Triangles with integer sides and given perimeter

In this paper we first study the isoperimetric problem in the case of integer triangles, as well as Alcuin's sequence and how it relates to the number of different integer triangles with a given perimeter. We then present and compare two different approaches to the above problem. The first approach is due to a university professor (a working mathematician), who uses a well-known technique of calculating the number of elements of finite sets, while the second one is due to a high school student with a special interest in mathematics and is purely number-theoretic. Furthermore, some related instructor perspectives are discussed.