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Over here it was previously discussed whether one can end up with all nine queens.

There is a variant of chess called Crazyhouse in which captured pieces may be reintroduced as the capturer's. In theory, then, a player may have two queens and sixteen pawns which all may be promoted.

Is it possible to get eighteen queens without checkmating before that, or else putting the opponent into a draw?

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If that is the goal of both players, should be easy to achieve. Simply move weak side king away from his 1st rank and build a wall of minor pieces between him and the passed pawns. With a little more effort a player with a huge lead could force the king out and achieve the same thing.

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