Kevin, 

Again, this is off the main topic of the specific endgame in question.  I agree that computers still have difficulty playing certain endgames well.  However, if they can force a position that they have in their databank, they play flawlessly.  Also, just ask Mickey Adams, Vladimir Kramnik, or Garry Kasparov, and I think you will find that they all believe computers have improved their endgame play dramatically over the last few years and are now as good as any human in most endgames.

Woofwoof,

The easy question first:  The Lucena position is the basic winning position for the side that has a rook, a pawn on the seventh, and controls the queening square with his king.  It involves Nimzovich's famous "bridge-building technique" between the King on the 8th rank and a spot on the 5th rank (as white) which will be protected by the rook on the fourth rank.  Just about every general endgame book, "My System" and most beginners' books have this position somewhere.   

(Imagine having a King on d8, Pawn on d7, Rook on e1 vs. K on f8 and Rook on c2.  White to move wins by forcing the Black king further away (1. Rf1+! Kg7 2.Rf4!! Rd2 3.Ke7 Re2+ (and to hide from the checks...) 4.Kd6 Rd2+ 5.Ke6 Re6+ 6.Kd5! Rd2 7.Rd4 and White wins.  Notice, if White had reached the same position without chasing the Black king away, Black could have drawn by taking the Rook and playing Ke7.)

The main reason most programs play at about the same strength after a certain point regardless of the handicap setting is that either a) they have the position in an endgame hash, making the play nearly perfect or b) the program's handicap feature has already been used in the middlegame (for instance, Chessmaster personalities that value Bishops over Rooks doesn't matter in a R+P ending).   

There is another common handicap that programs use, which is to limit the time it thinks.  This is done either by  making it use very little time on its move, or by not running its analysis engine on your time.  These methods will allow you to find deep refutations if there are any, but the computer will still usually catch one- and two-move combinations.  The problem with the endgame is that the refutation in an endgame is often not purely tactical in the first place, so the computer makes the move it would "naturally" make even if it was given longer to analyse.   

Of course, there are many computer experts online here who could give you much more information than this, but I hope this gives you a general (and generally accurate) answer. 8)

Kevin, 

Thank you for providing the analysis here.   I will take some time in the next couple of days to try to break it down.   However, as you know, very long lines of analysis such as you have provided (while necessary in this case) are particularly suspect.  I will focus my attentions on three specific areas:  a) whether Gurevich-Glek line is actually won; b) Dvoretsky's 4...Ra1; and c) the most thematic idea for Black, 4...f6.  I think these three are the most likely places for Black to find serious improvements.  If you happen to find anything, let us know!

