9/27/2009  We will get to know each other and go over the written assignment. Then we will solve several fun problems on dividing some number of objects between two people, or sharing some number of objects. Handouts: Sharing and Dividing We will work on a variety of entertaining problems, brainteasers and puzzles. Handouts: Mathematical Potpourri  Pictures Fermat's Last Theorem has been baffling and intriguing mathematicians for over 350 years. We are going to trace the work of some of the amazing men and women who worked on this problem, and even prove the Theorem in a few cases ourselves! Handouts: Fermat's Last Theorem  10/4/2009  We will solve a lot of interesting problems dealing with balance scale. Handouts: Problems with balance scales We will do more problems on coloring maps as well as a variety of other topics. Handouts: Problem set We will discuss the continued fraction expansion
and talk a little bit about the golden ratio and
its occurrence in arts and nature. Then we will calculate the
continued fraction expansion of roots of small integers
and discover some interesting structure in these expansions.
This will lead to the understanding of a beautiful theorem
named after Fermat but proven by Euler, that characterizes
the primes that can be written as the sum of two squares. Handouts: Continued Fractions and a theorem by Fermat  10/11/2009  You have a balance scale and a lot of weights. All the weights are powers of two (that is, they represent numbers 1, 2, 4, 8, 16, 32, 64, ..., expressed in grams). You have just one copy of each of these weights. Using just these weights, can you balance any object weighing the whole number of grams on the balance scale? We will find out! Handouts: Weighing with Powers of 2 We will continue with the handout from last time and will solve problems on a variety of topics. Handouts: Mike\'s group handout  Carey\'s group handout  Carey\'s group handout  Carey\'s group handout We will continue with the topic started last time.  10/18/2009  We will continue exploring binary notation for numbers concentrating on the analogies with the decimal system this time.
Handouts: Binary (continuation) Handouts: Combinatorics homework (Carey)  Logic arrows (Carey)  Combinatorics (Carey) Going from real numbers (ordinary numbers) to complex numbers is like coming out of a tunnel. You can see much more of the mathematical landscape than you thought possible. Even some properties of real numbers that were mysterious before become clearer. This session will be an introduction to complex numbers, their basic properties, and some things you can do with them. In future sessions we'll discuss more applications, ranging from number theory to cell phones. Handouts: Unit circle  Complex plane  Geoboard  10/25/2009  We will solve a variety of problems involving inverse operations and backwards reasoning. Handouts: Backwards reasoning Handouts: NumberTheoryProblems This is the second part of the meeting (from 3 p.m. to 5 p.m.).
We will start reviewing the material for AMC 10 and AMC 12.
This is the first part of the meeting (23 p.m.): Some thoughts on using math and science thinking and math and science knowledge far outside math and the sciences, from Eugene Volokh, who?s a professor at UCLA School of Law. Eugene started as a math buff, shifted to computer programming, and eventually turned to law as well as popular writing about the law (he?s the founder of The Volokh Conspiracy weblog, http://volokh.com). Before going into teaching, he clerked for Justice Sandra Day O?Connor at the U.S. Supreme Court.
 11/1/2009  We will be working on a variety of problems, including backwards reasoning, binary/decimal, and elementary algebra, to solve the mystery of the missing candy! Handouts: The Mystery of the Missing Candy This week students will get more practice with doing calculations mod n, and using modular arithmetic in solving divisibility problems. We will continue going over examples of the addition and multiplication principles. Then we will learn about Venn diagrams and double counting in order to begin counting more complex sets.  11/8/2009  We will be building squares and cubes and examining some of their properties. Handouts: Squares and Cubes  11/15/2009  We will make several models of simple 3d solid bodies. We will use paper and glue for some models, and clay and toothpicks for the rest. Handouts: Making models of 3d solid bodies In Carey's group, we will go over the homework, and begin applying what we have learned so far to counting with repetitions and the idea of a "combination." Attached are last week's handouts and the homework. In Mike's group, we will review/introduce the notions of least common multiple and greatest common divisor. Euclid's method, and various realizations of gcd's will be covered. Handouts: Homework  InClass Problems We will begin with a review of basic number theory/abstract
algebra, discussing the "integers modulo N." We will move on to
discussions of how to determine if a number is a "quadratic residue"
modulo N, introducing the Legendre and Jacobi symbols. We'll conclude
by using these tools to build an encryption scheme, which we'll play
around with at the end. Handouts: Problem Set  Handout  11/22/2009  Polyhedra are three dimensional shapes that have vertices, edges and faces. We will use the models we have built last time as well as other examples to figure out if there is a relationship between the numbers of vertices, edges and faces for polyhedra. Handouts: Euler's formula In Mike's group, we will continue our study of greatest common divisors and least common multiples, and their application to problems in modular arithmetic and remainders.
