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Black to play - Draw (instead of Fine's White wins)

After showing a draw with White to move, Fine states, "If, however, Black must move, his King must go away, when g6 wins". Actually, Black draws with 1... Ke4! 2.g6 hxg6+! 3.Kxg6 Kf3! = Other tries don't help White. For example, 2. Kh6 Kf5! = and White has made no progress.

Analysis using computer generated endgame tablebase