the natural numbers, integers, and rationals.
Handouts: Blank Worksheet | Solutions
We'll discuss Einstein's theory of special relativity!
The students not finished with the Galois theory handout (4/23 posted on 4/30) will work on it with Oleg. The students finished with the Galois theory handout will work on the second Functions and Permutations handout (posted on 5/21) with Jason.
We wrap up the new topics of the quarter with an extremely beloved subject by all of the students, logic puzzles!
Handouts: Blank Worksheet
We will meet at 3:30 PM on the lawn next to the 5th floor vending machines and take pictures of the class. If you have a fancy camera, and know how to use it, please bring it with you. If you would be willing to share your pictures, please email them to Oleg. We will come to our classromm at 4:00 PM and work on finishing the handouts. If there remains some time, we will have a "bring your own problem" problem solving session. For that, please email a problem that you find super-cool, with a solution, to Oleg no later than Thursday, 6/8.
We finish the quarter with a fun competition, Math Dominoes!
Handouts: Math Dominoes