Gomoku is the game where Black and White take turns placing stones of their own color, and the winner is the player who first gets five of their own stones in a row. Black moves first.
In Gomoku on an infinite big board, I made the following claim:
If Black can win on a $15\times 15$ board, he can certainly win on an infinite board, since he can pretend to be playing on a $15\times 15$ board, and if White plays outside the imaginary boundary, so much the better for Black.
I am no longer happy with this argument. Here is a counterargument:
Suppose Black pretends to be playing on a $15\times 15$ board, making all his moves inside the imaginary boundary of a $15\times 15$ region. If White has the opportunity to make any five moves outside the imaginary boundary before Black wins inside the boundary, White will win first. Thus if White can find a strategy $S$ for the $15\times 15$ board that delays Black's win long enough for her to pass five times, then she can win on the infinite board by playing $S$ against Black, and using her pass moves to play outside the imaginary boundary.
My idea is that Black's winning strategy might take a long time to execute, say 103 moves, and perhaps pass moves by White only allow Black to speed up his win by a moderate amount; say by passing at the correct times, White can still force Black to take 37 moves to win. Then despite Black's winning strategy for the $15\times 15$ board, White can win on the infinite board if Black does not defend outside the $15\times15$ region.
One of the two arguments must fail. Right now, I think it's the first one that fails. My questions are:
Is there some reason that I've overlooked that the notional winning strategy $S$ for White can't exist?
If there isn't, does such a strategy exist?
Is there any way in general to convert winning strategies on smaller boards to winning strategies on a larger boards?