In a math class, I was playing the following $2$-player game on an online whiteboard and I became curious if there was an optimal method.
- Start with a triangle, divided into $n^2$ smaller triangles by line segments, as shown below. The top vertex is red, the lower right vertex is blue, and the lower left vertex is green. The others start out uncolored.
The two players take turns coloring in the other vertices, subject to the following constraint: any vertex on an edge of the large triangle can only be colored with one of the colors of the two endpoints of the edge. (So, for instance, vertices on the left edge can only be red or green.) A vertex in the interior of the triangle can be colored with any color.
At the end of the game, one player gets one point for each small triangle colored red, green, blue (in that order) in the counterclockwise direction, and the other player gets one point for each small triangle colored red, green, blue in the clockwise direction. The winner is the player with more points, or is drawn if the players have the same number of points.
Is there an optimal method for each of the two cases in which the first player gets points for the clockwise or counterclockwise color ordered triangles with red, green, blue? If so, what is it?