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I always used to play Gomoku in school on paper, and if we reached the edge of the field, we just put another one at that side.

And now I just saw that black can always win on 1 15x15 board. But what is about the way I used to play it? Is black also winning there, if both players play perfectly?

MJD
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Platurt
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2 Answers2

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There is discussion of this topic in volume 3 of Winning Ways (Berlekamp et al. 2003, pp 740–741). They discuss the general case of $n$-in-a-row on infinite boards. A strategy-stealing argument shows that each game is either a draw or a first-player win. It should be clear that if $n$-in-a-row is a first-player win, then so too is $m$-in-a-row for $m<n$, so the only real question is what is the largest $m$ that is a first-player win. They provide a proof that 9-in-a-row is a draw, and say “T.G.L. Zetters… recently showed that the second player can even draw 8-in-a-row”.

MJD
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  • Didn't thought that well about it, yeah ofc black can win then... but what about all this 9-in-a-row or 8-in-a-row... what is the board if you say that it is a first player win or a draw? Infinite board? – Platurt Oct 23 '14 at 14:11
  • @platurt I now think I was wrong about Black being able to convert his $15\times15$ strategy into a strategy for the infinite board; details at Strategy-stealing in Gomoku – MJD Oct 28 '14 at 02:35
  • This is an unsolved problem according to Beck's 2008 book on combinatorial games, see my answer. – bof Oct 28 '14 at 10:53
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This question seems to be an unsolved problem. On p. 60 of József Beck's book Combinatorial Games: Tic-Tac-Toe Theory, he states the following problem concerning "unrestricted $5$-in-a-row" (Gomoku on an infinite board):

Open Problem 4.1. Is it true that unrestricted $5$-in-a-row is a first player win?

bof
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