When the rook wins
The rook wins easily if its king can get in front of the pawn, or if the rook can win the pawn before the defending king arrives to support it. A pawn that is far advanced and escorted by its king, against a rook whose king is far away, is often a draw.
So count before you move. Here the white king on h8 is much too far away to help, and the black king on e4 is close enough to escort the b-pawn if it is allowed to.
Cutting the king off
Rd1 cuts the black king off along the d-file. The king cannot cross it to reach its pawn, and the pawn alone cannot outrun the rook: after Ke3, Rb1 stops it, and Rxb3+ wins it when it advances.
Rd1 is not the only win. The check Rh4+ and the blockade Rb1 also win, but the cut is the idea to remember.
The wrong cut
Rc1 and Re1+ look similar but only draw. After either move the black king steps straight onto the d-file and joins its pawn.
Cut the king off on the file right next to it. A cut one file further away leaves the king a file to walk onto, and a check that drives it towards the pawn helps the defender.
Remember
- Count the kings' distances first
- Cut the king off from its pawn
- Use the file next to the king
- A lone pawn falls to the rook