Mathematical Contest in Modeling

For one of my experiential learning activities, I participated in the Mathematical Contest in Modeling, an international competition where teams create a mathematical model of a real-world problem. We have two problems to choose from and four days to write a paper.

I probably learned more from the preparation than the contest itself. Over Christmas, Siyu, Shawn, and I formed a team. We studied various areas of math, read papers from previous contests, and did a mock contest. My readings introduced me to some important mathematics and I learned to make nice graphics.

Our mock contest gave us a sense of timing. When you focus on one thing for a long time, you notice how slowly and steadily time moves. Four days is a long time if you think about it. And yet time does not stop; the deadline marches inexorably toward you. You must be ready, before it arrives, to stop tinkering with your computer programs and write the paper.

When the contest began, we had to choose between two problems. The explanation for the first problem was far more thorough, but that did not make the problem easier. The apparent precise statements were actually quite vague after we thought about them for a while. That is typical of mathematical modeling. The real world seems simple, but when we try to describe it mathematically, we realize how complicated it really is.

The contest taught me about teamwork, perspective, and perseverance. Although we came up with some good thoughts when the contest began on Thursday evening, we lost much Friday and much of Saturday to dead ends.

I think we could have avoided some of our errors by keeping better tabs on each other. The problem had two parts; I focused on one while Shawn focused on the other and Siyu was either assisting or trying to write. We should have looked at each other’s work more in order to have more perspectives on the same problem. We could have warned each other that our methods were not going to work.

For my part of the problem–simulating the heat distribution in a brownie pan–I spent a long time trying to get various softwares to solve the heat equation numerically over a strangely shaped domain. They all failed, and on Saturday afternoon, there was no time to write our own program for that purpose. I was desperate and therefore creative. I realized that the heat equation also models diffusion, so I quickly wrote a program that simulated diffusion (which is a lot easier).

The key insight was that the same equation describes more than one thing. Essentially, we modeled one real-world process (heat) with another real-world process (diffusion). The two processes were connected by their mathematical similarity, but in the end, we took mathematics out of the equation! It required a wider perspective to do this, and it was only when I was desperate that I had that perspective. Then, the problem that had stumped us for two days vanished in about an hour.

By the end of Saturday, we had something resembling a solution, but we had to hurry to finish the paper. We persevered, staying up nearly all night twice, and I was happy to have finished.

But the day after the contest ended, I had some new insights. Although pressure can make you work harder and sometimes induces creativity, it is usually easier to have perspective when you are not stressed. And there is only so much you can think of in four days.

I also learned I have a great capacity for obsession. For four days, maybe fourteen hours a day, we did nothing but work on that problem. Humans have a great ability to focus. It can save us, but it can also destroy us. After only four days of this, I was exhausted and hungry; I felt barely able to think or move. Doing this over a long period of time could ruin my body.

Here is our paper.  Here is an even more in-depth discussion of our problem-solving process.

Art of a Mathematician