So, 56 Kf4 Re1 (....Rh5, 57 d7+ Kd8 58 Be6 ) 57 d7+ Kd8 58 Rxh7 (preventing Rf1 +) Re7 59 Rh8+ Kxd7 60 Rxb8 Kc6 and white should win.

P.S. "Second best" isn't always bad (especially in reference to chess moves), but I thought I'd share a few quotes anyway:

Dale Earnhardt -- "Second place is just the first loser."

Ljubojevic vs E Torre, 1975 (56.?)

White to play and win.

Material: B for R. The Black Ke8 has 1 legal move. The White Rf7 controls the 7-th rank, squeezing the Black Ke8. The White Pd6 is a candidate for queening. It controls the light squares complementary to Bb3, creating some possibilities for back-rank mates. The White Kg3 requires activation.

Candidates (56.): d7+, Kf4

The following variation does not quite work:

56.d7+ Kd8 57.Rf8+ Kxd7 58.Rxb8 Re3+ forking Bb3 and Kg3

The variation suggests that White should remove his Kg3 from the 3-rd rank to avoid …Re3+. Immediate interpolation of the threat 56.Kf4 is most forcing:

56.Kf4 (threatening 57.Kxe5)

<[Aware that without Ph3, the R+B vs. R endgame is a draw, I focused on the following variation]>

(1) 56…Rh4 57.Kg4 Rh6 58.d7+ Kd8

59.Rf8+ Kxd7 [Kc7 60.Rxb8 Kxb8 61.d8=Q+] 60.Rxb8

White now captures Ph7, advances Ph3, and forces the Black R to sacrifice itself to prevent h8=Q.

If White uncritically uses the idea in Variation (1) after 56…Rc5, however, Black has a trap:

(2) 56…Rc5 57.d7+ Kd8 59.Rf8+ Kxd7 60.Rxb8 Rc3!

< <JG27Pyth> wrote: I think my variation is every bit as good as the text... >

[after 56.Kf4 Rc5]
<The win is problematic for White after the premature 57. d7+ Kd8 58. Rf8+ Kxd7 59. Rxb8 Rc3 60. Bd1 Kc7 61. Rxb5 Rxh3 62. Bg4 Rc3 63. Rh5 >

57.d7+ Kd8 58.Rf8+ Kc7 59.Rxb8 ?

Yeah, if we credit the GMs with playing tablebase accuracy the premature d7 does indeed throw the win away! The follow up question here is, how difficult is the technique for holding R + B vs R to a draw? I know that Q vs. R is a theoretical win for the Q, but it's a pretty tricky technique for the Q to win if the guy with the rook knows what he's doing. So, is R + B vs R an easy draw?

I'm realizing there's a bunch of fundamental endgame knowledge I'm deficient in (*and I was so proud of myself for learning how to mate with N+B, too)

... what's the theory on N + R vs R. I imagine there's quite a bit to know about all the different unbalanced endgames... time for a crash course in the Chessgames endgame explorer.

@JohnLSpouge< <JG27Pyth> wrote: I think my variation is every bit as good as the text... >

<Sorry, <JG27Pyth>, White cannot hold both the Bb3 and Ph3, so Black draws. I would have fallen for the trap, too.>

Yes indeed, the sad reality of this has sunk in. I knew it was all too easy for Friday... but I never learn and quite reliably I just get all excited that I've found something the GMs missed. Only to be disappointed, yet again... LOL.

Of course the move that first leaps into view is 56.d7+. It's check, it's forcing, it's dangerous, what more could you want?

We then have 56...Kd8 (forced) 57.Rf8+ Kxd7 58.Rxb8 and we win a rook. Too simple for a Friday, though. Looking around for black's reply, I noticed the fairly obvious: ...Re3+ forking K+B and it looks like a draw.

What we need then -- before we play d7+ -- is a move that evades that rook fork, and yet doesn't give black time to defend against the d7+ threat. One doesn't have to look hard to find 56.Kf4!, which does exactly that. With one move, we move the king off the bishop's rank and attack a rook -- how resourceful!

Unfortunately for black, he has no place to move his rook where it helps him against the d7+ threat. He can't defend his b8 rook since that Pb5 is in the way, nor can he attack the d-pawn since the bishop hits d5.

What I failed to notice until afterwards is that black can try 56...Rh5!?, which disrupts the 57.d7+ plan a bit.

Carlos Torre vs Lasker, 1925

One more quote for you: