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The fastest possible checkmate

Starting from the normal initial position, what is the smallest number of moves in which checkmate can occur? (Count a move as one turn by White followed by one turn by Black.) Who delivers it, and what are the moves?

Then: what is the fastest checkmate that White can deliver?

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The only piece that can reach the enemy king quickly is the queen, and she needs a diagonal to be opened by a pawn move. Which pawn moves open the diagonal to the king, and which open the diagonal from the queen?

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