Running a self-written program I've found the square-less end-configurations to be listed (because of the 1000 characters limit) in a comment of this comment for the grid sizes (n) from 3 to 11.
If I understand the note in the problem text right, this proofs that there is no sure winning strategy for those sizes.
On my 1 GHz Celeron/PIII the search for n=11 took 65 seconds.
But I gave up the search for n=12 after circa 12 hours of calculation estimating the 130-fold duration for the complete search. On the other side: In the first 8 hours there was a square-less configuration just before the occupation of the last field.

## Drawn games for grid sizes from 3 to 11

Running a self-written program I've found the square-less end-configurations to be listed (because of the 1000 characters limit) in a comment of this comment for the grid sizes (n) from 3 to 11.

If I understand the note in the problem text right, this proofs that there is no sure winning strategy for those sizes.

On my 1 GHz Celeron/PIII the search for n=11 took 65 seconds.

But I gave up the search for n=12 after circa 12 hours of calculation estimating the 130-fold duration for the complete search. On the other side: In the first 8 hours there was a square-less configuration just before the occupation of the last field.