Lines-By-Lines

This is a game for 2 players.
Two rounds are played. In each round one player is offense while the
other is defense.
Players switch who is offense and who is defense for the 2nd round.
Start a round with an n-by-n grid.
Both rounds are played on the same sized grid drawn carefully on
paper.
(It is preferable to use graph paper.)
The offense player moves first in a round.
The first player to move draws a straight line-segment (with a
straight-edge, preferably) from any vertex of the grid to any other,
provided that no intermediate vertexes are crossed. (The only vertexes
to coincide with the line-segment are at the segment's end-points.)
Players take turns each drawing a straight line-segment (from the
vertex where the other player last drew a line-segment to) to another
line-segment. So all the line-segments together form a continuous
path.
Line-segments must not cross or coincide with each other.
Line-segments must not coincide with any vertexes of the grid, with
the exception of at the line-segments' end-points.
Line segments must not travel horizontally or vertically.
Play continues until it is not possible to draw a line-segment to any
other vertex, given the rules.
The offense player gets a point for every square of the grid the path
of line-segments passes through.
(A player gets at most one point for each square, no matter how many
line-segments pass through that square.)
Players switch who is offense, then play another round.
Highest score wins.
Note: It may be more interesting if it is required that the first
player to move draws his/her line-segment from the center of the grid.
Thanks,
Leroy Quet