In Carey's group, we look more at combinations and several cases with choosing 2 or 3 objects from a larger set. We also do a colored picross where different colors do not necessarily need to be separated by squares.
Handouts: Colored Picross  Combinations Problems  11/29/2009   12/6/2009  We will be doing holidaythemed problems dealing with Euler's formula, Gauss's formula for summing up integers 1 to n, and other topics we have covered this year. Handouts: Holiday Math Many surprises arise as one tries to apply the familiar notions of size of sets to infinite sets, or compare the sizes of two different infinite sets. We will explore some of these surprises in a series of classic examples.  1/9/2010  In Mike's group, we will begin a series of sessions loosely centered around geometry.  1/10/2010  Pirates are searching for buried treasure on Treasure Island and encounter many math problems along the way, including logic problems, magic squares, and games. Handouts: The Treasure Island Mike (6221): Proof Techniques in Number Theory
Clint (6201): Counting with Combinations Handouts: Clint's group handout  Mike's group handout We will see how to glue surfaces out of polygons and learn about distinguishing properties of various surfaces.  1/12/2010   1/16/2010  We will take 75 minutes to solve a Math Kangaroo contest from one of the previous years.
Note: class will meet 23:15 in MS 6627 (both groups).
 1/17/2010  We will be working on basic logic, including the negation of statements and finding counterexamples. Handouts: Meeting Mr. No and drawing conclusions In Clint's group, we will continue with the handout from last time. In Mike's group, we will continue to study linear congruences, and discuss proofs. Handouts: Clint's group handout  Mike's group handout This is a continuation of the previous meeting.  1/24/2010  We will solve a series of problems about math circle students who take various classes, travel to different places and play various sports, as well as some logic puzzles. Handouts: Venn Diagrams Clint: We will begin looking at topics in number theory, starting this week with parity. Mike: We will discuss how to prove some statements in number theory, building on our discussion of logical propositions from last time. Handouts: Clint's group parity handout Handouts: Trigonometry  Logarithms  1/31/2010  We will work on the problems that can be solved using a simple invariant (a notion that we will introduce), as well as discuss several twoplayer games. Handouts: Invariants and Games Mike: We will continue to practice formal proofs in number theory. Clint: We will continue our study of parity. (Note that the handout below is different from previous week's.) Handouts: Mike's group handout  Clint's group handout We will solve a variety of problems on Graphs and Colorings  2/7/2010  We will be playing games with coins, a chessboard, and a binary card trick! Handouts: More fun and games! Clint, 6201: Our group has been studying parity, or divisibility by 2. This week we'll enlarge our focus and start looking at divisibility in general. Mike, 6221: We will go over the worksheet "Welldefinition of addition modulo n", and continue our study of modular arithmetic from a rigorous logical perspective. Handouts: Clint's group handout  2/14/2010  We will be solving some fun Math Kangaroo problems. Handouts: Math Kangaroo Practice In Clint's group we'll continue divisibility, with special attention to the role of prime numbers. As Mike is out of town Olga Radko will lead Mike's group. See last week's attachment for the handout "Welldefinition of multiplication modulo n". Handouts: Clint's group handout  2/21/2010  We will be examining two basic mathematical operations: rotations and translations. We will also be discussing symmetry. Handouts: Rotations and Translations In Clint's group, we'll see that there are infinitely many primes and use prime factorizations to solve a variety of problems. In Mike's group, we'll try to apply some ideas we've learned about modular arithmetic to solve a variety of problems. Handouts: Mike's group warmup  Mike's group handout I  Mike's group handout II  Clint's group handout  2/28/2010  We will be examining compositions of translations, reflections, and rotations. We will also study symmetry with respect to a point and symmetry with respect to a line. Handouts: Rigid motions of the plane Clint, 6201: We tie up some loose ends from previous sessions on primes and divisibility, and also try our hand at some problems involving measurements. Mike, 6221: This week in Mike's group we will investigate writing numbers in binary (base 2) notation, as well as other number bases. Handouts: Mike's Group Handout  Clint's group handout Handouts: Maps, areas, and kissing numbers  3/7/2010  We will be learning about implications, converses, and contrapositives, as well as doing some fun problems with reflections and mirrors. Handouts: Logic and Mirror Problems Clint, 6201: In Clint's group, we'll look at problems involving the GCD and LCM of numbers. Mike, 6221: In Mike's group, we'll continue to practice working with binary numbers by solving problems and playing Nim! Handouts: Clint's group handout We will discuss various questions typically of the form:
What is the shortest path with prescribed properties? This will lead
us to some consequences in optics. We will end with a discussion of
the isoperimetric inequality.  3/14/2010  We will be working on a Math Kangaroo test from a previous year. Handouts: Math Kangaroo This week Clint's and Mike's groups will combine for a team problem solving contest called Relays!
