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How can be it proved that tic-tac-toe on an infinite grid (winning with 12 in a row, a column or a diagonal) can always end in a tie (with optimal strategies of both players)?

There is a hint: to use a "magic square of 4x4" and a "tessellation".

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Tic-tae-toe on an infinite grid can never end in a tie. Presumably you mean that neither player has a winning strategy. –  Peter Taylor Apr 15 at 10:39
@PeterTaylor I define tie as "not making 12 in a row, a column or a diagonal", so the infinite play is a tie. You are right about the inappropriateness of the verb "end", though. –  John Smith Apr 15 at 18:42
There's a proof in one of the later volumes of Winning Ways that the 9-in-a-row game is a draw on an infinite board. A fortiori the 12-in-a-row game is too. –  MJD Aug 26 at 14:10

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