user-1: Why did Carlsen resign in the position shown below? (blitz game played in the London Chess Classic GCT Finals)
Can't white help the pawn queen after Ke3 and Kd3 then Kd4? I must be missing something.
[Title "Magnus Carlsen-Maxime Vachier-Lagrave, London Chess Classic GCT Finals, London England 12/02/2019"]
[FEN ""]
[startply "108"]

1. d4 Nf6 2. Nf3 g6 3. Nbd2 d5 4. b3 Bg7 5. Bb2 a5 6. a4 O-O 7. e3 Bf5 8. Be2 Nc6 9. O-O Nb4 10. Ne1 Ne4 11. c3 Nxd2 12. Qxd2 Nc6 13. Nd3 e5 14. Nxe5 Nxe5 15. dxe5 Bxe5 16. Rfd1 Qd6 17. Ba3 c5 18. Qxd5 Qxd5 19. Rxd5 Bxc3 20. Rad1 Bb4 21. Bxb4 cxb4 22. Bc4 Be6 23. Rb5 Rfd8 24. Rd4 Rac8 25. Rxd8+ Rxd8 26. Kf1 Bxc4+ 27. bxc4 Rd2 28. Rxb7 Ra2 29. c5 Rxa4 30. Ke2 Ra3 31. c6 Rc3 32. c7 Kg7 33. Ra7 b3 34. Rxa5 Rxc7 35. Rb5 Rc2+ 36. Kf3 b2 37. Rb6 h5 38. h3 Kf8 39. g4 h4 40. g5 Ke7 41. e4 Kd7 42. Ke3 Kc7 43. Rb5 Kc6 44. Rb8 Kc5 45. f4 Kc4 46. f5 Kc3 47. Kf4 Rg2 48. fxg6 fxg6 49. e5 Re2 50. Rb6 Kc2 51. Rc6+ Kd1 52. Rd6+ Kc1 53. Rc6+ Rc2 54. Rb6 Rc4+


user21810: Imagine there'd be no other pawns on the board, except for the b2 pawn and the e5 pawn. Then ask yourself the question: is black's position won? If the answer is yes, there is no need for calculation, right? 
The answer is yes, because of an elementary (yet crucial) manoeuvre: keeping the black rook on the fourth rank, thereby cutting off the enemy king. This is seen very often in rook vs. pawn endings and important to bear in mind. 
Explanation
So black promotes the pawn and white plays Kd3 after. Then black goes Ra4. After e6 black plays Ra6 and white can not defend the pawn and thus loses it.

user15543: While it may appear that White can whisk around Black's rook and promote, there is a indeed problem that you have missed: Black can promote first!
White's king is currently in check from Black's rook, so the monarch must be moved to either the f3 or e3, cutting the king off from half of the board.
Since White lost a tempo moving their king, Black now promotes their passed b-pawn to a queen, threatening a variety of mating sequences. White must give up their rook for the newly born queen, a practically forced move. After Black recaptures, they are a full rook up. White has a passed pawn, but it will be soon be taken down by Black's superior firepower, cementing a total loss for White and a victory for Black.
Here is a sample game, Stockfish generated at Depth 53, that shows how Black will be able to systematically suppress and restrict the White king to eventually win.
[FEN "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

1. Kf3 b1=Q 2. Rxb1+ Kxb1 3. e6 Rc6 4. Kg4 Rxe6 5. Kxh4 Re4+ 6. Kg3 Kc2 7. Kf2 Kd2 8. Kf3 Kd3 9. Kg2 Ke3 10. Kg3 Ke2 11. Kg2 Re3 12. h4 Re4 13. Kg3 Ke3 14. h5 gxh5 15. Kg2 Rg4+ 16. Kh3 Rxg5 17. Kh4 Kf4 18. Kh3 Rg4 19. Kh2 Ke3 20. Kh1 Kf2 21. Kh2 Rh4#

I hope that this helps you understand the endgame better!

user19877: In short, the plan you proposed is possible, but it is just too slow, even if black allows it, which is not mandatory. If black permits it, here is a simple win that, although I checked it with a computer, I could see it in my head fairly easily, which means those two could see it that much more easily.
 [FEN "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

