How do I develop fast "shortcuts" in math?

Question

In order to define my question, I will demonstrate what I consider an appropriate answer. My question is as follows: How do I develop fast "shortcuts" in math? What I consider a shortcut is a means to solving a problem in a fast and creative way. My own answer would be to look for patterns and model a formula to solve the statement in an appropriate and timely manner. Maybe one could also introduce something brand new to the problem, like the perfect square technique. As one can see, I am not asking how to approach the problem, but what to do to find new and fast ways to solve it. This is obviously getting back to my question. How do I develop fast "shortcuts" in math? I value the community's intelligent feedback on this question. Thank you for your time and patience with me.

Answer 1

Obviously the methods will be different for everybody, but here are a couple of thoughts:

• Solve lots of different kinds of problems on your own. The more branches of mathematics you explore, the better you will get at noticing starting points.
• Always try to do as much of the problem as you can on your own, referencing previous resources if necessary. When you become stuck on this, learn how to google similar questions to see techniques.
• TAKE A REAL ANALYSIS TYPE COURSE. This did a ton for me and helped me transition into upper level mathematics. It taught me different proof techniques, and supplied me with enough problems to give me a solid intuition of how to approach proof-based mathematics.
• If you want to take a more computational / logic puzzle type route (as opposed to a proof based one), then I recommend looking at competition problems. You can find these in lots of different places.
• Finally, spend some of your time following along lecture notes / proof / problem-solving methods as opposed to trying them on your own. This is a great way to build additional skills in solving problems quickly and efficiently.