Grid Game Of Differences

This is a game for two player.

Draw an n-by-n grid on paper, where I suggest that n is at least 8.

The players take turns placing x's in the empty squares of the grid, one x per turn.

No two or more x's may be placed in the same row or in the same column of the grid.

After n total moves (when there is exactly one x in each row and column), the game is over.

Now to determine the score:

Reading left to right, write down the (n-1) absolute values of the differences between the consecutive x's' vertical coordinates, in terms of number of squares.

In another list, reading bottom to top, write down the (n-1) absolute values of the differences between the consecutive x's' horizontal coordinates, in terms of number of squares.

Player 1 gets a point for every distinct numerical value occurring in the first list of differences.
Player 2 gets a point for every distinct numerical value occurring in the second list.
If a particular difference occurs at least once in a single list, then the player gets one point for that particular difference.

Largest score wins.

We may need an example here:

n=9:

. x . . . . . . .
. . . . . . . . x
. . x . . . . . .
. . . . . x . . .
x . . . . . . . .
. . . x . . . . .
. . . . . . . x .
. . . . . . x . .
. . . . x . . . .

Player 1's (vertical) differences (reading left to right) are:
4,2,3,3,5,4,1,5
The unique values that occur are:
1,2,3,4,5
Player 1 gets 5 points.

Player 2's (horizontal) differences (reading bottom to top) are:
2,1,4,3,5,3,6,7
The unique values that occur are:
1,2,3,4,5,6,7
Player 2 gets 7 points.

In another variation of this game, count ONLY those differences that occur exactly once (and no more than once).
In this variation, player 1 would have gotten 2 points, for the differences 1 and 2.
Player 2 would have gotten 6 points, for the differences 1,2,4,5,6,7.
(Since 3 is the only difference in this list that occurs more than once.)


Which variation is more fun?

Thanks,
Leroy Quet