[Title "Why is Ra1 the only move?"]
[fen "2k5/p5r1/R7/8/7K/8/8/8 w - - 0 1"]
According to the tablebase at lichess.org, the only drawing move in the above position is 1.Ra1. After playing various lines according to the tablebase, I can't figure out why 1.Ra1 draws, and why all other moves (like 1.Ra2) lose. How should this position be understood?
Black's aim is to push the a pawn and queen. The first step is trying to play a6 or better still a5. This is going to require help from the king because the rook can't get behind the pawn.
Let's look at the two possibilities Ra1 and Ra2.
[Title "1. Ra2"]
[fen "2k5/p5r1/R7/8/7K/8/8/8 w - - 0 1"]
1. Ra2 Kb7 {Threatening to push the pawn} 2. Rb2+ Ka6 3. Ra2+ Kb5 4. Rb2+ Ka4 5. Ra2+ Kb4 6. Rb2+ (6. Ra6 Kb5 7. Ra1 a5 {Also mission 1 accomplished. The pawn is going to queen}) Ka3 7. Rb1 a5! {Mission 1 accomplished! The pawn is going to queen} 8. Ra1+ Kb4 9. Rb1+ Kc3 10. Ra1 Ra7 {Mission 2 accomplished. The black rook is now ideally placed behind the pawn. Queening is guaranteed}
Now Ra1
[Title "1. Ra1"]
[fen "2k5/p5r1/R7/8/7K/8/8/8 w - - 0 1"]
1. Ra1 Kb7 {Threatening to push the pawn} 2. Rb1+ Ka6 3. Ra1+ Kb5 4. Rb1+ Ka4 5. Ra1+ Kb4 6. Rb1+ Ka3 7. Ra1+ Kb2 8. Ra6 {and the black is too far away from the white rook to threaten it and support the pawn advance to a5} Kb3 9. Ra1 {and black cannot make progress}
Bottom line: on a1 the rook is far enough away from the black king and the a5 square that the black king cannot force through the pawn advance. On a2 it is just close enough that the black king can threaten the rook and support the pawn advance.
