Instructive Rook Endgame

The position shown was reached yesterday (Round 7 of the San Sebastian) between GM Granda and GM Nakamura. There is a very instructive winning solution posted at the Daily Dirt Chessninja by Mig Greengard. With his acknowledgement, I would like to post it here for our readers with some extra notes. More than the variations the ideas to win are very nice. Lets study this in a top down approach breaking it from highest level ideas to broad techniques and then concrete variations. Here are the broad themes about the position.

1. Restriction of black rook . If the black rook were not on the a file (or if it were to leave without giving a check), then white wins by moving his own rook along the 8th rank (say Rb8), protecting the queening square a8 and promoting a8=Q next move.
2. Restriction of black king . If the black king were to move to a checkable square on the 6th rank (f6 or h6 -- not possible in this position though), then white wins by checking Rf8+ or Rh6+ or similar, and then queening a8=Q next move.

3. Exchanging to winning pawn endgame. White is currently up a pawn, but imagine an equal material king and pawn endgame shown in next figure. This pawn endgame is winning as white can win the e4 pawn. The reason is white can attack e4 from two squares d5 and d4, but black can defend only from f5 and will fall into zugszwang (...Kf6 Kd6 Kf5 Kd5 g5 Kd4 g4 Kd5 Zugswang). Also the black king is too slow to abandon the e4 pawn and win the h2 pawn. Hence it is critically important that the white pawn remains on h2.
4. Making the h pawn a passer From the first initial position, if white king can win the e4 pawn, then he can exchange his f pawn for black's g pawn. After this, white has a second passed pawn on the h file.

So with the above themes, it turns out that black cannot prevent all of these. Now we can note that, without themes 3 and 4, it is difficult for white to win since black rook will never leave the a file and if white king tries to come for support of the a pawn, then the black rook gives infinite checks and goes back to a file. So lets look at some techniques to achieve themes 3 or 4 (keeping in mind black has the key restrictions of 1 and 2 that he cannot fall out of).

1. If the white king can come to e6, then pawn e7 is lost as it cannot be defended.
2. The only way to reach e6 is through Kf4-e5-e6. But Kf4 leads to Rxf2+ (with check). But we know from the second figure that this is OK as long as white can win e7 and exchange rooks. So this gives us the first concrete variation 40. Kf4 Rxf2+ 41. Ke5! (aim is to win e7 by going Ke6) Ra2 42. Ke6 Ra6+ 43. Kxe7 Ra1 44. Kd7 Ra2 45. Kc6! . It is good to note here that without the e7 pawn, the blacks king is forced to remain at h7 and g7. If it leaves the 7th rank, we have theme 2 in play. If it remains on 7th rank but goes to f7 then we have tactic (Rh8! threatening to promote Rxa7 and Rh7+ skewer and black rook on a7 is lost). So continuing after 45.Kc6! Ra1 (...Rc1+ Kd5 and white picks up e4 after which two passers will win. White has made great progress and will leave it to reader to figure remaining out) 46. Rc8 (Idea is to exchange rooks) Rxa7 47. Rc7+ Rxc7+ 48. Kxc7 * and we have reached position 2.
3. If black does not take pawn on f2 and tries to stop white kings march to e6, then e4 will fall. This gives us second variation 40. Kf4 Ra4 41. Ke5 Ra6 (cutting of king and stopping Ke6, but cant defend e4) 42. Kxe4. Now we have achieved theme 4. Now white will advance his f pawn up, trade f for g pawn and have the h2 passer. Atleast white has made significant progress since the initial position. I shall leave it to the reader to win that position.

4. The last variation is if black tries to defend e6 with the King. 40.Kf4 Ra4 41.Ke5 Kf7 (instead of previous ...Ra6 attempt). The difference here is black has not captured on f2 and hence now he can play 42.h4! (Zugzwang). If black moves rook somewhere else on a file, he loses e4. If black moves king, he allows Ke6 after which similar king and pawn end game is reached (see last figure) after taking on e7 and exchanging rooks. And here although white pawn is on h4 and black can win it, BUT, white still has f2 pawn and hence he wins. Will leave it to the reader to solve the variations.

In the game GM Granda missed Kf4 and went for h4 instead trying to reach theme 3, but for that his pawn must be at h2. So he only obtained a draw.