Rook vs. Bishop: Ending 1
The task was left unstated: White to move and...? One might reasonably wonder if White can win this, given Black's two passed pawns, but in fact he can. All he has to do is avoid the move actually played in the game Hanken-Fries, USA 2004:
This seems logical, but it's actually very bad - Black can now draw even without his pawns! As Pal Benko explains in his "Endgame Lab" column in the March 2005 Chess Life (page 46), "All we have to do is chase away the king with check when it steps either to e6 or g6. In case the pawn is pushed to f7, we have to play either Ke7 or Kg7 [DM: which square is appropriate depends on the location of the White king - Black doesn't want to allow the White king to protect the pawn] and take the pawn only afterwards. Therefore White needs the f6-square for his king to win..."
Therefore, White should have played 60.Rg6+ Kf7 61.Rh6 Be2 62.Rh7+ Kg8 63.Rd7 h4 64.Kf6 h3 65.Kg6 and wins (Benko).
Of course, given what we know from the Benko quote, 60...Kf8 led to a simple draw. Now the Black king gets cut off from the f-pawn, so White's winning chances rise dramatically, though by sacrificing the h-pawn (in order to return the Black king to its proper defensive post on the f-file) the draw is still available. Nevertheless, since Black rejected that idea when it didn't cost anything, it's unlikely he'll reconsider at the cost of a pawn.
61.Rb7 h4 62.Rg7+ Kh6 63.Rg4 Bf7?
63...Kh5? lost to 64.Rxc4 (64...bxc4 65.f7), but Benko rightly notes that 63...h3! draws, as 64.Rh4+ Kg6 65.Rxh3 Kf7 allows the king to return to his roost.
64.Rxh4+ Kg6 65.Rf4??
A blunder, and we all know why at this point: it allows the Black king to return to f7! 65.Rg4+ followed by 66.Rg7 was a very easy win, but some days, nothing seems to go right.
Vacating f7 for the king, right?
Wrong. Even with the threat of 67.Ke7 hanging over his head, when the pawn clearly queens or costs Black the bishop, Black STILL avoids ...Kf7.
It's easy for us to look at this and feel superior, feel Schadenfreude, wonder how they could be so slow, etc. Even Benko expresses his exasperation, asking rhetorically after Black's 66th move "Does Black want to lose?" Really though, there's just one relevant idea, and neither player got it. White didn't fear the Black king's reaching f7/f8, and Black had no interest in its reaching those squares. Presumably both thought the king would be in a mating net if it became stuck on the back rank, but neither realized that without the White king's safely reaching e6 or g6, there's no mate to be had. Thus, since Black (with correct play) can check the White king away the instant he reaches either of those squares, the Black king is safe.
So let's be smart and learn from others' mistakes: the strong side is typically best off with his king leading the pawn, as that facilitates the crucial process of driving the defender away from the queening square. The goal is to cut the defender off from the queening file (or in some cases, to cut it off from the pawn horizontally, though that's rarer).
Conversely, the defender wants to stay in front of the pawn and to prevent (if possible) the strong side's king from getting in front of the passer. In such cases, the position is rather like an opposite colored bishop ending: the defensive side has a very strong grip on the squares of one color, and despite the strong side's superior firepower, it's basically impotent to break the blockade: the pawn covers the wrong-colored squares, and when the king tries to help fight for that color complex (light squares in our case), he gets checked off immediately.
This is useful, but we can learn even more by playing around with the position (without the Black pawns, perhaps - at least at first). Try moving everything over a file or two either way, or down a couple of ranks. Will it make a difference? Which side, if any, benefits from the changes? Are certain pawns harder for one side to handle?
Further, we can reflect on what we've learned here for more complicated positions. We can see that the rook is relatively impotent to break the blockade by itself, and can perhaps start to think about the implications of exchange sacrifices in positions where the bishop's side has nearly full control over one color complex and not too many worries about squares of the opposite color. And how much leeway does it provide? One way to proceed is by adding pawns for each side. If the result with the new material is a draw, then add a pawn to White, or at least a further pair of pawns for each side. If the result, on the other hand, is a loss for the bishop's side, give him another pawn.
By playing around like that (remember the Cycle World post?), you'll learn something about rooks and bishops. It won't be some sort of dull theoretical ending you're trying to learn from a book (not that there's anything wrong with that!), but something fun, something you'll have taught yourself. Better yet, it won't be some sort of isolated chess factoid, but a case of genuine know-how with applications extending well beyond the initial exercise.
As they used to say when I was a kid: Try it, you'll like it!