Philosophy of learning mathematics (combinatorics)

How does one learn mathematics in depth and well? I think I may be studying mathematics incorrectly, or I could be doing it a lot more efficiently/effectively. Sometimes I feel like I need every argument repeated to me at least three times (this can be exhausting and it takes a lot of time when I have so much to learn). I think I am learning which areas of mathematics that I like best and am starting to partition my time to focus on the areas that I enjoy the most. With that in mind,

1. How can I become proficient in Combinatorics? Most of what I hear and have read from other people is to do a lot of problems. I am looking for a magical set of problems that will make me a lot better at combinatorics (I am aware that this is unrealistic). Can someone give me some references though or some sort of reading list?
2. Is there a common book that a serious student in combinatorics should have? I am planning to listen to the advice of doing a lot of problems and going through an entire book. With that in mind, does anyone know of a combinatorics book which is well written (details of combinatorial arguments are spelled out clearly one step at a time).
3. Lastly, how should I study combinatorics? For example, drawing out pictures is always helpful, but when should I enumerate, look for inductive patterns and what should I do when I am stuck? I feel like I waste time looking for silly patterns rather than stepping back and looking at the big picture. Sometimes, I have trouble finding the big picture, but once I see it, then the problem is easy. Combinatorics seems to be more of a problem-solving field than a deep theory, how does this impact study?

All comments and advice are welcome. I am hoping to learn from your experiences so I can become a better mathematician.

Solution 1:

sometimes I get frustrated by this too, I might have learned something and after not using it a lot I forget it. One thing that I think is very important is patience: you are not going to become a master in anything after a year.

A problem that I face a lot is that if I don't use something I forget it, I try to solve this by using a combination of techniques, number one is simple, periodic practice, every time you can you should try to review material, do problems and think about things that you have already studied to keep the material fresh. The second technique is trying to use your new knowledge in as many ways as you can and thinking about everyday activities as mathematically as possible.

For example, I was in my chemistry class last week and I realized that every non-cyclic alkane can be uniquely determined by a tree ( graph theory).

Now, regarding how to study I agree that you should practice doing a bunch of combinatorics questions, you should check out the book concrete mathematics by Knuth,Graham and Patashnik which has lots of these problems.

If you are looking for a bigger challenge I recommend R.M Wilson and Van Lints book titled a course in combinatorics although this one is intended for a graduate course, or you can also try the classic and excellent book by Cameron called Combinatorics: Topics, Techniques, Algorithms.

Combinatorics is the topic that I like the most in mathematics, and really the best recommendation I can give is take your time, savour every bit, do problems by your self and don't try to stuff as much possible into your brain.