The strategy doesn’t seem, however, to prevent a column win for the first player, but rather ensures that any such winning configuration is duplicated for the second player, forcing the draw. If the first player always plays on one of the paired columns, and you copy on the other, then both will make the winning configuration.

Incidentally, I think you probably mean $3m+1$, $3m+2$, instead of $2m+1$, etc. I’m not sure if users can edit comments here, but if you are not able to edit, I think I can change this for you.

]]>Maybe the players can force draw already when it is sufficient to arrange k number of coins in a row, where k is finite? If the board is Z^2 without gravity and the winning configuration is a 2×2 square, there is an easy proof that both players can force a draw (arrange cells in pairs like bricks, if the first player plays X, the second player plays its pair). Similarly, if the winning configuration is a vertical or horizontal line of five. I think this is also the case for a vertical or horizontal or diagonal line for some k, but I do not remember the details. With gravity it could be similar.

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