Slice Through The Boundaries

This is a game played by any number of people.
It is played on an n-by-n section of grid taken from graph paper. (I
suggest an n of about 12 for beginners, if there are only 2 players.)
Players take turns who is the offense player. (The other players play
defense on a round.) A round is played for each player playing the
game. An empty n-by-n grid is used each round (with the same n as in
the other rounds).
Players take turns each filling in the empty squares of the grid, one
empty square filled in each move by each player.
If there are m players, then each player fills in floor(n^2/(2m))
squares. (That is a total of m*floor(n^2/(2m)) squares filled in all
together.)
Then the offense player draws a straight line (with a straight-edge)
from any side of the n-by-n grid to any other side.
The line must not be perfectly vertical or perfectly horizontal.
The offense player gets a point for every boundary between a filled-in
square and an empty square that the line passes through.
Highest score wins.
Example:
Filled-in square = *. Empty square = o.
n = 6. (View with fixed-width font.)
\ 1 2 3 4 5 6
A o * * * o *
B o * * o o o
C * o * o o *
D * * o o o o
E o o * * o o
F * o * * * *
Let us say that the line goes from just below the upper-left corner of
the grid to just left of the lower right corner. (The line meets the
perimeter of the grid less than one square's length from each of these
corners.)
This is kind of hard to depict here, because the squares of the grid
are literally squares, while they are not in my diagram; but hopefully
it is clear anyway.
The line first crosses the boundary between 1B and 2B. Then it crosses
the boundary between 2B and 2C. Then it crosses the boundary 2C to
3C. Then 3C-3D. Then 4D-4E. Then 4E-5E. And finally, 5E-5F.
Note: Technically we are concerned about the number of times the line
crosses from a filled-in square into an unfilled-in square, or vice
versa. So if a line crosses from a square to a diagonally adjacent
square via the vertex that joins them, then what we are concerned
about is the two squares' status. In other words, the vertex is
considered the "boundary" in that case.
Thanks,
Leroy Quet