Max Score For A Game

[Leroy Quet]

Below is a game I invented (as posted to sci.math and rec.puzzles -- it 
is just one of many stupid silly simple games I have posted there).

I post it here to seq.fans because I wonder what the sequence is where 
the nth term is the maximum possible score for an n-by-n grid. (We could 
also have a table where the element in the mth column and nth row is the 
maximum score for an m-by-n grid.)
Of course, if there is a way to always get the score n^2 -1 for each 
n-by-n case, or m*n -1 for each m-by-n case (n or m equal 1 excluded), 
then there is be no need to calculate the sequence.

Thanks,
Leroy Quet

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Here is a solitaire game played on a square grid.
(I hope this game is more fun than my last game.)

Start by drawing an r-by-r grid on paper.
(I suggest an r of 5 or so for beginners.)
 
Put a "1" in each square of the left-most column
and in each square of the top-most row of the grid.

On each move the player chooses any one of the grid's
empty squares. This square is (m,n), which is the
square in the mth column from the left side of the
grid and in the nth row from the top.

The player then sums up all the integers already
written in the mth column and nth row.
So, if the grid looks like this, where the * is
the square the player has chosen to fill in next
with an integer, 
1 1 1 1
1 3
1 * 6
1 6

then the sum of the column 2's terms and row 3's
terms is 1+3+6 +1+6 = 17. Let this sum be s.

Next the player counts the number of integers in the
squares to the left of (m,n) and above (m,n)
(ie, the squares with coordinates (j,k), 1 <=j <=m,
1 <=k <=n) which are coprime to s.
If the count is c integers in the given rectangle
which are coprime to s, then the player writes c
in square (m,n).

Play continues until there is an integer in every
square of the grid.
The player's score is the number in the last square
he/she fills in.

Example game:
(r=4)
(View with fixed-width font.)

1 1 1 1   1 1 1 1   1 1 1 1   1 1 1 1 
1         1 3       1 3       1 3 5  
1         1         1   6     1   6
1         1         1         1

1 1 1 1   1 1 1 1   1 1 1 1
1 3 5 6   1 3 5 6   1 3 5 6
1   6     1   6     1 5 6
1         1 6       1 6

1 1 1 1   1 1 1 1   1 1 1 1
1 3 5 6   1 3 5 6   1 3 5 6
1 5 6     1 5 6     1 5 6 11
1 6   10  1 6 1110  1 6 1110

So the player gets 11 points.

(It seems that a good strategy is to
{unlike in my example} finish in the
lower right square, and to {as in my
example, 1+6+10+1+5+6=29} have a row
and column sum on the last move which
is a prime.)

Question: Is it always possible to get,
for an r-by-r grid (r >= 2), r^2 -1 points?

Thanks,
Leroy Quet