The famous four-color-theorem states that every map can be colored with no more than four colors, such that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point.

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?

1. and 2. I have known forever, as I was an avid drawer in my youth.  Both indicate what I call a checkerboard pattern and can be done in two colors.

3.  Two concentric circles with a chord in the smaller will take three colors.  There may be a way of needing four with this arrangement but I haven't come up with one.

4. and 5. are just like 1. and 2.  and should still require only two colors.

6. can require an unlimited number of colors.  Picture a bunch of pieces of colored paper laid out on the floor.  From each a 'tendril' reaches out and branches to touch all the others.  There's too much room in three dimensions to have a n-color theorem.

Posted by Jer on 2006-09-19 12:38:19