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Too hard for me, but Lomonosov gives a mate in 32!!
I must say that Rh7 was difficult to find (and I had to check it with a computer). Nice puzzle, Susan!
Hear, hear Mr Carlsen!
My instinct tells me Rh7 has to be the first move. What white wants to do is to bring the king over to support the forward h-pawn, but to do this white must find a way to shelter the king from long distance checks- this is particularly troublesome when the passed pawn itself can't be used to do so. Just to see the problem, let's look at what happens if white takes at d6 first:1. Kd6 Ra4!This is certainly the only drawing move for black- black must not give white time to put the king on the 7th rank before he begins the checks. To do otherwise allows white to move the rook over to shield the king, or to protect the pawn from the side should black return the rook to h4. For example: [1. ...Rb4?? 2.Kc7! Ra4 3.Rb8! Ra7 4.Rb7 Ra3 5.h7 Rh3 6.Kd8 and the king will walk over to the h-pawn and get a shield from the rook at g7 when needed], or in this line [3. ...Rh4 4.Rb6 with the same basic structure, but with the pawn moved one square closer to the first rank- still won for white]. So, let's continue from 1. ...Ra4 above:2. Rc8Planning to shield the king....2. .....Rh4And white has to return the rook to h8 for no progress. Or, at move 2....2. Ke7Getting the king over to guard the pawn.2. .....Ra7 (or Ra6)3. Kf6 Ra6!And now it should be clear that white cannot escape the checks without walking the rook down on the a-file- neither his rook nor his pawns offer a haven, so....4. Ke7 Ra75. Kd6 Ra66. Kc7 Ra77. Kb6 Rf7 (Or Re7/d7)Planning to check from the other side now, and white cannot move the rook to c8 to shield now...8. Rc8 Kh2 (Rf6 now loses to Rc6) The black king gets off his butt to aid the defense:9. Rc7 Rf6 (a tempo ahead now)10.Rc6 Rf7!11.h4 Kh3 and I will stop here- the white king is out of the game, the forward h-pawn is stopped by the rook on the seventh rank, and the backward h-pawn is about to run out room to move- both pawns are going to fall to the rook/king combo.In my next comment, I will try to show the effectiveness of moving the rook to h7 on the first move.
In my previous comment I outlined. in a general way, the problems with taking at d6 on the first move. Let's look at why 1.Rh7 might be superior:1. Rh7 Rh3 (alternatives later)2. Kd6! Ra3Planning checks from the a-file as in my previous comment:3. Rc7!And now hopefully the difference is obvious. In my previous comment where white had taken at d6 on the first move, the analogous line (with the white rook on c8 and the black rook on a4), black simply returned the rook to the h-file forcing the white rook back to h8 for no progress, but now white can advance the pawn to h7 to protect it:3. .....Rh34. h7! Rh6 (what else now?)5. Ke5 Kg2 (Rh5 6.Kf6)6. Rf7Allowing the king to go through e6-e7-f8-g8 to support the pawn if necessary:6. .....Kg37. Kf5 and black is basically out of plausible options- even a duffer should win this ending now.So, let's back up to black's first move and see if there is any plausible defense:1. Rh7 Kh22. Kd6 Rh3 (alternatives later)3. Ke5 Rf3 (Ra3 4.Rd7)The idea of Rf3 is to cut the white king off from supporting the pawn, but this is doomed, too:4. Rg7 Kh3 (what else different?)5. h7 and the pawn can't be stopped.Now, at move 2 in this variation, black can take at h3 with the king:2. .....Kh33. Ke5 Kg44. Rg7!The only winning move. The black king does best to retreat, but we have already seen similar positions lose for black, so...4. .....Kh55. h7 and black is mated on the next white move.Now, back at the first move:1. Rh7 d52. Kd5 Rh33. Ke5 and this is no different than what we have seen before.Obviously, I haven't covered every single black move at move 1, but I don't see how most of them could possibly be better given white's basic plans after 1.Rh7.
Nakamura could break Carlsen's nose (that's what Lomonosov means in Russian) with Mendeleyev's table :)
Rd8 preventing queening of the d pawn
Sound study. In the main line, White's third and fourth move are just incomprehensible for me so far. I hope Yancey will explain me the whereabouts of this rook ending, because it is way beyond my skills.Complete sources after the right answer is posted.
