Grid Subdividing Solitaire Game

Here is a solitaire game.
Start with an (n-1)-by-(n-1) grid (of n-by-n lines) lightly drawn
on paper.
You also need 2 sets of cards, each deck of n cards labelled 1 through
n.
The player draws on each move one card from each deck.
Don't replace the card. Let the value of the cards be A and B.
>From the last position (at a vertex) drawn to on the grid, the player

either draws (along the grid's lines) a vertical line then a horizontal
line, or draws a horizontal then vertical line, drawing to either
vertex (A,B) or vertex (B,A). The line changes directions at most once
(taking a right angle turn). The line may coincide with lines which
were drawn earlier.
(Vertex (1,1) is the upper left corner of the grid. Vertex (n,n) is
the lower right corner of the grid.)
After n moves (when all the cards have been drawn from both decks)
the player gets a score equal to the *product* of the numbers of
squares
in each section the grid has been subdivided into by the player's
lines.
(Imagine the lines drawn during the game as a knife cutting the
grid-cake
into pieces. Multiply the sizes of each piece to get score.)
An example will hopefully make this clear:
(Ignore the ~'s.)
Cards on each move (for n = 9): (A,B) =
(4,2) (1,3) (6,5) (2,4) (8,6) (5,1) (9,8) (3,7) (7,9)
1.~~.~~*= = =+~~.~~.~~.~~.
~~~~~~~~~~!~~!~~~~~~~~~~~
2.~~.~~.~~S~~!~~.~~.~~.~~.
~~~~~~~~~~~~~!~~~~~~~~~~~
3.~~.~~.~~.~~!~~.~~.~~.~~.
~~~~~~~~~~~~~!~~~~~~~~~~~
4.~~* == == =+==+~~.~~.~~.
~~~~!~~~~~~~~!~~!~~~~~~~~~~~
5*= +== == ==+==+~~.~~.~~.
~!~~!~~~~~~~~!~~!~~~~~~~~~
6!~~+== == ==*~~!~~.~~.~~.
~!~~~~~~~~~~~~~~!~~~~~~~~~
7!~~.~~*== == ==+ == =+~~.
~!~~~~~!~~~~~~~~!~~~~~!~~~
8!~~.~~!~~.~~.~~*~~.~~!~~.
~!~~~~~!~~~~~~~~~~~~~~!~~~
9+== ==+== == == ==F =*~~.
.1 .2 .3 .4 .5 .6 .7 .8 .9
Start at S. Finish at F.
*'s are other vertexes that are either (A,B) or (B,A).
Score: 13 * 23 * 3 * 1 * 3 * 11 * 10 = 296010.
thanks,
Leroy Quet