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So far in my blog I’ve described various areas of mathematics, discussed common stereotypes and misconceptions about mathematicians, and even interviewed several of them. However, I’ve never quite managed to give you a sense of what it is that professional mathematicians actually do. Today, I’m going to try to do just that, using a problem I came across recently as an example. I’ll describe my own process from beginning to end; keep in mind that this process may be quite different for other mathematicians.
Let’s start with the problem choice. I was at a workshop on entrepreneurship where the workshop leader made us play a simple game. We were given a bag with 10 light and 10 dark chocolates, and had to call the type of chocolate we were going to draw next. We always got to keep the chocolate that we drew, and if our guess was correct, we also got to continue playing; otherwise, the game ended there. This game was used as an example of a fully predictable situation. The best strategy also seemed pretty clear: guess at random if there are as many light chocolates as dark ones, and guess the more abundant kind if there is an unequal number to stay in the game longer. When I saw this game, though, a new question immediately came to my mind: what was the expected value of this game? In other words, if I value a chocolate at a dollar, how much should I be willing to pay to play the game?
I took out a pencil and a sheet of paper and within a few minutes, I was able to calculate the value of the game, which was slightly above 2 dollars. This calculation was pretty simple; it used a recursion relating the value of the game to the values with either one less dark or one less light chocolate (which were equal since all the chocolates had equal value), and relating that value to the value of the game with one less dark chocolate and one less light one. While I was glad to be able to solve this particular question, another one immediately came to mind. What if I valued dark chocolates more highly than light chocolates? In that case, my recursion showed something slightly unexpected: the optimal strategy is still to guess the more abundant type of chocolate, but if there are equally many of each kind, it’s better to guess light even though I prefer dark. This becomes obvious when there is only one of each kind, but less obvious when larger numbers are involved.
At this point, the game had piqued my interest enough for me to try to find a general formula for the value of the game for different relative preferences over light and dark chocolates. At first I computed some values by hand, but couldn’t see a pattern. I sent the problem with my preliminary results to a few of my fellow mathematicians, but none of them saw a pattern either. Then I decided to ask MAPLE, the symbolic algebra software, for help. It didn’t find a simple pattern, but by working with it, I eventually saw a pattern myself, which it helped me confirm. I then checked by hand that the formula I had found satisfied the recursion. At that point I was ready to write down a proof of my formula, involving recursion and some case analysis.
Interestingly, the formula involved a famous sequence of integers known as the Catalan numbers; they count various mathematical objects, such as tied two-candidate elections where one candidate is never behind the other in the partial counts, the ways a polygon can be divided into triangles by joining vertices, and the ways an expression involving only one operation such as addition or multiplication can be bracketed. I was intrigued by that and had a hunch that something interesting would happen in a more general scenario, say with three kinds of chocolates involved. This turned out to be the case.
There were actually two ways of generalizing it, of which I picked the more natural one (where you had to guess the right kind among the three to stay in the game). Strategy-wise, the same result turned out to be true; namely, you should always guess the most abundant chocolate, breaking ties in favor of the one you like the least. The new formula, discovered with the help of MAPLE and a lot of trial and error, involved a new sequence of numbers that did not seem to be known, which I tentatively called the poly-Catalan numbers. Proving the formula turned out to be rather challenging, not only because of a much larger number of cases to analyze, but also because of the complexity of the formula itself. However, I finally managed.
I believe my work on this problem illustrates several common approaches to mathematics. First, I found an interesting situation which I could turn into an equally interesting mathematical problem. Second, I spent time figuring out a solution to this problem, unsuccessfully at first, and successfully later when I got a computer involved. Third, I found an interesting feature of the problem’s solution that made me interested in making it more general. Fourth, I developed the tools needed to solve the more general version, and discovered a previously undiscovered sequence of integers. Finally, I wrote up my solutions and I’m currently putting the final touches on it so I can try to get it published in a journal. In the process, I also raised a number of new questions that I hope will be answered by other mathematicians someday.
As for whether this kind of work is actually useful, I don’t know for sure. It was certainly useful for me on many levels: it taught me to be more patient, made me learn some new tools, and involved the help of a computer. The new sequence of numbers I discovered in the process may well turn out to be important in a completely different area of mathematics or the sciences; however, even if it doesn’t, I really enjoyed this work, and ultimately, that’s the most important justification for doing it.