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,
- 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?
- 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).
- 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.