Alexei Kadyrov talks about his online course at AwesomeMath, where he is studying Combinatorics 1.5. “This first day was focused on the basics to combinatorics: definitions of common terms like sets and subsets, ways of counting like complementary counting, rule of product, and rule of sum,” Alexei explains.
“After the lecture, I took the 10 minute break to quickly eat dinner before returning to the problem session where we were put in private groups with other students to solve three problems. We worked our way through them, returned to the main meeting, and I presented one of the problems to the class,” Alexei adds.
During the first few days, Alexei says the course continued with a lecture on combinatorial objects, talking about permutations and combinations, followed by homework, and a counting technique called stars and bars.
“Today we discussed a counting technique called stars and bars, which is the way to find the number of ways to arrange some objects, “stars,” into other objects with dividers between these objects being called “bars”. The basic type of these problems are questions like finding the number of ordered pairs (x, y, z) so x + y + z = 12 (where x, y, z are greater than 0), and the more complicated problems build on this idea to make it harder and harder to find this form. The lecture was pretty easy to follow, the examples were challenging, and the problem session was another nice finish with me and some classmates figuring out the assigned questions, and I presented one of the solutions that I found. The homework was more difficult than the past couple of days, with me only able to get one of them, and I had to go to our class’s office hours to understand the second problem,” Alexei reports.
The week continued with lessons on Pascal’s triangle, the Binomial Theorem, and more difficult homework assignments.
“Pascal’s triangle and the Binomial Theorem are, at the simplest competition level, a way to find the coefficients of expansions like (2x + 5y)10, but during the lecture we focused on proofs for different properties of the triangle and extended the concept to cover how to expand “multinomial” expressions like (a + b + c)7. This lecture was harder than the previous couple to follow as it was based on proofs, but with the help of our T.A. I was able to understand, and the problem session once again improved my confidence of understanding the topic, and I presented a solution,” Alexei says.
“The homework was very difficult, again based on proofs, and I was able to get one of the proofs right with a nice symmetrical argument, but the other one was much more difficult. The proof the instructor wanted us to do for that question involved thinking of two different ways to count the same idea, and I couldn’t think of a way to justify the way that I was using to count. Still, it was a good day,” Alexei adds.
Alexei’s week ended with a lecture on the Principle of Inclusion-Exclusion (PIE), and an exam which he completed over the weekend.
“Today the lecture covered a topic known as the Principle of Inclusion-Exclusion (PIE). This is used to count things that have overlap, saying that if you have a group A and B and they have an intersection, you can count the total number of things as the size of group A plus the size of group B and subtract the size of their intersection. This can get extended to have three groups, four groups, and so on. Then, we briefly covered derangements which is calculating the ways where things don’t happen, like the number of ways to give n homeworks to n students so that no student gets their homework back. The examples were challenging, but, despite this, I still got every question in the problem session right, and I presented one of the questions to the class later on,” Alexei says.
“To fit within the exam due date, I had to wake up pretty early to start on the test. This early wake up made me work slightly slower, but I was able to get 3/5 questions completely right, 1 question almost right except for a dumb mistake on the very last step where I used the wrong method to remove overcount, and I had two decent starts to the last question and the bonus question which gave me some points there. This gave me a 27/35 or 77%, which is 15% better than what I got on the Algebra 2.5 test last year where I got an above average grade, so I’m pretty happy with my performance. It was a very good week, and I’m looking forward to the next one,” Alexei exclaims!
Week Two — Bijection and the Pigeonhole Principle
Alexei Kadyrov’s second at AwesomeMath began with a counting technique lecture, followed by more difficult homework.
“We discussed a more difficult counting technique known as bijection, which is a description of a function if every output value has an input value and each output has only one input. This is used in problems to turn the given problem into one of a different form that can be easier to solve. This idea led to something called the Catalan numbers, which are the number of ways to find a path from one corner of an n x n square to the opposite while never going above the diagonal, and they show up a lot for some reason like Fibonacci numbers,” Alexei explains.
“The homework was pretty difficult, and I needed a starting hint on both of the questions to finish them (it was still very complicated to solve them even with the hints provided). Overall, it was a challenging class but I feel confident I understood the material after the problem session and my conversation with the T.A.,” Alexei adds.