I have to rethink my comment from yesterday. I overlooked something pretty serious.
I was definitely hasty yesterday in declaring this won for white in the following line:1. Rh7 Ra4To get a firmer grasp on this problem, I have verified that moves like 1. ...Rh5, 1. ...Rh3, 1. ...Kh2, 1. ...Kg1/g2 all lose to 2.Kd6 by consulting the 6 man Nalimov Tablebase. However, I found that of the plausible rook moves along the 4th rank, only 1. ...Rc4+ and 1. ...Re4 lose to 2.Kd6, while black can draw with the rook moves to a4, b4, f4 if white continues with 2.Kd6. Let's look at the losing move of 1. ...Rc4:1. Rh7 Rc42. Kd6Now, black has no other immediate check from the side of the white king, only the check from d4 is available. Black can return the rook to h4, or he can play the rook to a4/b4 to threaten a lateral check- first, let's look at Ra4:2. .....Ra43. Rc7!To shield the king as I discussed yesterday. Now black can check from a6, but after white blocks with Rc6, black will be unable to keep the white king from marching over to support pawn who is now protected by the rook from the side. The other defense is going to be returning the rook to h4:3. .....Rh44. h7And now the rook can never leave the h-file unless it is to check the white king. Black can, of course, check the white king from the h-file:4. .....Rh65. Ke5Heading towards the rook and the pawn. If black doesn't bring the king forward, white will eventually put the king on g8 and block any check from the lower ranks with Rg7. So...5. .....Kg26. Kf5 and surely I don't need to continue along this line- black is totally lost now. I think I covered this yesterday, though my notes are a jumbled so I am not certain. Black loses in a similar way if he plays 1. ...Re4 in this line above. So, we need to deal with the lines 1. ...Rf4, 1. ...Ra4, and 1. ...Rb4I will discuss those in greater detail than I did yesterday in my next comment.
So, let's look at the following line:1. Rh7 Ra42. Kd6Since my comment from yesterday hasn't yet appeared, I am not really certain whether I even covered this line. So, black will harass the king now:2. .....Ra6!3. Kc5What else can white do but try to walk the rook down. There is no shelter to be found on the kingside from the lateral checks from the a-file. Continuing:3. .....Rf6With the idea of continuing the harassment from the other side. Also, it keeps white's rook tied down to the h-file to protect h6. Can the white king successfully walk the rook down again?4. Kd5 Ra6So, I don't see any way for white to make progress in this line, so if 1.Rh7 is the right first move, then after 1. ...Ra4, white must find a different continuation than 2.Kd6.Now yesterday, I think I discussed playing 2.Rb7, but I am not certain any longer. From the top:1. Rh7 Ra42. Rb7 Ra6 (alternatives later)3. Rb6 Ra3 (what else?)4. Kd6 and this is pretty clearly won for white now. So, let's look at the 2nd move alternatives in this line for black:1. Rh7 Ra42. Rb7 Rh43. h7I am pretty sure I discussed this line, and I am pretty certain I thought white had a win, but I overlooked a line. I think black loses to 4.Kd6 on every single move here but 3. ...Rh6, a move I completely overlooked yesterday:3. .....Rh6Now, here, my instinct tells me white should play Rg7 to keep the king on the h-file. At this point I can see a potential theme of a skewer at some point from h1, the idea being to limit black's options for moves:4. Rg7Now, if the black rook moves, white either queens the h-pawn or wins at d6. Black can push d5 here at move 4, but white captures and that ending is also clearly won. In addition, the rook moves Rh5/h4/h3 all lose to 5.Kd6. So, black must play Kh2 now:4. .....Kh2So, now what does white do? My very first question is, does white have time to run the king to g8, then cover d1 with the rook?5. Kd7Puts the question back to black of what to do. Again, the rook moves of Rh5/h4/h3 all lose to 6.Kd6. Black has available the moves of Kh1, Kxh3, and d5. If black plays Kh1, white will just continue the march to g8 to support the forward h-pawn as he will if black plays Kxh3. Let's look at the latter of these moves before dealing with the pawn push of d5:5. .....Kh36. Ke7 Kh4 (for d5 see later)7. Kf7 Kh5 (any better?)8. Kg8 and this is clearly won for white since Rf6/e6 won't help since h8Q will occur with check. From move 5, black must try pushing the h-pawn for counterplay:5. .....d56. Ke7 d4 (in for a penny...)7. Kf7 d38. Kg8Threatens to win the rook by playing h8Q. Black can play d2 here, but white plays 9.Rd7 followed by h8Q, and moving the black rook over to a6/b6 for skewer won't help since white just queens the pawn anyway and then plays Kh7 guarding her. Black is lost here.I think black must find another defense on the first move #1. Right now I am still puzzling over them.
