I think this soft question may be marked "opinion-based" or "off-topic", but I really do not know where else to get help. So please read my question before it's closed, I am really desperately in need of help… Thanks in advance for all people paying attention to this question!
Background
I am a high school student who has participated in mathematical olympiads, where there are tons of tough combinatorics problems. I already know the basic counting techniques (binomial coefficients, inclusion-exclusion principle, etc.), some notions and theorems in graph theory (trees, Ramsey numbers, etc.), yet I do not know much beyond the definitions. Accordingly, I plan to (self-)studying this subject systematically. (I actually study for fun, not for better performance in olympiads.)
Context
A few days ago I stumbled upon the book A Course in Combinatorics and started working through this book with dilligence. It was not long before I found the problems way too hard. I am just halfway in chapter 2, and there are three problems (out of a dozen or so) that I cannot solve even after I read the hints and literally thinking for hours. I am unsure about whether I should pursue reading this book but a more important question arises:
Why do I find the problems too hard? Is it that I lack the so-called "mathematical maturity" required for an undergraduate text like that, or that I am not gifted enough, or something else?
I am not sure whether I do have the slightest degree of mathematical maturity, but I have been reading mathematics texts for at least half a year. I have worked through the first seven chapters of baby Rudin (and solved all the exercises, which would look "easy" to me compared to the problems in that combinatorics book), and I am learning linear algebra from Hoffman & Kunze and abstract algebra from Herstein, and having fun reading the three volumes Analysis from Amann & Escher. Then there are other books like Folland's Real Analysis or Artin's Algebra, which I have read only several chapters. Of all the books in mathematics I have read about, the problems of this book are the hardest. However, no one has ever mentioned that the book has tough problems. (Maybe it is just me?)
By the way, I posted a problem that I literally thought for an entire afternoon here.
Questions
All in all, my question boils down to these:
- Is it normal to find it hard to solve the problems in that book?
- Can you possibly tell me why I am finding it hard? I will surely provide more details if necessary. (This question seems impossible to answer, so feel free to ignore it.)
- What should I do if I still want to study combinatorics? For example, what books are recommended for my situation? And, more importantly, how can I improve my skills in solving combinatorics problems?
Please tell me what else I should add to get better answers. Again, thank you for reading this!
Best Answer
A soft-question should receive a soft-answer. This is primarily opinion-based, but I want that if this question gets closed, you get at least one opinion.
Combinatorics have always had the reputation of having hard and interesting problems, that's why they're so used as "contest" maths. I understand you're used to solving combinatoric problems and hitting this "wall" must be hard, but please keep in mind the following:
I would disagree if somebody says that book is intended to be solved completely (I mean, its exercises). Many of its problems (as in many other books) will be probably questions that came to the author's mind and never were answered, or problems with a really, really, REALLY brilliant idea that only comes to the mind of the person who actually invented/discovered the problem. I'm not saying that the problems you're leaving unsolved are of this kind, but trying to solve all of them may be hard! It's not only about learning the concepts in the chapter, it's also about getting use to the notation, concepts, practices, and so on. Hence I recommend you to leave these exercises for now (without feeling guilty for doing that!) and come back to them.
This book has a very different level than "contest" maths. Of course, it is intended for a deeper study of combinatorics, which often can be heavy and overwhelming. It is great you have studied many other math books, but please take combinatorics very carefully, since it differs from analysis and other branches in many aspects. For instance, creativity and a good memory are often required in combinatorics more than in analysis or algebra (I feel guilty by saying the latter!), and many other practices are very different! but I'm sure you probably know this already. Just make sure you understand you're working with something new rather than the same kind of things you were working on before.
By hearing your background, IMHO it's just a matter of practice and time. Mathematics can be deceiving, one day you'll be in love with them, feeling like you can master them, and the other day you feel like the most stupid person in the world. Just be modest, and understand that Maths will always be one step ahead of you, and that with some effort you will reach that step someday (and then it will move ahead again... yes, like the infinite Hilbert hotel... these are maths at the end)
Why are you throwing it to the ground already? I already mentioned my opinion: you should keep studying, omitting (by now) what you think is taking too much time from you. I know this is hard, but this practice will become handy some time in your (math) life! Please do not leave the topic, it seems it really encourages you and you still can learn a lot from it!