Move By The Numbers

This game is for two players, and it involves (as do many of my games)
an n-by-n grid drawn on paper.
First, each player makes a list of n positive integers (where n is
the number of squares along a side of the grid), each integer in the
list being <= n-1. (The order of the integers is significant.) Each
player makes his/her list without knowlege of the other player's list.
Players then write their numbers, in order, along the edges of the
grid.
Player 1 writes her/his integers above the top row of the grid,
exactly
one integer above each square. Player 2 writes his/her integers to the
left of the left-most column of the grid, exactly one integer to the
left of each square.
One of the players plays offense, the other defense. After the round
is
complete, the players switch who is offense and who is defense, using
the same numbers in the same order, but using a new unmarked grid.
To start, player 1 places a "1" in any of the grid's squares.
Players take turns placing integers in the grid. Player 1 places
1,3,5,7,..the odd positive integers, in order, in the grid. Player 2
places 2,4,6,8,... the even positive integers, in order, in the grid.
A player, on move k of the game, places the integer k in any EMPTY
grid
square. He/she places the k directly to the right, to the left, above,
or below the square with (k-1) in it. ((k-1) is in the last number put
in the grid by the other player.)
Let the integer written above the column the integer (k-1) is written
in
be c. Let the integer written to the left of the row (k-1) is written
in
be r.
The player placing the integer k in a square must place that integer
EITHER c or r squares (in any of the 4 main directions) from the
square
the (k-1) is written in.
(The side of the grid that an integer is written next to does not have
anything to do with what direction the k-square is from the (k-1)-
square.)
The players act as if the top and bottom of the grid are connected,
and
act as if the left and right sides of the grid are connected.
(Toroidal
topology.)
So if a move, say, is off the grid to the right, the players acts as
if
the row forms a circle, and continues counting from the left side,
counting to the right.
A move to the left off the grid continues from the right side on the
same
row, continuing to the left. A move upward off the grid continues from
the
bottom of the same column, continuing upward. And a move downward off
the
grid continues from the top of the same column, continuing downward.
And remember. All moves must end up on empty squares.
If a player can move, she/he must.
The round is over when the players can't move anymore.
The offensive player gets a point for every square that has a number
in it.
(Ie the offensive player gets a score equal to the largest number in
any
square of the grid.)
The players switch who is defense and who is offense, as I said above,
with
the same numbers along the edges of the grid, but with a new unmarked
grid.
Highest score wins.

Thanks,
Leroy Quet