Unique Sum Game

Here is another game. It it inspired by my recent game, "Prime Sums
Game",
but with a little less math (primes not an issue here).
For 2 players.
(Actually, the game can be adapted for any number of players.)
Start with an n-by-n grid drawn on paper. (like in so many of my games)
Players alternate writing the integers IN ORDER 1, 2, 3,..., n^2
into the empty squares of the grid.
So, for a 2-player game, one player writes odd integers,
the other writes even integers.
Players try to avoid having the sum {of the integer they are
writing} and {of any of the adjacent squares' integers
(in the direction of left, right, up, or down)} from equaling
the sum of any pair of already-written adacent integers.
So, for example,
if we have the grid:
. 7 6 4
1 . 3 .
2 . . 5
we would not want to put the 8 next to the 1 because there is
already the adjacent pair 3 and 6 in the grid, and 1+8 = 3+6.
So, if a player, player A, writes down an integer which, when
summed with an adjacent integer, gets a sum which exists somewhere
else in the grid, it is up to player B to notice this.
If player B notices that player A is making a sum which already
exists, then player B wins the game.
If player B does not notice, then play continues as if the
duplicate sum was not made.
And if the grid is filled in completely without anyone noticing a
duplicate sum, the game is a tie.
What would be a good strategy for this game?
thanks,
Leroy Quet