112 Combinatorial Problems
From the AwesomeMath Summer Program

I was looking for a combinatorics book that would discuss topics that often appear in math olympiads, a test that this book passed with flying colors. It provides a clear and thorough solution to every problem. The induction chapter was especially helpful, giving a clear and concise explanation to both normal induction and strong induction. I quickly got the hang of it after reading that chapter. The invariants chapter was also quite nice; it is usually quite hard to spot an invariant, but the examples provided in that chapter were well-motivated.

As someone who wanted to improve my skills in combinatorics from pre-Olympiad to Olympiad level, I thought that this book suited that purpose very well. In particular, I really enjoyed the section on counting in more than one way. Although the idea presented was rather simple in nature, I loved reading through the examples, as they were very nonroutine and helped see the technique in a more general light. For instance, the example on proving a identity involving the floor function was very enlightening, as I had never really been exposed to the combinatorial idea of counting lattice points, which made reading the solution very informative; I thought it was extremely elegant, and it added another idea to my combinatorics toolbox. I also liked the section on invariants, as it explained in detail the motivation behind the technique and how it is applied in combinatorics problems involving games and algorithms. Often, I would read solutions to these types of problems in other resources and have no idea how I would come up with those solutions myself. However, I believe this chapter definitely provided enough explanation on how such elegant solutions can be found on one's own, which I appreciated. Lastly, I really liked the section on combinatorial geometry. When preparing for competitions, I would often get stuck on problems that combine the ideas of geometry and combinatorics, but I never really found appropriate resources that covered the ideas behind combinatorial geometry in an informative and intuitive way. This section was what I was looking for, as it outlined the basic ideas and techniques at the beginning, and then went straight to worked examples with detailed and motivated solutions, which I found to be very helpful and effective. I thought that the discussion on convexity was very helpful for the purposes of the chapter, as I never really got an idea of what convexity was until I read that part. Enough theory behind convexity was provided, and that helped me get an idea of what was going on in the solutions to the examples provided. Furthermore, I thought that the examples presented in this section were sufficiently challenging and also interesting and nonroutine. It was truly a joy reading through the solutions to these examples, as they methodically dissected the problem and showed the power of the math discussed by applying the ideas in very beautiful ways. Overall, I think that this book was a very enjoyable read, and the experience of working through the problems at the end was extremely rewarding, and the solutions to them were written with the right audience in mind. The problems at the end incorporated the right balance between computational problems and proofs, and I found that working through them at my own pace helped me bridge the gap between pre-Olympiad combinatorics and Olympiad combinatorics.

This book uses lots of neat ways to explain ideas and concepts clearly in a way that is understandable by a lot of people. The book also features many alternate solutions that are worth learning especially when they save you time such as solutions utilizing complementary counting. The book also explains when and when not to use certain concepts sometimes using visual aids like tables. When explaining why a certain method works this book doesn't skip over details as much as the other books may do like when proving the binomial theorem. Also, the introductory problems that come later in Chapter 15 are a very good representation of what you would likely see on math tests on the harder end of the spectrum because most of them are varied in interesting ways that sometimes require a bit of creativity. If I have any doubt or I am really confused about how to finish a problem the solutions at the end are really straightforward and I often find myself thinking why I did not realize this or that could be done.

The book, 112 Combinatorial Problems - From AMSP, is a great book that focuses on problem solving. In each chapter, the book provides the important formulas and follows up with many examples to support the content. One chapter that was particularly helpful was Induction, as it not only stated the basic idea, it went into great detail explaining the validity of each step, as well as the precautions that need to be taken when using Induction. The examples start with a very simple question that clearly shows the steps of Induction, and the problems gradually become more challenging and unique, branching into sets, combinatorial games, as well as into strong induction to prove harder formulas. The formula itself is useful in many proofs, but the detailed explanations behind it reduced many errors in my solutions.

"I am a grade 7 student. When I was selected for AMSP the last year, I decided to take a course for Math Counts with Proofs. This book was very helpful in making me understand basic concepts to get ready for the summer program. Book had really good theory to explain concepts and lots of problem to practice. Thanks to this book I really enjoyed AMSP and was became the best course I have ever taken.

I still need to finish harder problems from the book. It's undoubtedly the best book for combinatorics for every student who aims to compete. "

This book provides me with beneficial knowledge and methods of combinatorial problems. Lessons from this book are collected from various sources and help me to get detailed and sufficient math problems which I and my students can use to broaden their knowledge. I was very impressed at chapter 14: Probabilities and Probabilistic Method. This chapter surprises me by a lot of new and interesting things that I have never discovered before, its content is rarely mentioned in our math program.

"This book is truly amazing. My favorite chapters are combinatorial geometry and probabilistic method which is quite difficult to understand by reading other book. Combinatorial geometry chapter provides me a lot of essential ideas of this problem genre from the basic to challenging problem and also gives my students the advantage to solve modern combinatorics problem which tends to be combinatorial geometry. For the probabilistic methods, it provides fundamental knowledge of such method, new aspects to the combinatorics problem and new tools to solve combinatorial problems. It is quite difficult to find such a book that cover this topic and this book suit me in that point."

It is a great book to learn the Combinatorics, from basic to advanced knowledge. It covers many combinatorial topics that would be tested in AMC or AIME. The theorems are explained very well and the examples match the topics closely. It greatly improve my understanding of combinatorics.

In 112 Combinatorial Problems, you learn the basics of Counting and Probability, taught at a beginner level. You start with learning ways to solve computational problems, then go on to learning about proofs, required in Olympiad. I used to be bad at this subject, but after reading this book, I learnt much more in this subject. I'd recommend this book to anybody starting competition math, as all you need as a prerequisite is knowledge of basic arithmetic.