Junior Mathematical Olympiad 2025 Section A Q5

For which $n \ge 3$ can an $n$-sided polygon be made by joining several one-by-one squares together edge-to-edge? For instance, the given diagram shows an $8$-sided polygon formed in this way. Note that a shape with holes is not a polygon and should not be considered.

This is an Olympiad problem, so you must show that all such values of $n$ are achievable, and that no other values are.