A Möbius strip can be regarded as the configuration space of pairs of (indistinguishable!) points on a circle.
A dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it touches. Sometimes a new dot connects many components, and in this case, the new component takes on the color of the largest among the old components.
After drawing n dots, how many components should we expect to see? When you draw only a few dots, most of those dots are isolated and have their own color; but after drawing a ridiculously large number of dots, they are all connected and the same color.
Reflecting triangles in the plane
Start with a triangle in the plane; reflect that triangle across its three sides; repeat, reflecting the resulting triangles through their sides, over and over again.