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Imagine playing tic tac toe, but rather than the standard 3 by 3 grid, the board extends indefinitely in every direction. When playing the usual game, one player must get three squares in a row to wind the game. However, with an infinite grid, three in a row become pointless, as the first player is guaranteed to be able to get two in a row with nothing on either end. This would allow for a win on the next turn regardless of what the second player does. This could be fixed if the required consecutive squares was increased, such as to four in a row. What would be a winning strategy for this rule set? What about for five in a row? Is there a point where it becomes impossible to win given perfect playing?

Michael T
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PiGuy314
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    This seems to also be Connect-4 on an infinite grid (this might help with searching). What have you tried? 4-in-a-row seems to be easily possible. – Michael Burr Nov 11 '21 at 18:11
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    $4$ in a row still looks like an easy win for the first player. Not sure what length stops that, or even if there is such a length. – lulu Nov 11 '21 at 18:11
  • Some useful references might be found here – lulu Nov 11 '21 at 18:14
  • Fascinating. Is there a proof for 4 in a row, or maybe 5? Anything would be appreciated. – PiGuy314 Nov 11 '21 at 18:14
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    And here is an app that lets you play the $5$ in a row version. For $4$ in a row, I expect you could just break down all possible early move strategies. Should be "easy" but tedious. In practice (a few minutes with pencil and paper) it is always effortless. – lulu Nov 11 '21 at 18:16
  • That is very cool. – PiGuy314 Nov 11 '21 at 18:21
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    In fact, even in "five wins" on an infinite grid , the first player has a winning strategy. For "four wins" , I think, it is easy to find all cases without computer help. – Peter Nov 11 '21 at 18:37
  • Any answer of this sort would be likely accepted, especially if it contained a proof of some kind. – PiGuy314 Nov 11 '21 at 18:39
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    The stealing argument at least shows that the second player cannot have a winning strategy. Intuitively, there should be a number for which the game ends in a draw with perfect play, but a proof of this won't be easy. – Peter Nov 11 '21 at 18:42
  • Very true. Thank you for all of this wonderful information. : ) – PiGuy314 Nov 11 '21 at 18:45
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    The case "5-in-a-row" on an infinite grid is considered on page 738 of the excellent "Winning Ways for Your Mathematical Plays", Volume 3, by Elwyn R. Berlekamp, John H. Conway, Richard K. Guy (exists as a Google book) – Jean Marie Nov 11 '21 at 20:58

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(m,n,k)-game: get $k$ in a row on a $m\times n$ board. There is a wiki page that talks about this exact problem: https://en.wikipedia.org/wiki/M,n,k-game

"Computer search by L. Victor Allis has shown that (15,15,5) is a win"

"$k \geq 8$ is a draw on an infinite board... It is not known if the second player can force a draw when k is 6 or 7 on an infinite board."

So we know that for $k\leq 5$ is a win for the first player, for $k\geq 8$ is a draw, and for $k = 6,7$ we don't know.

dgadjov
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