So, let's see if black can successfully defend with 1. ...Rf4:1. Rh7 Rf42. Kd6?I already know this is drawn, but let's see why it is drawn to reinforce the point:2. .....Rf63. Ke5 Ra6 (Rg6,b6,c6 ok too)4. Kf7 Ra75. Kg6 Ra6 and the king will find no shelter, and if the king walks over to the rook, black transfers the rook out of danger always on the sixth rank to keep white's rook tied down to defending h6.So, from the top of the line:1. Rh7 Rf42. Rg7White must use the opportunity to get the rook off the edge of the board. Rg7 just looks logical to me as it traps the black king on the h-file to limit black's options. Continuing:2. .....Rf6 (alternatives later)3. h7 Rh6And we saw in a previous comment that white wins with 4.Kd7 with the threat of marching the king to g8 and queening the pawn. So, in this line, let's back up to black's second move:2. .....Rh43. h7 and we saw this position earlier as well.Right now, unless someone can show me where this is wrong, black cannot draw with either Ra4, or Rf4. Now let's look at 1. ...Rb4:1. Rh7 Rb42. Rb7 Rh4 (what else now?)3. h7 Rh64. Kd7 which we know white wins.Ok, have I left anything out, or am I just missing something here?i
One last item I forgot to include earlier (because I skipped over a variation by simply verifying white won in the Nalimov Tablebase). One might ask why black must threaten lateral checks instead of trying to threaten checks from the lower ranks 2 and 3. One interesting thing is that the starting position without the pawn on h3 is drawn, and it is drawn for precisely that reason- without the pawn, the black rook can play to h2 on the very first move and the king can't find shelter after taking at d6- the point being that the white king cannot walk the rook down on the g and h-files since the rook is protected by his king, and if the king walks the rook down on one of the other files, the rook transfers to the sixth rank to make sure white's rook doesn't get off the h-file. However, with the pawn on h3, the black rook can get no lower on the board than the third rank, and will undergo a deadly attack that allows white the time he needs to guard the h6 pawn and the tempo he needs to get the rook off the h-file:1. Rh7 Rh32. Kd6! Rd3 (else white plays Rg7)3. Ke5 Re34. Kf4 Now black has a problem. There are no safe checks on the white king, and the white rook is going to get off the h-file unless black plays Re6 or Rh3:4. .....Rh35. Kg5Again threatening to move the rook, which requires black to check the white king:5. .....Rg36. Kh4And this is the critical difference between having the rook on the second rank and the third rank (with the king on h1/g1)- the rook is unguarded. To keep the white rook tied down, black must play Rg6...6. .....Rg67. Kh5! Ra6 (for lateral checks)8. Rf7 and white will win with ease from here. Or at move 4:4. .....Re65. Kg5Threatening get the rook off of the h-file.5. .....Re56. Kf6 Rh57. Kg6 and white has protected the pawn and gained the time to move the rook off the h-file.
