Without it, the King might have been able to break out through f1 and e2, possibly forcing my rook off the back row and ultimately enabling the king to reach the imperiled pawns. If that should happen, White will likely win. Now, he is cut off and too far away to be of consequence in the coming fight for their lives.

"Rooks behind passed pawns" is a time-honored strategy that works in many situations.

White has to watch those rooks: there are two of them and only one queen. If they succeed on ganging up on his pawns, White will loose them.

It seems that the advanced free pawn are not enough to force white's win, since they are not connected.

Agree with damalfi. It is a very very hard work for White to win this endgame!

I would take with the other rook, though. Black should guard the 8th rank.

@isomorphism: correct. My reasoning in capturing with Rd8 was to keep the rook on White's 1st rank. By being there it can operate against the b-pawn from the "check-safe" field b1 and it further limits the "teleporting" ability of the queen by supervising a1. Based on this I regarded the 8th rank as less important; the fields the queen can actually reach by means of checks are few and easily calculated, and there is no danger of it gaining entry to Black's K-side position. However this misses a neat trick. After 35...R1xd6! 36.Qxa7 R6d7 White actually has no shelter for his queen! There are no checks available for him, and the rook on d7 will follow him up and down and threaten him. The -only- refuge he can find is in the shadow of the b-pawn, and after 37.Qa5 Rd5 he either has to move out again - and face that rook - or allow Rd8-b8 and the loss of his pawn. Moving the queen to g3, to shelter it behind an advanced f-pawn, also loses the b-pawn, thus: 35...R1xd6! 36.Qxa7 R6d7 37.Qb6 Rd6 38.Qc7 R6d7 39.Qg3 Rb7 followed by Rdb8. The line preferred by me in the game is needlessly complicated.

Preparing to swing over to the b-file by means of the "check-safe" fields b1 and b3. Placing either rook on the a1-h8 diagonal would be a fatal mistake, as Qb8+ would either capture a rook (after Kg8-g7, Qb8-e5+) or weaken my K-side position (after Kg8-h7, Qb8-f4) by forcing me to either give up the f-pawn or advance it.

This is White's sole, remaining ploy. By latching on to both rooks at once he prevents them from moving out of sync, thus keeping them on the d-file. Now, however, another aspect of this position comes into play: the queen is unable to find shelter from the rooks anywhere on the board, except behind the b-pawn - and from there it will be unable to keep the rooks from ganging up on the pawn. Fundamentally the same situation as Black could have entered into directly by following isomorphism's suggestion on move 35. Iso's suggestion is still stronger however, as White does keep the option of 8th row checks in the present position, which complicates matters a little.

This is the "horizon effect". The computer overevaluates the position because White retains the b-pawn as far ahead as it can see. But White cannot make progress: now that ...f6 has shielded the king, Black threatens ...Rb2. If White plays Qa1 or Qa4, Black just attacks the queen again. Draw.

It seems draw to me, too. White can't make any progress while avoiding the rooks to grab the pawn.

On 41.Qe4 Rd4 42.Qc2 (GK's suggestion) I would have continued 42...R4d2 after which I cannot see that White has anything but 43.Qa4 Rd4 44.Qa5 Rd5 left. The b-pawn is clearly lost now (eg. 45.f4 Rb1 46.Qc3+ Kg8 47.Qc8+ Kg7 48.b6 Rxb6), and any K-side pawn advance looks not only futile but positively dangerous.