The purpose for having students play the game of Criss-Cross is to motivate them to explain the underlying mathematical reason governing who wins or loses. This exploration should lead the students to form, test, and ultimately prove conjectures about how to win at Criss-Cross. The game illustrates a beautiful application of the Euler characteristic and gives them practice at elementary counting techniques as well.

The purpose of this activity is to introduce students to elementary concepts in
graph theory in a hands-on, accessible manner. Students will create their own graphs, count
certain quantities related to their graphs, make conjectures regarding these quantities, and
learn how to explain why their conjecture is true. In the process they will discover the Euler
characteristic, a powerful tool for understanding planar graphs.