We will meet in our usual rooms (Clint's group in MS 6201, Mike's in MS 6221) to organize before moving to the Graduate Lounge (MS 6620) for the competition. Students will work in small groups on a series of fun problems ranging from divisibility and modular arithmetic to estimation and combinatorial games. We will discuss the three common coordinate systems associated with a triangle (trilinear, tripolar and barycentric coordinates) and explore some of their applications.  4/4/2010  We will look at maps of "Insect Countries" consisting of cities and tunnels and explore their properties. (This is a first glimpse into the basic graph theory). Handouts: Life in an Insect World Mike, 6221: In Mike's group we will finish our discussion of 3pile Nim, and other games.
Clint, 6201: Clint's group will examine the Pigeonhole Principle and see how it applies to a range of problems. Handouts: Pigeonhole Principle  Pigeonhole Principle Solutions  Nim Handout In combinatorics, we are not only concerned with the study of
combinatorial objects (such as graphs, permutations, partitions, and
the like); we are also interested in how we can apply methods from
other areas of mathematics to help us understand these objects. In
this lecture, I will present one of the most common ways of applying
algebra (and some calculus) to combinatorics: the generating function.
A generating function is a way of encoding a sequence into a
polynomial. With generating functions, we can use the algebraic
operations of polynomials to greatly simplify calculations and (in
some cases) prove marvelous identities.  4/11/2010  We will continue looking at Insect countries (consisting of several cities some of which are connected by tunnels). This time, we will decide what's the best way to build railroads (in addition to tunnels) in the most economical ways.
Handouts: Railroads and Trees Clint's group, MS 6201: We will turn to graph theory and, in particular, look at a number of problems whose solution can be found using trees. Mike's group, MS 6221: This week we will play more combinatorial games! Handouts: Trees and Trees  Trees and Trees Solutions  Game Theory In the classic book ``Alice in Wonderland'' many strange things happen that are left unexplained by the mathematician author Lewis Carroll. Similarly, in this math circle session at UCLA, reflections will ``mystically'' become rotations, rotations will turn into translations, and translations will transform into reflections! Is this possible and mathematically sound? Come to this talk to find out what happened just a month ago at the Bay Area Math Olympiad and how three different brilliant solutions to the same geometry problem were created by student participants.  4/18/2010  We will be making paths and circuits around graphs, as well as understanding the ideas of an Euler Path and Euler Circuit. Handouts: Circuits and Paths Mike's Group, MS 6201: This week we will take a look at some other mathematical games.
Clint's Group, MS 6221: This week we look at problems that can be solved by thinking about graphs. (Last week we looked at trees, a special kind of graph with no cycles.) Handouts: Graph Theory 1  Graph Theory I Solutions  Games Groups are algebraic structures that are used, for example, to
study symmetries of geometric objects, the invariance of laws of nature,
conservation laws, roots of polynomials, combinatorial counting problems
and many other questions. We are going to take a look at examples of such
structures taken from those various applications.  4/25/2010  We will be finding the chromatic number of graphs and also testing the 4 color theorem. Handouts: Graph Coloring Mike's Group, MS 6221:This week we will study proofs by mathematical induction, and look at some games from the perspective of induction.