1. Ke3 b1=Q 2. Rxb1+ Kxb1 3. Kd3 Rc8 $1 4. Kd4 Kc2 5. Kd5 Kd3 6. e6 Ke3 7. Kd6 (7. Ke5 Re8 8. Kf6 Kf4 9. Kf7 (9.Kxg6 Rxe6+ 10. Kh5 Kg3-+) Ra8 10. Kxg6 Rg8+ 11.Kf7 Rxg5 {and Re5 next.}) Kf4 8. e7 Kxg5 9. Kd7 Ra8 10. e8=Q Rxe8 11. Kxe8 Kf4 12. Kf7 Kg3 13. Kxg6 Kxh3 14. Kf5 Kg3 15. Ke4 h3 16. Ke3 h2 17. Ke2 h1=Q $19

As I was checking my line with a computer, another line it suggested contained a common valuable theme in some rook endings where the king is cut off on a rank, but the pawn is not advanced enough: If the pawn moves forward, you can attack it with the rook, it advances, and then you can get behind it, and take it the next move.
After 3...Ra4!, there are only two plans: To move the king around until black gets close to g2 with the king, or to advance the pawn, which shows the common theme I mentioned.
 [FEN "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

 1. Ke3 b1=Q 2. Rxb1+ Kxb1 3. Kd3 Ra4 {What can white do here? There are only two plans: Move the king back and forth until black brings the king to g2, or to advance the pawn eventually.} 4. e6 (4. Ke3 Kc1 5. Kf3 (5. Kf2 Re4 $19) 5... Kd1 6. Ke3 Ke1 7. Kf3 Kf1 8. Ke3 Kg2 $19) 4... Ra6 5. e7 Re6 $19


user3594: Intuitive plan
Notice that black's last check is pushing the white king one row away from their passed e-pawn. Once black promotes the b-pawn, white will have to give up the rook for it. After which white's king is simply not in position to cover the advance of the e-pawn and therefore additional tempi will have to be spent with king moves to try and reach the pawn, during which the rook can first simply capture white's h-pawn and only then sacrifice itself for the e-pawn. Then black's h-pawn promotes and wins trivially. Now let's see how all this pans out concretely (no matter how strong the intuition might be, in endgames one must always primarily rely on concrete calculations in order to realise whether our ideas or plans can actually be optimally implemented).
Annotated diagram
 [title "A simple continuation showcasing why white is lost here"]
 [fen "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

 1. Ke3 b1=Q 2. Rxb1+ Kxb1 3. Kd3 (3. e6 {blunders the e pawn away} Rc6 4. e7 Re6+) Ra4 4. Ke3 {again, the pawn advance would simply leave it hanging, so white has to make king moves while not allowing black to get behind the pawn} Ra3+ 5. Kd4 Rxh3 6. e6 Rh1 7. Kd5 h3 8. e7 Re1 9. Kd6 Rxe7 {though black doesn't even have to sacrifice the rook here, but it's simply a practical way of going about it, and in-line with our original plan} (9...h2 {much more accurate continuation as we can in fact keep our rook since white's tactically losing their queen immediately upon promotion, in fact you might find this line easier to calculate} 10. Kd7 h1=Q 11. e8=Q Qd5+ 12. Kc8 Rxe8+) 10. Kxe7 h2 {and it's easy to judge that black's winning here since the promoted queen is well in time to get to white's promotion square g8 and then simply bring the king over. I'll include this sequence} 11. Kf6 h1=Q 12. Kxg6 Qb7 {cutting off the king from accessing the 7th row} 13. Kf6 (13. Kh6 Qf7 14. g6 Qg8) Qh7 14. g6 Qg8 {and now black simply brings the king and wins the pawn}


-1: <p>Why did Carlsen resign in the <a href="https://www.chessgames.com/perl/chessgame?gid=1983200" rel="nofollow noreferrer">position</a> shown below? (blitz game played in the London Chess Classic GCT Finals)</p>

<p>Can't white help the pawn queen after <code>Ke3</code> and <code>Kd3</code> then <code>Kd4</code>? I must be missing something.</p>