For those still following at this late date, I think maybe I have finally found the main line, or at least the line of most resistance for black:1. Rh7 Rh32. Kd6I discussed this position in a couple of the comments above, but the critical comment is the one at 5:16 PM CST where I discussed the problem black has in not having the rook protected when it is checking on the g and h files, however, I left one idea unexplored simply because I overlooked it. Continuing from 2.Kd6 above:2. .....Kh2Connecting the rook and king. I simply overlooked that black could move the king up to h2/g2, and I overlooked that black could also move the rook to h2. I will cover those alternatives, too. Continuing from 2. ...Kh2 above:3. Ke5A difficult position to assess even if you spend inordinate amounts of time trying to understand rook endings. It is probably worthwhile to see what fails for white here in order to understand why this is the correct move (not the only move that wins, but is the key move). Let's look at 3.Kc5 which only draws for white:Variation with 3.Kc53. Kc5 Kg3!This is the only move that draws for black. I hope it will be clear why other black moves fail here, but for construction of an argument, I think it better to just show how black draws from this point. Continuing:4. Rg7Using the opportunity to get the rook off the edge and to support the pawn from the side. White could bring the king back to d5, but black would just continue to march the king forward to g4. Continuing:4. .....Kf4 (Kf3 ok, too)5. h7 Kf5Black has other drawing moves like Ke5 and Rh6 (which cuts the white king off from crossing the sixth rank for the moment). In any case, it should be obvious that black is going to double attack the h-pawn, and the white king is unable to assist now. One of the points of 3.Ke5 is to keep that black king out of the game. Now, let's return to the main line. Continued in my next comment.
Back to the Main Line With 2. ...Kh21. Rh7 Rh32. Kd6 Kh23. Ke5What should black do here? Does it help to bring the king forward?3. .....Kg34. Rg7Again, using the opportunity to get the rook off the edge and out of front of the pawn. Of course, now the black king cannot go to f4 (one of the points to 3.Ke5). If the black king goes to h4, he will lose very quickly as white pushes h7 and will support the advance to h8 with a subsequent Rg8 if necessary. The longest defense possible here is to put the king on f2, keeping the third rank clear for the black rook:4. .....Kf25. h7 Rh6There is no better defense. White can easily maneuver the king to g8 otherwise. Continuing:6. Kf5 Ke3 (nothing better)7. Kg5 Rh18. Kg6Heading for g8. Continuing:8. .....Rg1 (if Kf4, 9.Rg8+-)9. Kf7 Rh1 (if Rf1, 10.Kg8+-)10.Kg8 and it is over.So, we need to discuss black's alternatives at move 3 in the main line.1. Rh7 Rh32. Kd6 Kh23. Ke5 Rh54. Kf6 Rh3To keep the white rook on the h-file and planning to harass the white king to keep him from supporting the pawn while white transfers the rook off the h-file. However, white is now in a position to deal with this with a counterintuitive move:5. Rh8!Not the only move, but a required move at some point. For the rook to shield the white king from checks along the third rank, the white king would have to go to g8. For example, if white plays the king to g7, black transfers the rook to the queenside and will harass the king endlessly while ensuring to transfer back along the sixth rank as I discussed in an earlier comment. Let's just see how 5.Rh8 solves white's problems now, even though getting the rook off of h8 was important at move 1. Continuing from 5.Rh8 above:5. .....Ra3Planning the lateral check strategy:6. Rg8 Ra6 (Rf3 7.Kg7)7. Kg7 Ra78. Kh8 Rb7 (nothing holds)9. h7 Re710.Rd8 Re3 (so Rg3 if Kg7/8)11.Kg7 Rg312.Kf6 Rf313.Kg5 Rg314.Kf4 and white wins.Now, the question to ask is, why did the lateral check strategy black employed at move 5 in this line fail here when it succeeds in the line where white has played 1.Kxd6? There, the white rook was also on h8, and the forward h-pawn was still on h6, too. The line in question is the following:1. Kd6 Ra4!And I showed that the plan to shield the king by playing 2.Rc8 fails to the simple 2. ...Rh4 forcing the white rook back to h8 for no progress. The analogous move in this present line would be 2.Rg8, but that fails to the simple 2. ...Ra6-in the winning line above, the white king is on f6 instead of d6 and this skewer along the 6th rank is no longer an issue.Ok, now back to the 3rd move alternatives for black in the main line:1. Rh7 Rh32. Kd6 Kh23. Ke5 Re34. Kf4Keeping the pressure on black's rook. Continuing:4. .....Rh3 (Re6 5.Kg5 Re5 6.Kf6)5. Rh8!Again, not the only winning move, but the required one. White can't effectively put pressure on black with 5.Kg4 since black can check from g3 safely. Continuing:5. .....Rh5 (nothing better now)6. Kg4 Rh37. Kf5 Rf3 (Ra3 8.Rg8 like above)8. Kg6 Rg39. Kh7The other point to the counterintuitive move of a later Rh8. It should be clear that this is won for white now.
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