> Here is a game for any plural number of players. > > Needed: Blank piece of paper and a pen/pencil. > > In the first part of the game, players take turns placing n dots on > the paper. No 3 dots are to be co-linear. n is determined ahead of > time amongst the players and is a multiple of the number of players. > > The first player to move connects any (almost any) 3 dots with 3 > straight line segments to make a triangle, such that no dots are on > the interior of the triangle.
Why only "almost any"?
> Thereafter, the players continue to take turns. On a turn, a player > makes a triangle either by connecting (with two straight line > segments) a single dot to two dots that are at the ends of the side of > a triangle already drawn, or by drawing a single line connecting two > dots that are two vertices of two adjacent triangles that come > together to form a concave angle (a concave angle along the perimeter > of the polygon containing all the triangles drawn so far). > > No line-segments shall coincide with line-segments already drawn. > > The triangles drawn in this game shall contain no dots in their > interiors. > > After a player moves, she gets a point for each concave angle along > the perimeter of the bounding polygon, the polygon containing all the > triangles drawn so far.
My English may be a bit weak, but I'm not completely sure what you mean by "concave angle". I know obtuse and acute angles and concave and convex polygons, but I don't recall seing the word "concave" being used about angles. Are they angles of more than 180 degrees?
> The player gets an addition point every move he completes a triangle > with only one line-segment.
Would this not be the same as saying that a player scores the number of acute angles either before or after his move, whichever gives the most points? If a player closes a triangle with one line segment, the number of acute angles (as I understand the term) is reduced by one, so 1 plus the number of remaining acute angles is the same as the number of acute angles before the move.