Game of Criss Cross
Activity: The game is played by two players. The game board is created by drawing three points at the vertices of a large equilateral triangle, along with two to seven additional points anywhere in its interior. Players alternate turns drawing a single straight line segment joining any two points, as long as the segment does not pass through any other points pr segments already appearing on the game board. The winner is the last player to make a legal move.
1. How many different moves can the first player make on the game board on the left?
2. Will the first or second player win on the left-hand game board? Explain why any game on this board always lasts for the same number of moves.
3. Play three games of Criss Cross using the right-hand game board. Make a conjecture regarding the outcome of any game played on this board.
4. By trying games on several different boards, come up with a method of predicting the winner of any game of Criss Cross based on the board configuration.
5. For each of the games played in the previous problems, count the number of vertices (points), edges (segments) and faces (regions) appearing in the completed game board. In clude the area surrounding the game as one of your regions. For example, you should find that the left-hand game board has five vertices, nine edges and six regions.
6. Compare the number of edges and faces on each completed game board, then make a conjecture about these two quantities. Finally, prove your conjecture.
[Hint: First explain why every region on the completed game board is triangular. Then imagine cutting out all the regions with a pair of scissors. How many edges must have appeared on the completed game board, in terms of F?]
7.The expression V - E + F = 2 is known as Euler characteristic. Prove that the Euler characteristic of any completed game board is equal to 2.
8. Use the relationships between V, E, and F developed above to predict the number of edges and faces that will appear on a completed game board which starts with a total of 99 points. Will the first or second player win this game?
9. Use reasoning similar to the previous problem to prove that your method of predicting the winner on any game board is valid.
10. In a certain small country there are villages, expressways and fields. Express ways only lead from one village to another and do not cross one another and it is possible to travel from any village to any other village along the expressways. Each field is completely enclosed by expressways and villages. If there are 100 villages and 141 expressways, then how many fields are there?
Note: The activity and all of the questions in this handout are taken from Chapter 5 of Sam Vandervelde's book titled, Circle in a Box.