Sum Of Products, Number Of Numbers Game

Here is a game for 2 players played using an n-by-n grid drawn on paper.

Players take turns. On a turn a player writes any one integer, 1 to n, into any empty square of the grid.

After n^2 total turns, the game is over.

Player 1 gets the sum of the scores for each row. The score for a row is (the number of 1's in the row) * (the number of 2's in the row) * (the number of 3's in the row) *...(the number of m's in the row), where m is the largest integer such that all integers 1 through m occur in that row.

Player 2 gets the sum of the scores for each column. The score for a column is (the number of 1's in the column) * (the number of 2's in the column) * (the number of 3's in the column) *...(the number of m's in the column), where m is the largest integer such that all integers 1 through m occur in that column.

A player gets 1 point for a row/column if there are no 1's in that row/column.

The player with the largest score wins.

And example game:

1 1 2 1 2 5
1 3 2 1 3 5
5 1 1 1 3 1
1 2 1 2 1 2
2 1 1 2 1 1
2 3 5 3 4 1

Player 1 gets:
3*2 + 2*1*2 + 4 + 3*3 + 4*2 + 1*1*2*1*1
=
33 points.

Player 2 gets:
3*2 + 3*1*2 + 3*2 + 3*2*1 + 2*1*2*1 + 3*1
=
31 points.

Player 1 wins.

Thanks,
Leroy Quet