All Roads…

One of the great joys of mathematics—and one of the things that I think is often lost in lower level math classes—is that problems, especially the interesting ones, rarely have a single solution. When a given question is asked, there may be many routes to a single correct answer, or there may be a multitude of correct formulations and answers.

As a brief example, let’s consider the homework assignment for an upper division applied math course that I just finished grading. One of the problems on this assignment presents a production situation: a dairy buys two kinds of milk, which it processes and uses to make two kinds of cheese. These two cheeses have differing production costs and sell for different amounts, and there are a plethora of other details that constrain the precise mixture of cheeses that the dairy should produce and sell. The homework problem asks how the dairy can maximize its profits given all of the constraints presented.

A naive approach to the problem might be to simply try different combinations of cheeses, and see which ones are even possible given the constraints, and which of those is the best possible solution. We have powerful computers which can crunch numbers with great speed, and there is a solid body of statistical research which indicates that we would get pretty close to the correct answer using that approach. Of course, this method is pretty time consuming and computationally intensive.

Fortunately, it turns out that there is a better method, which is one of the topics of the course. This method, called linear programming, requires that all of the various constraints be written as linear equations and inequalities. Once this is done, there is a fairly simple algorithm which can be employed to determine the best possible outcome.

While the exact details are not relevant, it is important to note that there are often many ways of expressing the various constraints as linear equations. There are literally infinitely many combinations of variables and equations that will ultimately lead to the correct result. Some formulations use large numbers of variables and equations, while others use few variables and equations. The equations themselves can be relatively simple, or incredibly complicated.

What is really fascinating is that one can look at different formulations, and gain some insight into the cognitive process that lead to that formulation. When I took the class several years ago, I attempted to break the problem down into the smallest possible pieces, then build up a solution from those pieces. All of my equations and inequalities are very simple—most of them contain only one or two variables—but there are a lot of equations (21, to be precise), and a ton of variables (I count at least 15).

Compare this to the solution given by the professor who is teaching the course. His solution has only five variables, which are related together in only eight equations and inequalities. Rather than come around from the side and break the problem apart into smaller pieces, the professor tackled it head on, packing as much information as he could into the smallest space possible.

Even more interesting is the variety of formulations given to me by the students. Of the 12 assignments I graded, only two of them contained nearly identical formulations. Some of the students managed to break it down into even smaller atoms than I, while others managed a breathtaking level of compression. Moreover, the ways in which different students broke down the problem revealed quite a bit about their thinking. One student started by looking at how much cheese was sold, and worked backwards to determine how much milk to buy and process. Anther student started with purchasing, and worked through the process to determine how much cheese to sell.

While getting into the students’ heads and figuring out whether or not their formulation is going to work can be a royal pain (especially when the students are incredibly terse and don’t explain what they are doing), I love seeing this aspect of problem solving. There is something beautiful about the fact that many minds can follow many paths, yet ultimately all arrive at the same destination.

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