This is an exceptionally instructive study I found in GM Paul Motwani’s book, C.O.O.L. Chess

1.Kf6 If 1.d4 Kf8 2.d5 (2.Kd7 g5 3.Kc7 g4 4.d5 g3 5.d6 g2 6.d7 g1Q 7.d8Q+) 2...Ke8 and the game is drawn as White can't afford the black pawn running and by capturing, it would lose his own . 1...Kh7 2.d3!! Awful would be 2.d4 g5 3.Kxg5 Kg7 4.Kf5 Kf7 5.Ke5 Ke7 6.Kd5 Kd7 and Black keeps the opposition and draws! 2...Kh6 And now you get a demonstration of why the move 2.d3 was so brilliant. It saves a tempo and allows White to gain the opposition, which is not available with 2.d4. Compare with the g5 line above. 2...g5 3.Kxg5 Kg7 4.d4 Kf7 5.Kf5 Ke7 6.Ke5 Kd7 7.Kd5 Ke7 8.Kc6 Kd8 9.Kd6 3.d4 g5 4.d5 g4 5.d6 g3 6.d7 g2 7.d8Q g1Q 8.Qh8# This is why studying king and pawn endings, and especially the concept of opposition, is so critical to being good at chess.

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