Triangular Numbers and an interesting Equilibrium.

A simple repeated action on any partition of 10 objects always eventually leads to the same equilibrium, 1, 2, 3, 4. An analogous claim holds for 15 objects, 21 objects, in fact, for all triangular numbers.  We'll show why this is true and scratch our heads a bit. Time permitting, we'll throw in a completely unrelated (brief) discussion of an unrelated topic and an open question associated with it.