Vertical/Horizontal Guessing Game

Here is a game for two players, and, you guessed it, it uses an n-by-n grid drawn on paper. (I suggest an n of at least 8.)

First, the players each secretly guess how many squares will be filled in before the game terminates. Each player writes their guess down and hides the guess from their opponent.

Next, fill in with a pen/pencil the center square (if n is odd) or one of the 4 center squares (if n is even).

(*)The players then both secretly choose a direction. Player 1 chooses either up, vertically-steady, or down. Player 2 chooses either left, horizontally-steady, or right. The players each write down their choices.

Then the players both reveal their choices simultaneously.

The next square filled in has the vertical direction chosen by player 1 and the horizontal direction chosen by player 2 from the last square filled in; and this next square filled in is adjacent (touching on a side or on a corner) to the last square filled in. (There are 9 possible combinations of directions, including not changing the square at all {when both players pick steady}.)
Note: If the last filled in square is on the edge of the grid, then this limits what directions can be stated by one or both of the players.

If that adjacent square is already filled in, then the game is over.

But if that square is empty before being filled in on the current move, then the game continues. (Go to *.)

When the game is over, the winner is the player whose guess for the number of squares filled in is closest to (either greater than or lesser than or equal to) the actual number of squares filled in.

(By the way, if both players choose the direction 'steady' at the same time, the game ends then, of course.)

Thanks,
Leroy Quet