The problem-solving skills in this question can be interpreted broadly as the skill or intuition to solve mathematical problems in general, or they can mean the skills in a particular field such as analysis, algebra, combinatorics, etc. However, it should not be interpreted narrowly as one’s familiarity with a specific subject.
In mathematical logic and real analysis, one learns to think rigorously. In combinatorics, one discovers ways to build connections. Whether it is cryptography or Ramsey Theory, learning any mathematical subject can always more or less alter one’s way of thinking and improve one’s problem-solving skills. However, since I have limited time and energy, I can not learn every subject in mathematics. If increasing my problem-solving skills is my motive, what courses or subjects can have the greatest impact on the way I think about problems?
Currently, I’m reading about Thinking Mathematically, which is a book about the ways to approach mathematical problems. It led me to wonder what processes in learning actually contribute to one’s growth in problem-solving skills. Sure, reflecting on obstacles when attacking problems and trying to challenge oneself can result in growth. However, to what extent should one learn? One can independently prove all the theorems in books and even do every exercise. One can always learn more about a subject, but I wonder if the marginal return will decrease as one learns more. So if increasing my problem-solving skills is my objective, to what extent should I learn a subject?
Thank you!
From the comments and answers, I see that this answer might vary because of one’s focus. However, I agree from the answer that understanding logic is beneficial in almost every subject. Since I don’t have a focus right now, what is something similar to logic that will be beneficial to know for all subjects?
This is not a complete answer, but I write it here because it is too long for being a comment.
The answers to your questions can be a matter of taste. According to https://www.ams.org/notices/200902/rtx090200212p.pdf, we have birds and frogs among mathematicians. I always disitinguish mathematicians somehow similar to the mentioned pdf file with the following example: If you want to solve $\int_{0}^{\infty} \frac{1}{1+x^4} dx $ you can consider $1=-i^2$ and do partial fractions, or you can see like Caushy from a higher dimension and use complex integrations. So, we can say some mathematicians have a geometric view and some others have an algebraic view, and of course, some others have both. I suggest if someone wants to have both perspectives, Algebraic Geometry or Algebraic Topology would be useful. Although these two subjects differ totally from one another, they are superior tools of mathematics in my opinion. So, they may give you a better intuition for daily life problems and the approaches to solving them.