# Our Lessons

The math camp students get two three-hour lecture sessions everyday. Each week, the morning and afternoon sessions each have their separate “themes”. Here’s the lineup:

### Week One

##### Morning

Basic Mathematical Background-Professor O’Nan
Our first lesson at math camp (and 4 more later that week) involved those things you don’t always learn in school, but are basic to any geeks toolkit anyway. Professor O’Nan introduced the logical topics of truth tables, contrapositive and negation of statements. Later lectures involved quantifiers and mathematical induction. On the last day, we had the first part of the Stage D’Euler, a competition among the groups in which puzzles had to be solved as fast as possible.

### Week One

##### Afternoon

Graph Theory-Professor Bergstrand
The afternoon sessions were on graph theory. No that's not properties of the x-intercept when the equation is such and such. It's about nice figures (called "graphs", surprisingly enough :) ) with very big vertices and edges or lines between them. They have all sorts of interesting properties, and a whole new world opened up for us. Professor Bergstrand saw to it that we were all very happy. Sorry, that's an inside joke. We learned about the different classifications of graphs, about Ramsey numbers (how many people do you need at a party to have at least x strangers or y mutually acquainted people) used in complete graphs (where every vertex is connected to all others). If you want to know all the other little things we learned, go to math camp at Rutgers this summer!

### Week Two

##### Morning

Algorithms in Graph Theory-Professor Wantland
Professor Evan Wantland was there to wake us up when we filed in after breakfast everyday during the second week, teaching us about such things as Kruskle's and Prim's algorithms (some solutions to the traveling salesman problem, or that of finding the spanning tree in a graph with the least total weight on it. Don't ask.) Other topics included Hall's Theorem (finding the perfect matching in a bipartite graph. See Kruskle's theorem if you have any questions.). Max flow and min cut were topics helping the potential engineers among us, and on the last day we got to learn how calculators work nice and efficiently with postflux, and how screwed up the human brain really is when doing arithmetic. Fun stuff.

### Week Two

##### Afternoon

Combinatorics -Professor McNulty
All of Monday morning, the group had been teased by comments about how Prof. Wantland would kick our a...butts in soccer together with Prof. McNulty. Off course, we couldn't think of anything else during lunch! :D. We sat through the oh-so-wonderful introduction on Montana, and then we got to learn more math!. Combinatorics is basically a nice word for counting. So we started out with addition, but our great intelligence helped us to quickly understand that topic and advance to permutations and combinations. Other topics included Bell's and Sterling numbers, binomial coefficients, about five hundred (all right maybe, 8) variations on the how can I put/arrange n balls in x boxes problem, and lastly ever-present Catalan, Fibonnacci, and other interesting types of numbers. We were supposed to have another competition at the end of the work, but it turned into a who-can-build-the-nicest-thing-with-blocks type of thing for about half of us.

### Week Three

##### Morning

Number Theory-Professor McNulty
During the third next week, the oh-so-exciting topic of number theory lay before us. Number theory, we learned on Monday, is very old. It seems to mainly talk about primality, but we also took a look at Fermat's and perfect numbers. Modular arithmetic is another topic in number theory. But again, the main topic of the week was prime numbers. How does one determine if a number is prime in the most efficient way? Can we find a formula for finding prime numbers? What are Mersenne primes? But most importantly, there is Goldbach's conjecture. This unsolved problem in mathematics states that any even number greater than two can be stated as the sum of two primes. Other weird questions that bothered us during the week included that of the magical witches, and the coding process Prof. McNulty must have had for them. Prof. Wantland also strolled in magically at exactly the right moment to perform some weird card trick with Prof. McNulty. How do you guess what card someone has if you only get to see four seemingly unrelated cards?

### Week Three

##### Afternoon

Robotics-Professor Coffey
The afternoon sessions of week three were more exciting than some of the other sessions, just because they were so different. Rather than staring at overhead projectors all day, every group got Lego kits. Yes, Lego! A program called Robolab helped us program our self-designed robots, which we were supposed to disguise as animals. On Friday, there was a competition, when Professor Rosenstein and others judged each robot, demonstrated at work in its environment by its builders, to determine who won what. For the 6 groups, there were 4 prizes: Most creative, Best engineered, Best overall, and everyone's favorite: The little engine that could!

### Week Four

##### Morning

Mathematical Applications – Professor O’Nan
By the fourth week, we have learned many different aspects of math. Right now, we are learning and solving many puzzles and games. It's a fun week considering we are solving puzzles and working as a group to finish the homework. This week requires a bunch of information that we learned in the previous weeks, such as various theorems and topics. An example of a puzzle would be slicing a square into various shapes to make a rectangle of a certain measurement.

### Week Four

##### Afternoon

Fractals - Professor Perciante
The afternoon of the fourth week consists of many visual math. The last week's afternoon classes are about fractals (and a bit of chaos theories). Fractals are visual images such that every part of them contains a complete down-scaled picture of the whole thing. They are often very pretty :). The professor is funny, for example calling fractals "totally now"; so our routine of 8 hours of math a day for four weeks ends on a good note.