Smyslov_Fan: Did you check out the suggested line in "Endgame Corner"? Here is a copy/paste of that analysis:
"The position in Kantorovich's study (30.02) from
1989 occurred in the game Tal-Gipslis 1983. Tal
played 1.Ra8 and agreed a draw after 2 more
moves. I would like to show now that the
position has been wrongly estimated until now,
as it is won for White: 1.Kd4! Rxf2 2.Rc7 Ra2
3.a7 Kf5 4.Kc4!! (Kantorovich gives only
4.Rxf7+ and ends with =) 4...Kg4 5.Kb3! Ra5,
and we have reached the game Gurevich,MGlek,
I, Vlissingen 2002, which was won by
White instructively: 6.Rc4+! Kxg3 7.Ra4 Rxa7
8.Rxa7 Kxh4 9.Kc3! Kg3 10.Kd2! g5 11.Rxf7,
etc
If this is correct then Black should avoid the Kantorovich position as it is probably
also lost with Black to move, e.g., 1....g5 2.Kd4! gxh4 3.gxh4 Ra5 (3...Rxf2 4.Rc7
Ra2 5.a7+-) 4.Kc4 Ke6 (4...Ke5 5.Kb4 Ra2 6.f4+!) 5.Kb4 Ra2 6.Kc5+-. But this
requires further investigation.
For a safe draw, Black should play 1...g5 in time instead, when White has the set up
Pa5, Ra7: 2.hxg5 (2.Kd4 gxh4 3.gxh4 Rxf2=) 2...Kxg5 3.a6 Kg4=. So Black's
defence is even more difficult than originally thought."
Mark Dvoretsky found further defensive ressources, but Steckner managed to deal
with them:
1.Kd4 Rxf2 2.Rc7 Ra2 3.a7 Kf5 4.Kc4!
A) 4...Kg4 5.Kb3+- as we have already seen.
B) 4...Ra1 (Dvoretsky,M) 5.Kb5! Rb1+ 6.Kc6 Ra1 7.Kb7 Rb1+ 8.Kc8 Ra1
9.Rxf7+! (9.Kb8? Kg4 10.Rxf7 Kxg3 11.Rf6 Kxh4 12.Rxg6 Kh3 13.Kb7 Txa7+
14.Kxa7 h4=) Kg4 10.Rg7 Kxg3 (10....Kh3 11.Kb8+-) 11.Rxg6+ Kxh4 12.Kb7!
The difference, saving an important tempo (12.Kb8? Kh3= would transpose to the
line 9.Kb8 ...) 12....Rxa7+ 13.Kxa7 Kh3 14.Kb6+-
C) 4...f6!? (Dvoretsky,M) The most interesting try. 5.Kb5! Rb2+ 6.Kc6 Ra2 7.Kb7
Rb2+ 8.Kc8 Ra2 9.Rg7! (9.Kb8? Kg4 10.Rc6 Kxg3 11.Rxf6 Kxh4 12.Rxg6 Kh3=)
and I do not see how Black can save himself:
C1) 9....Kg4 10.Rxg6+ Kh3 11.Rg7 Ra3 (11....f5 12.Kb8 Rb3+ 13.Rb7 Ra2/Re2
both rook moves can be answered by 14.a8Q or 14.Rb3, winning in both cases)
12.Kb8 Rb3+ 13.Rb7 Rxg3 14.Kc7! Ra3 15.Rb3+.
C2) 9....g5!? 10.Kb8 gxh4 (10....Rb2+ 11.Rb7 Ra2 12.a8Q Rxa8 13.Kxa8 Kg4
14.Rg7!+-; 10....Kg4 11.a8Q Rxa8 12.Kxa8+-) 11.gxh4 Ke4 12.a8Q Rxa8 13.Kxa8
f5 14.Rg5+-
C3) 9....Ra1 10.Kb8+-"

Woofwoof: 1. Ra8 is not the right way, as mentioned in the analysis above. Computers will not play this position correctly. As an example of how bad computers are at endings sometimes, I gave Fritz a position where it could prevent me from reaching lucena position in 1 move, and it didn't find it. And this position is much more complicated than lucena.

Wow.   

I've looked at this position and ones similar to it (with different k-side structures) all week and I still don't know what's going on.  I think I could draw as White and lose as Black.  

Having said that, there seems to be a pretty general consensus in the endgame books I've seen (Dvoretsky, Levenfish and Smyslov, Flear, Keres, Speelman, Mueller and others) that Black should draw this.   

I have had experience with this twice, once with the extra pawn and won (even though I thought my opponent could have drawn) and another being a pawn down against a 2300, and drawing it.  After that last game, my opponent and another master insisted I was dead lost, but IM Ibragimov gave it a cursory glance and declared it drawn.   

I still have no idea what's going on.  I'm gonna go get a rum-and-coke and read the new Harry Potter book to clear my head. 8)

U can find this exact endgame position and similar ones analysed on chesscafe :


http://www.chesscafe.com/text/dvoretsky39.pdf  and


http://www.chesscafe.com/text/dvoretsky38.pdf

So am I. Can someone explain me, why 1.f3 is wrong? I don't get it. The idea of course, is to play Klick's line in a better version.