TraverX

This is a game for any plural number of players. Start with a grid of n-by-n squares ((n+1)-by-(n+1) lines) drawn on paper, where n is even, and where I suggest that n is >= 12.

At the beginning of the game, a small x is drawn at the intersection of the middle horizontal line and middle vertical line.
Players take turns moving. On the k-th move (the kth move considering all the players' moves together) the player "traverses" j(k) = (k-1)(mod(n-1))+2 intersections from where the last player last put an x.
(So, for k = 1,2,3,4,...,n-1,n,n+1,n+2..., the number of intersections traversed is 2,3,4,5,...,n,2,3,4,..., repeating 2 through n.)
(On the first move, the first player traverses 2 positions from the central x.)

The player can "traverse" j(k) intersections in the direction of either right, left, up, or down, and then may change direction at any time at most once, and traverse perpendicularly to their initial direction for the remainder of the j(k) intersections traversed. The player then places an x at the intersection they land upon. It is only acceptable for players to land upon (at the j(k)th intersection traversed) an intersection without an x already drawn upon it. Players may, though, traverse over intersections with x's already on them, or not.

After a player writes down an x, he/she gets a point (points are bad in this game) for every other x already written on the same vertical line and same horizontal line as their x.

The game continues until any player cannot move anywhere (given j(k) and the lack of available intersections).

The player with the SMALLEST score wins.

Thanks,
Leroy Quet