Recently, the following Numberphile video on the Josephus Problem has been making the rounds on math-related social media. I watched the video, and I thought Daniel Erman did a remarkably good job at explaining how to solve a mathematical problem. Daniel’s approach is similar to the techniques described in Polya‘s “How to Solve It.” Yet the particular story that Daniel tells also has an appealing narrative arc.
Daniel’s video adheres to the following principles, which I think are fairly universal in mathematical problem solving.
- Start with a concrete problem. If the problem has a nice story to go along with it, all the better. The Josephus Problem is a great example of a concrete mathematical question. Given a method by which the soldiers kill one another and the number of soldiers, where should Josephus stand to be the last living soldier?
- Formalize and generalize the problem. What is special about the number 41? The mechanism by which the soldiers kill one another works just as well for any number of soldiers, so consider the problem for \(n\) soldiers.
- Consider simpler versions of the general problem. Now that we have the general \(n\)-soldier Josephus problem, we can easily work out a few examples when \(n\) is small. To quote Polya, “If you can’t solve a problem, then there is an easier problem you can’t solve: find it.” This process of finding simpler and simpler related problems until you find one you can solve is to me the most important general problem solving method.
- Solve enough of the “simple” problems until you see a pattern. Solving the simpler problems gives one both data and intuition that will allow you to conjecture about a general solution.
- Generalize the pattern as much as you can so that it fits the examples you’ve solved. Even if the pattern doesn’t give a complete answer (for example, Daniel’s observation that if \(n\) is a power of \(2\), soldier \(1\) is the last living soldier), even a partial solution is likely valuable to understanding a complete solution.
- Prove your generalization of the pattern to obtain a solution to the general problem. Often, this doesn’t happen all at once. The Numberphile video happens to give a particularly elegant solution in a very short period of time. Don’t get discouraged when not everything falls into place the first time you try to solve the problem!
- Apply your general solution to the original concrete problem.
In my own research, I follow the strategies above. In particular, Polya’s advice regarding finding and solving simpler problems (steps 3 and 4) is maybe the most valuable single piece of problem solving advice I know of. I think math could be characterized as the art of generalizing simple observations. Often, the simple observations arise by wasting a lot of paper trying to solve simple problems.
The narrative outlined in the steps above is also valuable from a pedagogic standpoint. By starting with a tangible (if slightly morbid) problem, the student/participant immediately has some intuition about the problem before beginning formal analysis. In my experience, one of the biggest challenges students face is connecting abstract statement and theorems to concrete problems. By introducing the concrete problem first and using this problem to motivate the formal development needed to solve the problem, students can begin using their imagination earlier in the problem solving process. This makes learning more interactive, memorable, and effective.