First with 30.Re5! (threatening Qf6 & mate-to-follow by blocking the black queens diagonal access to f6).

I think you are right. The move to Kg2 instead of Kh2 wins for white.

I'll respond to your comment under yesterday's game, below.

That's an interesting task... sort of a question for a computer chess programmer trying to fine tune his engine.

h6 is a very non-intuitive move IMO. I certainly don't have a method for moving it up the decision tree... I'm impressed h6 was played, I think the fact that it's correspondence chess had a lot to do with it... its a very accomplished tactician who spots this move with a clock ticking in his ear... to push a pawn rather than capture a piece... and ordinarily this kind of pawn thrust is being hurled at the castled position in order to open it up rather than lock it down. h6 is an 'exception to the rule' sort of move, exactly the sort of thing that makes a good puzzle.

<Today was particularly tough, because the other candidates <almost> work.> Ah There's today's advantage ... I'm too stupid to make those other lines _almost_ work... they all failed miserably for me, good thing I didn't spot the resources you did or I'd have never found h6 ;)

About using the dark-squares to find h6... ahem... well, it goes like this:

My actual decision making process made generous use of the ultra-BS "it's a puzzle" decision tree! ... I looked at the position and thought, "It's a puzzle but I can't see one sac that looks productive... I sorta want to take the bishop but if the pawn takes the bishop what kind of puzzle solution would that be?. ah... but if the pawn _doesn't_ take and instead pushes... now that would make me sit up in my chair and say, hoo-boy what a puzzler of a move... let's see if it works... hey hey it works!!! ...you know now that I think about it, it occurs to me black is weak on the dark squares... If I'd seen that earlier it might have helped me solve this puzzle in a less utterly non-chess BS way.. ah well, it always sounds smart when people talk that "weak on the dark squares" stuff, better mention it in my post about this puzzle..."

I found 28.h6 very unintuitive, too, so thanks for the reassurance. <dzechiel> also had trouble, so the move is clearly anathema to "method".

To promote 28.h6 in the decision tree, perhaps <"the harmony of tactics and strategy"> (my phrase) can help. <UdayanOwen> first suggested it to me as a criterion for halting calculations. To paraphrase him, if you win at least a P and <the resulting position furthers your strategic aims>, calculation can stop.

Today's position had as its strategic theme the weak dark-squares around the Black Kg8. There were tactical features as well (the pins), but they turned out to be secondary to the strategic theme. Of my first three candidates, 28.Bxf7+, 28.hxg6, and 28.h6, only 26.h6 directly addressed the dominant <strategic> feature. The rote rule of "checks, captures, and threats" takes care of tactics, but sometimes more is required...

<So there's my method... BS + Self-aggrandizement. Oddly enough, I'm not seeking elected office.>

Then, after h6, doesn't that leave black with the defense of ...Rg6?>

27. g5 bxc3 (unless you have a better move for black) 28. h6 Rg6 29. Rf3 Qa3 30. Rxf7 Rxf7 31. Qe8+ Qf8 32. Bxf7+. Maybe it isn't faster, but it is a win.

It’s because of where white king’s locates after 28…Qc1+, a subtle but crucial difference. Black has a devious threat after 28... Qc1+, 29 Kh2 Re6, 30 Bxe6 fxe6 that he cannot play after 28... Qc1+, <29 Kg2> Re6, 30 Bxe6 fxe6, below.

click for larger view

He wants to play …Rxf2+, which he cannot now do now that white’s king is on g2 and not h2. If the king were on h2, white would have had to play Ne4 to prevent Rxf2+. This knight move allows black’s queen to play Qb2 in order to access the a1-h8 diagonal, enabling the queen to assist in the defense of his king.

Now, after 28... Qc1+, 29 Kg2 Re6, 30 Bxe6 fxe6, white continues with 31 Qe7 and black is now out of luck.

click for larger view

I think you have the line wrong. Black would not play 29...Re6?? as 30.Bxf6 fxe6 31.Qe7 win easily (e.g. 31...Rf7 32.Qe8+ Rf8 33.Qxe6+ Rf7 34.Nd5 Qb2 <what else?> 35.Nd7+ Kf8 36.Nxg6+ hxg6 37.Qe8#).

27. g5 bxc3 (unless you have a better move for black) 28. h6 Rg6 29. Rf3 Qa3 30. Rxf7 Rxf7 31. Qe8+ Qf8 32. Bxf7+. Maybe it isn't faster, but it is a win. ...>

It looks like 29...Qc1+ spoils that plan: 30.Kh2 Rxh6+ 31.gxh6 Qxh6+.

Dynamic Evaluation. Bg6 guards an otherwise open King. White's Knight is threatened. Removal of the rook on d6 and the bishop on g6 should faciliate an attack on the black King.

Abstract Assessment: Rook moves to e6 attacking d6 and g6. f x e6 opens up the f file for a possible checkmate, so that's out.

As well there is no check of the King by Bd5 as Rook would take Bxe6. Bd5-e4 would give g6 being attacked 3 times and defended three times- no advantage there. f5xg6 if taken by hxg6, Qh6,g6-g7 and not so clear cut. hxg6, Rxg6 less so. Bxf7+, Kh8, Bxg6, Qxf2+, that's a no-goer.
N-e4, attacking one of the defenders of g6, Rf-e8 Ne4xd6, Rxr, QxR. Candidate move, Ne4
28.Ne4 ...Re8 29.Nxd6 ...Rxe3 ..30.Qxe3 , my best shot

PM=

My calculation was shot down in flames by the simple 28. ...h6 I'll leave it all in, warts and all, for future reference

Analysis by Fritz 11: Time 5min.

1. (1.27): 28.h5-h6 Qb2-c1+ 29.Kg1-h2 Rd6-e6 30.Bd5xe6 f7xe6 31.Nc3-e4 Qc1-b2 32.Kh2-g2 Kg8-h8 33.Ne4-d2 Qb2-d4 34.Qg5-e7 Qd4-f6 35.Re3xe6 Qf6xf2+ 36.Kg2-h3 Rf8-f7 37.Qe7xb4 Rf7-f4

2. (0.46): 28.c4-c5 h7-h6 29.Qg5-e5 Rd6-e6 30.Bd5xe6 f7xe6 31.Nc3-e2 Qb2xe5 32.Re3xe5 Bg6-c2 33.Ne2-d4 Bc2-d1 34.Re5-e4 Rf8-f6 35.Kg1-g2 e6-e5 36.Re4xe5 Bd1xg4 37.Re5-e7