1 · Try to break the pentagon

The five-person table, safe with two colors. Click any string to swap its color.

Try this. Swap a few strings and watch the count. Every swap that helps one triangle tends to hurt another; see if you can find a second safe coloring at all.

Up to rotating and reflecting the circle and renaming the colors, the ring-and-star coloring is the only safe one, which is why your swaps keep failing in the same ways.