It has been, shall we say, a little while since
part 3 was written; in fact, I wouldn't be surprised if only Steve B. (sbb1cpa) remembers this series, and by now even he might have forgotten. It all goes back to this position (see the
original post and its successors), from the game Karim-Zhu Chen, Gibtelecom Masters 2008:
Despite the even material and opposite-colored bishops, White is better and even managed to win. When I first replayed the game (unannotated
here; the endgame above starts after Black's 60th move), I was almost shocked by the result. Upon taking a closer look, my opinion changed, but not completely. It seemed on good old-fashioned general principles that such an endgame ought to be a draw. Over time, my opinion changed, though
proving a win is also quite difficult.
I offered this ending as an exercise to the readers, and they came through beautifully (and hopefully learned a lot in the process). I too learned something from the exercise - and from my readers - and would like to offer several general principles and provisional conclusions (to put it oxymoronically).
(1) An obvious reminder: opposite-colored bishop endings
can be won, even when the material is even and there are no passed pawns. (Of course, White starts with a passed pawn here, but the winning tries all involve giving it away.)
(2) A slightly less obvious corollary: opposite-colored bishop endings can be drawn, even when the weaker side is down a pawn or two.
(3) As in many other endings, and perhaps in chess in general, it's rarely possible to win without utilizing the "principle of two weaknesses". (A "weakness", in this context, refers to something the defender has to protect or defend: an enemy passer or a weak pawn, for example.)
(3a) If the defender's can stop multiple passers on a single diagonal, then those passers may only count as
one weakness.
(3b) Sometimes the "weaknesses" are static, like distinct passed pawns, but sometimes they're more dynamic, like potential king penetration. (As we'll see, both sorts of weakness pairs come into play in our ending.)
(4) Let's turn to specifics. When I was first trying to win this endgame against the computer, I found ways to break through with the king to the kingside, winning the h6 pawn doesn't necessarily win, even if the White king can find its way back out. (Example: White retains his a-, c-, g- and h-pawns, Black his a-, c- and g-pawns. As long as Black's king can get to h8 it's a draw: the bishop protects the g-pawn when it's attacked or goes to b4 if White's king journeys cross-country to the queenside.)
(5) Likewise, White can sometimes win the c- and even the a-pawn, and still not win! (You can find such a line in the first main analysis paragraph of the second game file linked at the end of this post.) The basic drawing setup is this: Black's king is on d6 and his bishop is on the a7-g1 diagonal. In this way, the pawns are stopped and White's king can't get to Black's kingside. If White's king goes to d3, Black plays ...Ke5. If White's king goes to a6, Black plays ...Kc7.
(6) The winning method, then, seems to be this: White must win a queenside pawn, but do so in a way that doesn't involve giving up the chance to occupy the d5 square. That square is key for White, because it allows him to play c5 (it's often important for White's c-pawn to break that barrier)
and because it gives his king access to Black's kingside pawns. And then, and perhaps only then, Black's defensive resources will be overtaxed. His bishop can control the c-pawn, but his king can't both deal with White's inevitable passed h-pawn and with White's king returning to the queenside to further advance the c-pawn.
You can find some specific analysis
here. The first game is a compilation of the readers' analyses, with an occasional comment of my own interspersed. The second game presents a bit of my own analysis of 61.d6. I present two independent paragraphs of analysis, both starting with that move. The first shows White failing to win, despite winning both queenside pawns, but the second presents a White win, utilizing the method outlined in point (6).