# Combinatorics looks incomprehensible, is it that I lack "mathematical maturity" and what should I do?

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:

1. Is it normal to find it hard to solve the problems in that book?
2. 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.)
3. 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?

• I am facing the same problem. Also a bit with synthetic geometry. Thank you for asking Commented Feb 18, 2017 at 8:03
• I'm almost willing say that fluency comes with practice. No different for combinatorics. I studied combinatorics via "Invitation to discrete mathematics" by Matoušek and Nešetřil. Deceptively easy very often and then suddenly BAM, you can't figure out for the life you how to solve some problem. Demons creep into mind, sure. Must be above that, must keep trying! Commented Feb 20, 2017 at 10:47
• I am now in a very similar position to yours; I am finding Combinatorics way harder than anything I have studied so far. Commented Sep 30, 2020 at 17:08
• Math is hard, and combinatorics is highly non obvious in difficulty. Changing one word in a question is the difference between a closed form solution, a generating function as the best we can do, or no known solution at all. Exercises are also hard. It's not uncommon for harder exercises in upper level textbooks to have an expected time of a week's rumination. Last, contest math is a different game than pure math. They intersect in places, but have entirely different strategies and aesthetics. Commented Jul 18, 2021 at 21:01

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.

Is it normal to find it hard to solve the problems in that book?

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.

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.)

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)

What should I do if I still want to study combinatorics?

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!

• Thanks for your answer! You offered insight into general problem solving and reading maths texts (to which I haven't payed attention). I actually thought no one will ever notice my question... I might have looked into another book as well (in fact I've started reading Bondy & Murty, whose exercises look much more "reasonable"), but I'm just somewhat eager to find out why I'm finding a book hard while it should be easy. I still hope there will be someone who has read that book and can tell me about the specific things about combinatorics, like what exactly is it that I lack. :) Commented Feb 11, 2017 at 6:44
• I'm glad you found my answer useful! :) Unfortunately, I cannot add too much technical details to it. I hope you get assistance with that part. By the way, you would like to take a look to Miklos Bona's book, which I find really instructive and well balanced. Commented Feb 11, 2017 at 6:59
• I guess it's like chess; chess is one of the most difficult games because it's very hard for us humans to be able to think with several movements ahead. We are not good "accumulating branches of cases" in our head, which is basically the definition of combinatorics. So the only way for us to manage combinatorial problems in an intuitive way is by looking at the problem from a different angle that bypasses, at least partially, the act of combine.
– ABu
Commented Dec 1, 2022 at 18:54

In my personal experience, I find reviewing the basics on subjects in which I am advanced will often settle my frustrations and allow me to continue to higher levels. In addition to discrete mathematics, I enjoy learning languages. Through high school,it was very mechanical, not so much fun but I was always st the top of my class, a nice reward. In college, my knowledge of basics helped me adjust to the speed of learning more advanced lessons. Not until I became a teaching assistant, did the world of language truly open up! Joy,understanding and confidence now we the reward of my toil!

As a high school student, you may yet to have applied combinatorics in personal setting. Additionally, the basics must become rote. Perhaps you've never had the opportunity to explain what you do understand to a student of math with a genuine interest or need to know. Try this.(maybe your old Gifted Teacher from junior high has a few promising student to whom you could present Pascal's Triangle). Again, when I became a TA, in grad school, I was showing students how to apply basic combinatorics in an economic model. And again, my challenge to present the basics allowed my pursuits to rise.

One person's road to increasing joy and understanding of a subject, for what's is worth...