UCLA Olga Radko Endowed Math Circle

ORMC Meetings Archive • Fall 2007–Spring 2024

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For 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 2020–2021 2021–2022 2022–2023 2023–2024
Mathematical celebration:
  • group B will be engaged in a Math Battle ;
  • group A will have a Math Relays contest;
  • Junior circle will enjoy playing several mathematical games.
    These events will be followed by the awards ceremony for all the wonderful things the participants have done throughout the year and a party!
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
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
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.
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
We will solve a variety of problems involving inverse operations and backwards reasoning.
Handouts: Backwards reasoning
Handouts: NumberTheoryProblems
This is the first part of the meeting (2-3 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.
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.
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.
We will be building squares and cubes and examining some of their properties.
Handouts: Squares and Cubes
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 | In-Class 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
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
We will be doing holiday-themed 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.
In Mike's group, we will begin a series of sessions loosely centered around geometry.
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.
We will take 75 minutes to solve a Math Kangaroo contest from one of the previous years. Note: class will meet 2-3:15 in MS 6627 (both groups).
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.
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
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 two-player 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
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 "Well-definition of addition modulo n", and continue our study of modular arithmetic from a rigorous logical perspective.
Handouts: Clint's group handout
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 "Well-definition of multiplication modulo n".
Handouts: Clint's group handout
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
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
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.
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.
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 3-pile 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.
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.
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.
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.
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)
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.
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 non-10 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/.)
We will be re-examining 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 P-positions | 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
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