The week continued with lessons on the Pigeonhole principle and proof by induction, followed by a problem sessions, and homework.</p”We continued the class by talking about the Pigeonhole Principle. It’s the idea of how some objects, pigeons, would be arranged in some other objects, pigeonholes. It generally comes up in proofs, but we saw some competition problems where it could be useful. The examples were difficult, but manageable, which finished off the lecture,” Alexei reports.
“Like usual in this class, the problem session felt a lot better than the lecture, and I was both able to do every problem and present one of the more difficult ones to the class. The homework was also easy, yet I somehow decided that the two digits in two numbers could, at most, sum to 10 (completely forgetting numbers like 99 which would sum to 18) so I got that problem wrong despite using the proper approach. Whoops,” Alexei adds.
The week ended with lectures on recursive sequences and probability, followed by homework and an exam on Sunday.
“The material was fairly easy as I had covered recursive sequences before, but the examples and homework made it slightly harder as it branched into combinatorics, not just algebraic recursive sequences (which I had learned last summer). The homework was challenging, but I was able to do both of the problems, with the second one being really satisfying to finally find the pattern to make the problem make sense,” Alexei reports.
“This class extended the basic idea of probability into a more combinatorial sense, making it a lot more difficult than usual. The examples were annoyingly difficult, but I got most of them on my own and all of them with some help from the TA, and the problem session made me feel better about the subject, again, where I was able to do all of the problems and present one of them to the class,” Alexei adds
The test was a lot harder than the other one, but I still performed well despite the enhanced difficulty (and, cool enough, my answer to the question involving recursion demonstrated great understanding of it),” Alexei says.
Week Three — Conditional Probability & Graph Theory
Alexei reports that during his third and final week, the class learned about conditional probability and graph theory.
“Building off of last week’s initial lecture on probability, today was conditional probability. This involves probability questions where some event B is dependent on the outcome of another event A. We talked about Baye’s rule, which is a way to calculate this, and then about additional probability concepts like random variables (assigning values to outcomes in a probability question), and expected values (frequency of the random variables assigned). At first confusing, the lecture’s examples made everything make sense, and we started the problem session. I was the only one able to solve one of the assigned questions, which I presented to the class,” Alexei explains.
Alexei says the lesson he enjoyed the best was graph theory. “It’s basically a way to define lines and points by their vertices and their edges. A pentagon, for example, would have Vertices = {A, B, C, D, E}, and Edges = {{AB},{BC},{CD},{DE},{DA}}. Graph theory, as usual in this class, extends far beyond this idea, yet I easily connected with all of the ideas being presented. Compared to everything else in this class, my strongest subject was graph theory. I participated a lot on the proofs where I contributed the main idea to a lot of them when we were prompted to. The problem session was also satisfying, where I was able to do all the problems,” Alexei reports.
As the week progressed, the class learned more about Graph Theory, Euler’s formula, graph isomorphism and generating functions.
“As if I was having a bit too much fun, today we talked about generating functions. This is the idea of creating an infinite sum of some anxn, with the an coefficients originating from some other sequence. This idea got slightly easier to understand in the examples, but I had to flip back and forth from the notes to the examples to try to understand everything that was happening. The proofs were extremely difficult to understand, as verified by my classmates when we went into the problem session, and I had to ask the TA a lot of questions to understand the material. However, the problem session went a lot better than expected,” Alexei explains.
The week ended with more lessons on generating functions, a Q&A session, and a certificate for completing the course.
“We talked about generating functions for a second day. The, already, hardest subject of the class made even more difficult. I understood most of the lecture, but the examples were extremely difficult to understand, only doable with the help of the TA. As a final day, however, instead of doing a problem session, we did a question and answer section where we could ask our instructor/TA questions about anything math/school-related to get advice. I got a lot of useful feedback from this, like best ways to study for upcoming math competitions, what careers in mathematics look like (if I wanted to go there), and general tips on problem solving, time management, and more,” Alexei says.
“Most importantly, however, was a private message I received from my TA where he stated that he and the instructor were very proud of how I did in the class, and said that they would submit a recommendation to the AwesomeMath office to let me apply as a TA for this class in the future. This was extremely awesome and made my view of AwesomeMath this summer much more enjoyable than it already had been. At the end of the class, our class said bye to each other, and I received my course completion certificate, finishing Combinatorics 1.5,” Alexei adds
We are happy to hear that you did so well in the course and were recommended to apply for a TA position for future classes. Great work, Alexei.
>> Read Alexei Kadyrov’s final report (PDF File, 30 KB)
>> Learn about the other students’ experiences in the GFF Scholarship Program.
