Fill-In A Grid: Game

Here is asimple and unoriginal, but possibly fun, game.

Any number of players >= 2.

Start with an n-by-n grid drawn on paper. (I suggest an n of at least 5.)

Before play each player *secretly* picks a positive integer.

Starting in the upper left square, players take turns filling in one square each move such that: * the filled-in square is next to the square filled-in most recently by the previous player, and the filled-in square is immediately either above, below, left of, or right of the square most recently filled-in by the previous player. * the filled-in square was not already filled-in by any player in an earlier move.

Play continues until there are no unfilled squares immediately adjacent to the last square filled-in.

When the game is over (when every square next to last filled-in square is already filled-in), the winner(s) is the player whose secretly picked number (each of which is now revealed) is closest to the number of squares of the grid which did *not* get filled-in.

So, players might try to guess, by observing which squares their opponents fill-in, what their opponents secret integers are. (For instance, is their opponent trying to end the game early, attempting to continue the game, bluffing,...?)

Sample completed game:

(Numbers are filled-in squares. *'s are empty squares.)

1 2 * * * 4 3 101112 5 8 9 1413 6 7 1815* * * 1716*

Player whose number is closest to 7 (the number of *'s) wins.

thanks, Leroy Quet

I forgot to say that each player writes down his/her secret integer. (Or otherwise he/she could claim that the number of unfilled squares just happens to be the number they had picked...)

A math question:

Considering all possible games, each of those games played until no more squares can be filled in, what is the most likely number of unfilled squares and the average of the numbers of unfilled squares?

thanks, Leroy Quet