Common sense tells me that white ought to be able to win with the extra rook. So do various online sources, which suggest that in most (if not all) positions, this is a win for white. But I've just spent a frustrating game failing to achieve checkmate. Eventually I blundered and lost the extra rook, but I'm confused about how checkmate could have been achieved when my opponent was unwilling to exchange rooks. I haven't been able to find anything about how to achieve mate with this combination on the board. The problems are:
1) To achieve mate, the black king needs to be driven to the edge of the board. However, from most positions, the king is close enough to the white rooks that it can move diagonally to intercept them as they advance up the board and thus prevent checkmate. If the rooks defend themselves by moving onto the same rank or file, the king is then able to slip between them when one moves; thus the king cannot be confined to the edge of the board.
2) The above problem would not apply if it were not for the extra black rook, since in a 2 rooks plus king scenario, the rooks can dart across the board in a couple of moves and continue to force the king towards the edge. However, when black still has a rook, the moves required to get white's rooks over to the other side of the board are enough for the black rook to move up and shield the king. The only way for white to force an exchange is to move both rooks onto the same rank or file, so that the rook can be attacked by a piece which is in turn defended. But that takes two moves to achieve, so as soon as white makes the first, the black rook can move out of the way again; white then has to move back onto adjacent ranks / files to have a hope of checkmating; if he does so, that gives black a chance to move the rook again and thus the process repeats itself. Usually black can spend some intervening moves chasing white's king around the board, wasting moves and leading towards a draw.
3) Lastly, if white tries to get his king up the board to restrict black's own king's movements, this gets in the way of the squares at least one of his rooks is attacking; and then the black king can sneak away from the edge of the board on the far side of the white king.
I'm not a great player so I am probably overlooking something obvious, but it looks to me that unless black makes a mistake and puts his rook into a position where a white rook can pin it against the king, the result is going to end up being a draw. But sources seem to agree that this game can be won by white. The position at the start of the endgame is shown here. I'd be very grateful if someone could tell me how I should have won this game!