There are certain types of maps however, where less than four colors are sufficient:

1. We partition the plane into regions by drawing a number of straight lines. In the resulting "map" there will be some infinitely large regions, but that shouldn't bother us. How many different colors do you need for such types of maps?

2. Instead of straight lines, partition the plane by drawing circles. How many colors are needed now?

3. What if you draw circles and chords?

4. Now consider 3-dimensional "maps", where you partition space into regions and two regions are called adjacent when they share a common (2-dimensional) face, not just an edge or a point. How many colors do we need if space is partitioned by planes?

5. As above, but with spheres instead of planes.

6. How many colors are required for general 3-dimensional maps?