the condition is satisfied iff $ n$ is even. the "only if" part is obvious. the "if" part: if $ n$ is even, then let $ A, B$ be two diametrically opposed crows in the initial configuration. if $ B$ is also diametrically opposite $ A$ in the final configuration, then clearly $ A$, $ B$ and any third crow will satisfy the condition. if $ C\neq B$ is the crow opposite $ A$ in the final configuration, then $ A, B, C$ form a triangle in the initial configuration as well as in the final configuration.