Clint's Group, MS 6201: We'll continue our study of graphs, looking at properties such as spanning trees, connectedness, and planarity. (See next week, 05/02/10, for handout and solutions.) Handouts: Induction Groups are algebraic structures that are used, for example, to
study symmetries of geometric objects, the invariance of laws of nature,
conservation laws, roots of polynomials, combinatorial counting problems
and many other questions. We are going to take a look at examples of such
structures taken from those various applications.  5/2/2010  We will be discussing the expected number of heads or tails after tossing a coin and calculating probabilities of certain numbers on dice. Handouts: Probability Clint's group, MS 6201: We will continue our study of some of the properties of graphs begun last week. Mike's group, MS 6221: We will study some simple examples of proofs by induction, moving on to more advanced problems if we have time. Handouts: Graph Theory 2  Graph Theory 2 Solutions  Induction Problems This is the first in a series of 2 meetings.
Mathematical probability emerged from the study of gambling, statistics, and the observed outcomes of experiments that are subject to some ?random? external in fluences. Here we will introduce some of its basic concepts:
1. Sample space and events; 2. Probability functions; 3. Random variables and expectation.
Handouts: Prelude to Probability (read before the meeting)  5/9/2010  We will continue examining probabilities with coins and dice, as well as understanding some elementary counting principles. Handouts: Probability and Reducing Fractions Clint's group, MS 6201: We do a little more graph theory, then switch gears and look at some problems that can be solved using logical reasoning. Mike's group, MS 6221: This week we will try to finish as many of the induction problems as possible from the last two weeks. Handouts: Logic Puzzles Handout  Graph Theory 3  Graph Theory 3 Solutions  Induction Problems We will use the background in probability from last time to explore Random Walks.
 5/16/2010  We will be doing some elementary counting problems, including combinations and permutations. Handouts: Multiplication Principle Clint's group, MS 6201: Clint is out of town, so assistants Alyssa and Liz will lead the session this week. We'll look at binary and other non10 bases, and play some Nim. Mike's group, MS 6221: This week we will take a look at some problems in Graph Theory. Handouts: Graph Theory Problems The Gaussian integers are a pretty set of numbers in the complex plane. Their properties resemble properties of the ordinary integers, but even better, they help to explain some properties of the ordinary integers. We'll discuss what the Gaussian integers are and why they work the way they do. Knowledge of the complex numbers is not assumed; we'll review what's needed. (For those who are interested, some notes from Math Circle sessions on complex numbers from Fall 2009 are at http://www.math.ucla.edu/~baker/circle/.)  5/23/2010  We will be reexamining reflections and rotations, but this time from the perspective of a permutation. We will also examine compositions of reflections and rotations and their commutativity. Handouts: Handout Clint's group, MS 6201: We'll continue our study of the property of alternative bases, esp. binary, and its application to the game of Nim. Mike's group, MS 6221: This week we will continue to study properties of graphs, particularly planar graphs. Handouts: Nim and Ppositions  Graph Formulas Geometers have been interested in the symmetry and aesthetic beauty of regular polygons and regular polyhedra since antiquity. Ancient Greeks even associated their 5 classical elements to the 5 convex regular polyhedra in 3 dimensions. In these two talks, we will use complex numbers and its close cousin, the quaternions, to study the symmetries of these beautiful objects and see how their symmetries can have an impact in our life  5/30/2010  We will continue studying geometric transformations, this time of a square. We will also find what the inverses are of these transformations, as well as which transformations commute. Handouts: Commutativity and Inverses Clint's group, MS 6201: We will conclude our study of Nim strategy and turn an eye to the strategy behind a number of other mathematical games. Mike's group, MS 6221: We will continue with the worksheet from last week (see last week on the Math Circle calendar), and discuss map coloring theorems (the six color theorem and possibly the five color theorem). Handouts: Graph Formulas This week, we will continue from last week by looking at symmetries of regular polyhedra and describing them using quaternions  6/6/2010  The format will be a little different this week. The group instructor will ask questions about topics that have been covered this year, and students who answer will get a prize! Handouts: Questions on topics this year Our last meeting of the Spring quarter will be a fun team problemsolving competition! We will break into small teams and have a friendly Math Relays competition. 
