Planar Graphs
When a game of Brussels Sprouts is completed, you are left with what mathematicians call a planar graph. This is a collection of points in the plane, connected by curves which do not cross each other. Planar graphs have many special properties: for example, take any finished game of Brussels Sprouts and four different colored pens. By being clever enough, you can always find a way to color each intersection with one of the four colors without ever giving two neighboring intersections the same color. Try it on your completed Brussels Sprouts games!
Planar graphs also always have a very particular relationship between the number of intersections, edges, and regions. Take any finished game of Brussels Sprouts. Add up the number of intersections, subtract the number of curves, and add the number of regions (don't forget to count the outside as a region!) and call the result E. E is called the Euler characteristic of the planar graph. Try computing E for many different finished games of Brussels Sprouts. What do you find each time?

