Summary

The Josephus problem is addressed in many discrete mathematics textbooks as an exercise in recursive modeling, with some books (e.g., [1] and [3]) even using it within the first few pages as an introductory problem to intrigue students. Since most students are familiar with the use of simple rhymes (like Eeny-meeny-miney-moe) for decision-making on the playground, they are comfortable with the physical process involved in this problem. For students who may wish to pursue this topic independently, [4] and [5] provide nice surveys and bibliographies, and the website [2] provides web-based tools for exploring the problem directly. The activities presented here are intended to be completed by students in a single class period early in the semester. We find that an opening student-centered problem can get the class involved and set a good tone for the semester. Moreover, we find that many issues arising from this particular problem can be built upon throughout the course. The next section provides some suggestions for connections to other parts of the course.

Notes for the instructor

The Josephus problem can be explored through role playing or through carefully constructed pencil and paper activities, depending on the amount of time one wishes to devote to it. We list below some of the things we discuss just before the activity as well as some of the contexts in which we have students revisit the problem later on.