Polygonix

Here is a game for two players. It can be played on a grid, but this isn't necessary. (Although playing on a bounded grid eliminates a lot of the ambiguity that may occur about the positions of dots.)

The players take turns being the defense player and the offense player.

The defense player starts a round by drawing m (m is a positive integer determined ahead of time) dots on a blank piece of paper. If you are using a grid, the defense player draws the dots at some of the intersections of the grid-lines.

Then the offense player connects pairs of dots with straight line-segments. The offense player continues to do this until all of the dots are each on the perimeter of at least one CONVEX polygon, and none of the dots are on the perimeter of any concave polygons or on the perimeter of any polygon that is not simply-connected.

(By "polygon", here I mean a region bounded completely by line-segments with no line-segments through its interior {except possibly grid-lines}.)

The offense player gets a point for each polygon.

Players play an even number of rounds, switching who is offense and who is defense, then add their scores.

The LOWEST score wins. (So, players try to minimize the number of convex polygons they draw when they play offense.)

Thanks,
Leroy Quet