1. Draw three overlapping circles. Color the resulting regions using two colors, so that no two regions that share a curve get the same color. (This is known as 2-coloring the regions. Grey and white are popular colors for experimenters who use pencil on white paper.) Now draw two pairs of overlapping circles and a single circle overlapping none of the others; 2-color this configuration. Using the understanding gained from these experiments, prove that n circles drawn in the plane can be 2-colored, using induction.
2. Challenge: Analyze the proof you gave for the previous problem. Would it work for n overlapping squares? Triangles? What about for spheres in space?`