<pre><code>[Title "Magnus Carlsen-Maxime Vachier-Lagrave, London Chess Classic GCT Finals, London England 12/02/2019"]
[FEN ""]
[startply "108"]

1. d4 Nf6 2. Nf3 g6 3. Nbd2 d5 4. b3 Bg7 5. Bb2 a5 6. a4 O-O 7. e3 Bf5 8. Be2 Nc6 9. O-O Nb4 10. Ne1 Ne4 11. c3 Nxd2 12. Qxd2 Nc6 13. Nd3 e5 14. Nxe5 Nxe5 15. dxe5 Bxe5 16. Rfd1 Qd6 17. Ba3 c5 18. Qxd5 Qxd5 19. Rxd5 Bxc3 20. Rad1 Bb4 21. Bxb4 cxb4 22. Bc4 Be6 23. Rb5 Rfd8 24. Rd4 Rac8 25. Rxd8+ Rxd8 26. Kf1 Bxc4+ 27. bxc4 Rd2 28. Rxb7 Ra2 29. c5 Rxa4 30. Ke2 Ra3 31. c6 Rc3 32. c7 Kg7 33. Ra7 b3 34. Rxa5 Rxc7 35. Rb5 Rc2+ 36. Kf3 b2 37. Rb6 h5 38. h3 Kf8 39. g4 h4 40. g5 Ke7 41. e4 Kd7 42. Ke3 Kc7 43. Rb5 Kc6 44. Rb8 Kc5 45. f4 Kc4 46. f5 Kc3 47. Kf4 Rg2 48. fxg6 fxg6 49. e5 Re2 50. Rb6 Kc2 51. Rc6+ Kd1 52. Rd6+ Kc1 53. Rc6+ Rc2 54. Rb6 Rc4+
</code></pre>
 21810: <p>Imagine there'd be no other pawns on the board, except for the b2 pawn and the e5 pawn. Then ask yourself the question: is black's position won? If the answer is yes, there is no need for calculation, right? </p>

<p>The answer is yes, because of an elementary (yet crucial) manoeuvre: keeping the black rook on the fourth rank, thereby cutting off the enemy king. This is seen very often in rook vs. pawn endings and important to bear in mind. </p>

<p><strong>Explanation</strong></p>

<p>So black promotes the pawn and white plays Kd3 after. Then black goes Ra4. After e6 black plays Ra6 and white can not defend the pawn and thus loses it.</p>
 15543: <p>While it may appear that White can whisk around Black's rook and promote, there is a indeed problem that you have missed: Black can promote first!</p>

<p>White's king is currently in check from Black's rook, so the monarch must be moved to either the f3 or e3, cutting the king off from half of the board.</p>

<p>Since White lost a tempo moving their king, Black now promotes their passed b-pawn to a queen, threatening a variety of mating sequences. White must give up their rook for the newly born queen, a practically forced move. After Black recaptures, they are a full rook up. White has a passed pawn, but it will be soon be taken down by Black's superior firepower, cementing a total loss for White and a victory for Black.</p>

<p>Here is a sample game, Stockfish generated at Depth 53, that shows how Black will be able to systematically suppress and restrict the White king to eventually win.</p>

<pre><code>[FEN "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

1. Kf3 b1=Q 2. Rxb1+ Kxb1 3. e6 Rc6 4. Kg4 Rxe6 5. Kxh4 Re4+ 6. Kg3 Kc2 7. Kf2 Kd2 8. Kf3 Kd3 9. Kg2 Ke3 10. Kg3 Ke2 11. Kg2 Re3 12. h4 Re4 13. Kg3 Ke3 14. h5 gxh5 15. Kg2 Rg4+ 16. Kh3 Rxg5 17. Kh4 Kf4 18. Kh3 Rg4 19. Kh2 Ke3 20. Kh1 Kf2 21. Kh2 Rh4#
</code></pre>

<p>I hope that this helps you understand the endgame better!</p>
 19877: <p>In short, the plan you proposed is possible, but it is just too slow, even if black allows it, which is not mandatory. If black permits it, here is a simple win that, although I checked it with a computer, I could see it in my head fairly easily, which means those two could see it that much more easily.</p>

