Today we are going to look at queen vs. rook, an endgame that is trickier than it seems at first glance. The good news is that the player with the queen (White here) always wins except in some particular exceptions.
Let's start from the position below. The first thing we want to do is to bring the king towards one of the side of the board.
1.Qf3+ Ke5 2.Qe4+ Kd6 3.Kd4 Rc6 4.Qe5+ Kd7 5.Kd5 Rc7 6.Qe6+ Kd8 (diagram). Now, 7.Kd6? would be a terrible mistake, as Black has a stalemate idea, 7...Rc6+! and after 8.Kxc6 the game is a draw.
Therefore, White wants to move his queen far away. 7.Qg8+ Kd7 8.Qg7+Kd8 9.Qf8+ Kd7 10.Qf4! (diagram). This move poses Black in zugzwang (a lack of useful moves).
Clearly, Black's does not want to part from the rook unless he is forced to do so, 10...Kc8 11.Kd6 Kb8 12.Qe5 Rb7 13.Kc6+ Ka8. Now that we brought the king to the corner, we have to put Black in zugwzwang again, 14.Qa1+ Kb8 15.Qa5 (diagram).
Finally, Black's king is forced to part from his rook (as 15...Kc8 loses for 16.Qa6). He could try 15...Rb8 but it does not really matter, he is lost now. 16.Qd8+ Ka7 17.Qe5+ Ka8 18.Qh8+ Ka7 19.Qh7+ (diagram) and 20 moves after the initial position, the rook is lost!
Remember, (1) first, bring the king to the side and then to the corner [be careful to avoid stalemate]. Then, (2) put Black in zugzwang and once the rook is far from the king, (3) calculate a sequence of checks with your queen to capture it.
Have a good training!
Below, I provide an example of exception. Black moves.
Because of the unfortunate position of White's queen, the game is a draw after 1...Rh7+ 2.Kg2 Rg7+ 3.Kf3 Rf7+ (diagram). Now it's clear why the queen is placed in a bad place. White cannot go on the e-file because of Re7 pinning the queen.
Therefore, he has to try another way, 4.Kg4 Rg7+ 5.Kf5 Rf7+ 6.Kg6 Rg7+ 7.Kh6 Rh7+! (diagram) the saving idea. White has nothing better than 8.Kxh7 with stalemate.
Good luck with your endgames!
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