Triangles with integer sides and given perimeter

TL;DR

This paper investigates the enumeration of integer-sided triangles with a fixed perimeter, comparing a set-based method and a number-theoretic approach, and discusses related pedagogical perspectives.

Contribution

It introduces two distinct methods for counting integer triangles with a given perimeter and analyzes their differences and educational implications.

Findings

The set-based approach effectively counts the number of triangles.

The number-theoretic approach offers a simpler, more direct calculation.

Comparison reveals strengths and limitations of each method.

Abstract

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.