Let $\Omega$ be a simply connected two dimensional region, for simplicity a rectangle. Consider $n \geq 3$ points. Define once randomly triples of these points (not necessarily pairwise disjunct) in such a way, that each point $p_i$ watches two other points $p_j, p_k$ $(i \neq j , j \neq k,i \neq k)$ and quests to form an equilateral triangle with them.

To do so $p_i $ moves a step into direction to the target position needed. But $p_j$ watches two other points (not necessarily $p_i$ and/or $p_k$) and quests to form with them an equilateral triangle (within $\Omega$). The same holds for $p_k$.

What do you think is the outcome e.g. in the unit square?

Do we see after a time a collection of moving equilateral triangles or even do the equilateral triangles stop moving? Or do the points never become all arranged into equilateral triangles? For the sake of simplicity the velocity of all points is equal and constant (or zero). Also two or more points can hold the same position in $\Omega$, they do not collide and do not bounce each other. If a point is nearer to its target position in its equilateral triangle it does only this smaller step to reach the target position. If a points has a position as corner of an equilateral triangle it stops moving iff the two points it watches do also not move. The points do not bounce from $\partial \Omega$ but hold on it moving from there back into $\Omega$ or along $\partial \Omega$ with the next step.

What happens for $n=3$?