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If White begins with Q h8-h3, Black can respond by capturing the Queen with the Knight at g1. This costs White the Queen without improving White's ability to checkmate Black. Being able to capture the White Knight with the King does Black no good, since it leads, as your example showed, to immediate checkmate. When White's object is to mate in so many moves, Black's object is to prolong the game. After Q h8-h3, N g1-f3 prolongs the game, but K c3-d4 does not.