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I always find finishing off my opponent tricky in nine mens morris. If they have 3 pieces and can fly, how do I guarantee a mill to secure the win? What's the minimum number of pieces I can do this with?

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up vote 6 down vote accepted

You can't guarantee a win. In a 1996 paper called Solving Nine Men's Morris, Ralph Gasser describes how he used search techniques to determine that, given best play, the game of nine men's morris should end in a draw.

In figure 11 of that paper, he gives a graph of the percentage of won positions for the player to move, against the number of stones each has on the board. The highest percentage of won positions is 83% when the player to move has 3 stones against 3.

The data also reveals that a player with 3 stones left cannot force a win against an opponent with 6 or more stones left. Although many of those positions should still end in draws, rather than wins for the player with more stones.

A player is much more likely to be able to force a win when they have 7 or more stones left, than when they get down to 6. A player with 4 stones against 3 can only force a win if it is their go, and they can immediately make a mill.

All of this assumes two players who are playing completely optimally, which is not usually the case. In 1990 six games were played between the British champion Mike Sunley and a heuristics-based computer program. Two games were drawn and four were won by the computer.

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There is a winning opening: Any corner of any quadron. If opponent does not play same quadron, you follow the corners of the quadron until you have a double mill opportunity. If opponent plays same quadron, proceed to next quadron with same strategy. If opponent plays same quadron all 3 times, there is almost always a way to connect existing pieces to a double mill opportunity, unless opponent knows that, in which case he can block.

You are now way above beginner level.

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