17 May 2016

In high school I did well in math competitions and even qualified for the national team.

Part of the reason was that I spend a bunch of time on my own solving math problems. It didn’t feel like a compulsory work. Instead, it was often a compulsive behavior for me. I was often unable to un-glue myself from the math problem at hand.

When I saw a good problem, I attempted to solve it. The easier ones I could solve quickly. But some of them only gave in after hours of trying different approaches and strategies. It felt fantastic to solve a hard math problem. It was a puzzle, and I enjoyed solving it.

In the typical math problem, there were several facts given. An example problem might be:

Problem 1. Let a, b and c be non-negative. Suppose

a^2 + b^2 = c^2

and

2b >= c

Prove that

3c^2 <= a^2 + (b+c)^2 <= 4c^2

I just made that problem up. But I know it is true, so I’ll let you try to solve it.

When solving a problem like that, I am trying to connect the dots. Here, the dots are, on one side, the two given input conditions, and on the other side, the output requirement. Solving the math problem is a sequence of manipulations, trying to bring both sides them closer to each other. One approach would be to start manipulating the inputs into a different form, that is closer to the output. Or manipulate the output bring it closer to the inputs. Or do a transformation on the problem space into another, more comfortable space to work in.

Each of those steps translates the problem into a different problem, which may be easier, or harder. It may take hours of trying different transformations until I get on the path to solution, and bring the two sides of the problem together.

It is like playing a video game where I’m walking around a maze, trying to find the right combination of items to unlock the exit door. I may walk around for hours, without success, but the feeling of progress fuels me. Every time I collect a new item, every time I think of a transformation of the problem, I feel that I’m making progress. Even if that transformation itself is not useful at the end of the day, I still get the excitement. I’ve discovered a new corridor in the maze. It is thrilling, what might be inside it. Each of those thrills helps me sustain my excitement in solving the problem.

Every new problem is a new maze, a new puzzle to figure out. In the beginning, the act of exploration was what was thrilling. But after solving a large number of problems, just the existence of the maze was a reason for excitement. Even without having actually tried any problem solving approaches, I knew that I could try them. I knew that I could explore, and that I’d discover interesting paths to the solution.

Just knowing that this adventure awaits was the reason for me to jump on it. I had trained myself through repetitive exposure to the enjoyments of the process of problem solving, to love it.

The act of fully connecting the dots was the most rewarding part of the problem solving, but it wasn’t the most accessible. In the long run, reaching solution was the high octane fuel which fueled my desire to become better at problem solving and to do more of it.

But I needed a kindling, a tinder, a fire-starter to get to it. The fun of the process was part of it, but also the environment I was in provided additional boost. I was among people who considered math cool. I liked the competitive feeling of the math olympiads. I liked traveling to different cities for math competitions, and meeting other students who enjoyed solving math problems.

All this encouragement was the catalyst which got me to love solving math problems so much. It became a self-sustaining habit, to enjoy the pleasure of math. It became a compulsive behavior, to attempt to solve a problem whenever I saw an interesting one. This helped me become good at it.

Now, I’m looking to become good at writing. Writing helps to clarify and express my thoughts. I have the intuition that it a great exercise for my mind. And it feels awesome to complete a nice essay. And it helps me grow.

Still, it is not a solid habit yet. I am still adding fire-starters to it. I have a nice emacs-based writing environment. I reward myself with coffee when I’m writing. It is trivial for me to post a new essay to my online collection.

All these niceties help me reinforce this new habit until it is strong enough on its own.

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I write these posts to clarify my thoughts and improve my communication. They are mainly written for myself, and my future self, but I'd love to hear any comments you might have, especially if you disagree with what I've written, or if you've found it valuable in any way. Message me on Twitter @themitak or email me at comments@dimitarsimeonov.com.