The re-enrollment test will only cover the topics we went over this quarter: games, successive differences, graph theory, geometry, combinatorics. Attendance is mandatory.
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.
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.
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.
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?
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.
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.
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.
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 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.
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.
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.
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.
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?
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.
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!
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.
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.
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.
