This article presents several mathematics problems on the same theme that can be used in class or for independent study. It is very difficult to show students how mathematics research is done by mathematicians, so the author first introduces a problem that can be tackled solely by arithmetic and then gives an idea of how a mathematician might tackle such a problem. Additionally, attention is drawn to how students can engage in mathematical work, including experimenting, collecting data, simplifying problems, checking work, proving, extending, and generalising.