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LiChess puzzle 82753 involves a rook and pawn endgame where white sacrifices their rook to queen a pawn. The final position is the following (with black to move):

[FEN "2Q5/6p1/1K3pkp/8/8/4p3/r5PP/8 b - - 0 44"]

I'm not convinced white can win from here; it looks like black can swap off one pair of pawns, then set up a fortress. This seems to hold on against LiChess's version of Stockfish, but I don't think it looks especially deep. Maybe there's a winning strategy for white, though, that it didn't find.

Question: Can white win with a queen and two pawns vs. rook and four pawns?

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  • 1.b8Q+ looks like an easy win. – bof Jan 9 '19 at 8:36
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    I think the board is the other way around (there is no b pawn). I can’t see how the FEN is rendering from my devices. – Rebecca J. Stones Jan 9 '19 at 8:45
  • The puzzle had a solution – HelloWorld Jan 9 '19 at 9:06
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    Yes, this question is about the final position in the solution. – Rebecca J. Stones Jan 9 '19 at 9:10
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It looks very difficult for white to prevent the fortress.

The obvious sequence is something like -

[FEN "2Q5/6p1/1K3pkp/8/8/4p3/r5PP/8 b - - 0 1"]

1. ... Rb2+ (to kick the king one square further from the action) 2. Ka5 Rxg2 3. Qe8+ Kh7 4. Qe4+ Rg6 5. h4 h5 and black has 3 extra pawns, albeit the e3 pawn is likely to fall soon

If 5. Qf5 then 5... h5 anyway when 6. Qxh5 Rh6 leaves white struggling to save the h pawn.

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I think it's a draw. Don't have an access to an engine right now though 1... Rxg2 2. Qe8+ Kf5 if 3.Qxe3 Rxh2 I don't see a way to win black's rook immediately. Even if black can't capture the h2 pawn it doesn't seem that difficult for him to regroup and set up a fortress. 3. Qh5+ Rg5 leads to an immediate draw.

Missed that after Kh7 black can block Qe4+ with Rg6 as given by Brian. That variation is completely drawn.

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Yes, the endgame is winning for White. Take the position after these first moves:

  1. Qe8+ Kh7 2. Qxe3.

Note here that 2...Rxg2 allows White to win the Rook with the fork: 3.Qe4+

The endgame is now just 2 pawns vs 3 pawns. You assumed that Black can swap off a pair of pawns, but this is easier said than done. If Black pushes their kingside pawns up the board, the king will be exposed and the fortress weakened. White's queen can also help to prevent any unwanted exchanges.

So Black's best strategy to defend is to put the rook on e5, where it prevents White's king from crossing the e-file and reaching the kingside. However, notice how the rook must stay on e5 to accomplish this. If it moves anywhere else on the e-file, it's liable to be forked. Note also that it's not easy for Black to even bring the rook to the e5-square in the first place. But let's assume Black manages.

Therefore, White's winning strategy consists of putting Black in zugzwang. The ideal arrangement of White's pieces:

  • King on d6 or d7.
  • Queen on f8.
  • Pawns on h4 and g4

And then Black's pieces will be as follows:

  • King on h7.
  • Rook on e5.
  • Pawns as they are in the diagram position (moving just hurts Black's defensive chances).

If you set this position up on a board, you'll notice how Black's king can only move to g6. But then White plays Qg8 and Black's king is stuck in a vulnerable position.

Black's only other option is moving the rook somewhere else along the e-file, such as 1...Re1. But then White strikes with 2.g5!, since the Rook is no longer on the 5th rank preventing this push. After 2...fxg5 3.hxg5 hxg5 4. Qf7, Black's rook no longer has a protected square it can sit on along the e-file. This means it won't be able to prevent White's king from crossing the e-file and reaching the kingside much longer.

Once White's king crosses the file, it's easy enough to checkmate Black's king.

EDIT: For Black to move in the starting position:

  • On 1...Rxg2 2.Qe8+ Kg5 3.Qxe3+ Kh4 (...Kg6 or ...Kf5 lose the rook to a fork), Black can't really set up a fortress.
    • Placing the rook on g5 can eventually lead it to being attacked when White pushes h4 with the help of the queen.
    • Meanwhile, putting the rook on e5 leaves the g7-pawn undefended, since the king is stranded on h4. If White takes the g7-pawn, Black's fortress falls apart due to the weak f6-pawn.
  • On 1...e2 2.Qg4+ Kh7 3.Qe4+ Kh8 4.g4 Kg8 5.h4, White is going to round up the e2-pawn by bringing the king to b3. If Black's rook moves to d2, White plays Qe3 and Black's rook is unable to stay on the 2nd rank. White's queen then takes the e2-pawn, and see my main answer for how to break the subsequent fortress w/ rook on e5.
  • Other first moves for Black don't do anything special.
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  • 4
    Maybe I should have been clearer about this, but it’s black to move. – Rebecca J. Stones Jan 9 '19 at 9:15
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    Updated my answer for that case. – Inertial Ignorance Jan 9 '19 at 9:23
  • Let's say we reach the zugzwang position you mention and then we go for the line 1...Re1 2.g5 fxg5 3.hxg5 hxg5 4.Qf7, how does white win after 4...Rh1? Black is planning to go Rh6-f6 and if that happens it's a dead draw. Meanwhile, I don't see how white stops this simple plan from black's side without giving perpetual check, but that is hardly a win. – Scounged Jan 9 '19 at 21:01
  • After 1...Rxg2 2.Qe8+ the bK doesn't have to go for thre hills, it can go 2...Kh7 and if 3.Qe4 Rg6 probably reaching a fortress soon. – Evargalo Jan 10 '19 at 13:45

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