Conjecture: If one player is in a winning position (will win with optimal play) and opponent plays in a hex, lets call it X, there exist a move at another hex Y that is winning such that Y lies somewhere in the area spanned by all possible paths between opponents edges that uses the hex at X.

While I'm at it, I also like to see a proof that the templates below is valid in its generalization of larger sizes.

o

o . .

o . . . . o

o . . . . o . . . .

o . . . . o . . . . . . . . o

. . . . o . . . . . . . . o . . . . . .

How about it, can this be prooved?

Ok, not very successful “pictures” of templates since all double spaces were removed… here is a picture:

There is a simple counterexample: You have a winning position and your opponent plays inside a region which is completely surrounded by your stones. You do not stipulate that the opponent must play well.

Ok, you are right. Corrected conjecture:

Assume one player is in a winning position (will win with optimal play) and opponent plays in a hex X. Let the set A consist of all empty hexes that are members of any path between opponents edges that uses the stone at X. If A is non-empty, A contains a winning move. Otherwise any move are winning, even passing the turn.