<pre><code> [FEN "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

1. Ke3 b1=Q 2. Rxb1+ Kxb1 3. Kd3 Rc8 $1 4. Kd4 Kc2 5. Kd5 Kd3 6. e6 Ke3 7. Kd6 (7. Ke5 Re8 8. Kf6 Kf4 9. Kf7 (9.Kxg6 Rxe6+ 10. Kh5 Kg3-+) Ra8 10. Kxg6 Rg8+ 11.Kf7 Rxg5 {and Re5 next.}) Kf4 8. e7 Kxg5 9. Kd7 Ra8 10. e8=Q Rxe8 11. Kxe8 Kf4 12. Kf7 Kg3 13. Kxg6 Kxh3 14. Kf5 Kg3 15. Ke4 h3 16. Ke3 h2 17. Ke2 h1=Q $19
</code></pre>

<p>As I was checking my line with a computer, another line it suggested contained a common valuable theme in some rook endings where the king is cut off on a rank, but the pawn is not advanced enough: If the pawn moves forward, you can attack it with the rook, it advances, and then you can get behind it, and take it the next move.</p>

<p>After 3...Ra4!, there are only two plans: To move the king around until black gets close to g2 with the king, or to advance the pawn, which shows the common theme I mentioned.</p>

<pre><code> [FEN "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

 1. Ke3 b1=Q 2. Rxb1+ Kxb1 3. Kd3 Ra4 {What can white do here? There are only two plans: Move the king back and forth until black brings the king to g2, or to advance the pawn eventually.} 4. e6 (4. Ke3 Kc1 5. Kf3 (5. Kf2 Re4 $19) 5... Kd1 6. Ke3 Ke1 7. Kf3 Kf1 8. Ke3 Kg2 $19) 4... Ra6 5. e7 Re6 $19
</code></pre>
 3594: <h2>Intuitive plan</h2>

<p>Notice that black's last check is pushing the white king one row away from their passed <code>e-pawn.</code> Once black promotes the <code>b-pawn,</code> white will have to give up the rook for it. After which white's king is simply not in position to cover the advance of the <code>e-pawn</code> and therefore additional tempi will have to be spent with king moves to try and reach the pawn, during which the rook can first simply capture white's <code>h-pawn</code> and only then sacrifice itself for the <code>e-pawn.</code> Then black's <code>h-pawn</code> promotes and wins trivially. Now let's see how all this pans out concretely <em>(no matter how strong the intuition might be, in endgames one must always primarily rely on concrete calculations in order to realise whether our ideas or plans can actually be optimally implemented).</em></p>

<h2>Annotated diagram</h2>

<pre><code> [title "A simple continuation showcasing why white is lost here"]
 [fen "8/8/1R4p1/4P1P1/2r2K1p/7P/1p6/2k5 w - - 0 1"]

 1. Ke3 b1=Q 2. Rxb1+ Kxb1 3. Kd3 (3. e6 {blunders the e pawn away} Rc6 4. e7 Re6+) Ra4 4. Ke3 {again, the pawn advance would simply leave it hanging, so white has to make king moves while not allowing black to get behind the pawn} Ra3+ 5. Kd4 Rxh3 6. e6 Rh1 7. Kd5 h3 8. e7 Re1 9. Kd6 Rxe7 {though black doesn't even have to sacrifice the rook here, but it's simply a practical way of going about it, and in-line with our original plan} (9...h2 {much more accurate continuation as we can in fact keep our rook since white's tactically losing their queen immediately upon promotion, in fact you might find this line easier to calculate} 10. Kd7 h1=Q 11. e8=Q Qd5+ 12. Kc8 Rxe8+) 10. Kxe7 h2 {and it's easy to judge that black's winning here since the promoted queen is well in time to get to white's promotion square g8 and then simply bring the king over. I'll include this sequence} 11. Kf6 h1=Q 12. Kxg6 Qb7 {cutting off the king from accessing the 7th row} 13. Kf6 (13. Kh6 Qf7 14. g6 Qg8) Qh7 14. g6 Qg8 {and now black simply brings the king and wins the pawn}
</code></pre>
