My first idea was <43...Rh2>. The point is to play ...Rh4 inducing g5, and return ...Rh2, so that the pawn structure becomes nearly frozen with White's king still cut off. But I now see that White has a good answer to that plan:

After 36... a6 37. Rxb6 axb5 38. Rxb5 Ra2+ 39. Kf1 Kc6 40. Rc5+ Kd6 41. g4 fxg4 42. fxg4 Ke6 43. Rc8 Rh2 <44. Rd8!> Rh4 45. f5+ gxf5 46. gxf5+ Kxf5 47. Rxd5+ Ke4 48. Re5+ Kf3 49. Ke1 Rh2 50. d5, when in spite of all Black's activity, White still looks winning by the simple advance of the passed pawns.

I saw the possibilities you posted but overlooked White's winning chances in the first one:

36... a6 37. Rxb6 axb5 38. Rxb5 Ra2+ 39. Kf1 Kc6 40. Rc5+ Kd6 41. g4 fxg4 42. fxg4 Rh2 43. Rc8 Rh4 44. Rh8 Rxg4 45. Rxh7 Ke6.

At first glance, this is drawish because we expect White to either play for the e4 break, leaving too few pawns left to win, or to maneuver the king toward c5, allowing Black's rook to counterattack e3.

But actually, White can exploit the trapped position of Black's rook to lure Black's king away with 46. Kf2 Kf6 47. Rh1! Ke6 48. Kf3 Kf5, and then return to cut off the e-file with 49. Rh8 Rg1 50. Re8, winning another pawn and the game.

There, White can aim to win a pawn ending after a properly prepared Rh5. For example, 50...Ke6 51. Re5+ Kd6 52. Rh5! Rxh5 53. gxh5 is a win because if Black's king attacks the h-pawn, White applies the e4 break and queens the d-pawn first.

If Black's rook quits the h-file to avoid this, the position of the king on d6 turns out to be a liability in 50...Ra2 51. Rh5 Ra7 52. Rh6+, etc.

That, of course, suggests the possible improvement 47...Kc6 instead of ...Kd6 in <Gypsy>'s line above, so that if all goes as before, Black will have 52...Kb5. Then at least White would have to find another winning plan, although I'm sure <Gypsy> is up to the task.

But since Black's manuever of h2-h4 fails to achieve its aim, maybe Black would simply fall back on 43...Kd7. In this case, Black aims to prevent White's rook from reaching the e-file, leaves the rook in place to play ...Rh4 at some later point, and waits for White to try the f5 break.

I wanted to comment on the Rh2-h4 idea for Black discovered by <beatgiant & Gypsy> which seems about to be discarded. I think it draws.

beatgiant wrote earlier: <So at this moment, I don't see anything better for Black than to try Gypsy's earlier line 36... a6 37. Rxb6 axb5 38. Rxb5 Ra2+ 39. Kf1 Kc6 40. Rc5+ Kd6 41. g4 fxg4 42. fxg4 Rh2 43. Rc8 Rh4 44. Rh8 Rh2 45.Kg1 Re2 46.Re8 Kd7 47.Re5 with the attempted improvement of 47...Kc6. I have a hunch White can still win it, but I don't see exactly how yet.>

In the above variation with the Black rook very active on h4, I would suggest the unusual looking 44...g5 instead of 44...Rh2. The idea is to force White's pawns to commit before the King is ready to help from f1.

I only find draws in the King and Pawn ending

36... a6 37. Rxb6 axb5 38. Rxb5 Ra2+ 39. Kf1 Kc6 40. Rc5+ Kd6 41. g4 fxg4 42. fxg4 Rh2 43. Rc8 Rh4 44. Rh8

36... a6 37. Rxb6 axb5 38. Rxb5 Ra2+ 39. Kf1 Kc6 40. Rc5+ Kd6 41. g4 fxg4 42. fxg4 Rh2 43. Rc8 Rh4 44. Rh8 44...g5 45 f5 Rxg4 46 Rxh7 Rh4 <47. Rg7>.

It might go on 47...Re4 48. Kf2 g4 49. f6 Ke6 50. Re7+ Kxf6 51. Rxe4 dxe4, but Black seems to draw this King and Pawn ending too as he has the opposition (52. Kg3 Kg5, etc.)

<Probably it is a candidate for most heavily analysed ending if this keeps up!>

36... a6 37. Rxb6 axb5 38. Rxb5 Ra2+ 39. Kf1 Kc6 40. Rc5+ Kd6 41. g4 fxg4 42. fxg4 Rh2 43. Rc8 Rh4 44. Rh8 44...g5 45 f5 Rxg4 46 Rxh7 Rh4 <47. Rg7>.

47...g4 48 Kg2 Rh5 and the f pawn eventually goes. Then it is rook and two versus rook and one, but I don't see a way to convert.

It seems Rubinstein could have prevented this by playing <34. f3> instead of his 34. Ke2. Then 34...Rxc6 35. Rxc6 Kd7 36. g4.

Another thing I find truly amazing about this ending is how all the pawn endings are won for White if Black ever offers the rook exchange with ...Rc8. I think that's a major factor in White's win.

I am comming to the conclusion that Rubinstein played the endgame "optimally", in the sense that his winning chancess would not have been better, had he diverged earlier, say with 34.f3, but quite possibly worse. It is quite possible that, until <36...Re8>, Alekhine just had a hard-to-follow-but-already-real path to a draw.

I think Rubinstein held off playing <f3> as long as he could. I believe that he wanted to keep those f3-g2 hiding squares for his king, but he was against a drawn pawn endgame when he played <36.f3>. For instance, 36.Kf3 Rc8 37.Rxc8 (37.Rf6? Ke7) 37...Kxc8 38.g4 Kb7 ... looks good for Black.

Before we conceede White his victory, we should probably ask how does he win from the position after like 36...a6!? 37.Rxb7 axb5 38.Rb7! Kc6 39.Rxh7 Rg8 ...

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<Gypsy> In your second diagram, doesn't Kc3 win pretty easily?

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Take any computer engine, input the game moves, and then cursor rapidly through the Rook-Pawn ending. It looks so simple! Consolidate in the middle, march the King over to the King side, provoke the weakness and then mop up. Too easy...not.