Diagonal/Straight: A Cooperative Game

Here is yet another game of mine played on a grid drawn on paper.
It is a somewhat unusual in that the game is a cooperative game,
the 2 players get the same number of points.
Start with an n-by-n grid drawn on paper, where n is odd
(and I suggest n be at least 7, maybe much higher).
Players alternatingly take turns filling in, on each move,
an *empty* square immediately next to the last square filled
in by the other player.
Player 1 can only fill in an empty square immediately either
to the left of, to the right of, above, or below the last
square filled.
Player 2 can only fill in an empty square diagonally adjacent
to the last square filled.
Fill in the center square.
Player 1 then fills in a square above,below, left of, or
right of the center square.
Player 2 then fills in a square diagonally touching the
previously filled square.
Etc.
Play continues until no more squares can be filled in.
The score (for both players) is the number of squares filled
in before play must stop.
(And there should be very little communication between players.)
If you insist on playing a competitive game, you can do so
with 3 or more players.
Just have each pair of players play a game,
for a total (if m= number of players) of m(m-1)/2 games,
and add every player's particular scores to get a total
score for each player.
You can also play this game solitaire, but I believe
that perhaps playing with 2 people might be more fun.
Thanks,
Leroy Quet

Update: I suggest that, instead of simply filling in the squares of the grid, the players instead fill in the squares with the integers 1 then 2 then 3 then 4, etc. (Player 1 writes in the odd integers. Player 2 writes in the even integers.) Doing this has two advantages. One, it it easy to see where each player last moved. And, two, it is easy to see how many points the players each get, since this is simply the largest integer in the grid when the game is over.