Most tutorials the K+Q versus K+R endgame claim that at some stage, the key is to force the rook away from the king, then it is supposed to be easy: give a series of checks and fork/skewer the rook.
However, when training with computer, it turns out that it's a bit more subtle. For example, I often reach an "almost Philidor's position" below, from which I don't know what to do, so I learned the following sequence:
[FEN "8/8/8/8/8/4KQ2/6rk/8 w - - 0 1"] 1. Qc6 Kh3 2. Kf3 Rg3+ 3. Kf2 Rg5 4. Qe4! Ra5 5. Kf3! Ra3+ 6. Kf4 Kh2 7. Qc2+ Kh3 8. Qc1! Ra4+ 9. Kf3
After black's 4... Ra5, the rook got separated from the king, however, there's no way to win it with checks. Instead, we have to make several quiet moves allowing rook checks, but using the fact that checks run out, and mate threats prevent the rook from rejoining the king. In the final position, Kh2 is the only move that stops both mate threats, but then Qd2 and the rook is finally forked next move. This is all rather non-obvious (to me). A similar situation occurs when the computer suddenly abandons the third-rank defense.
So, my questions are about the position after 4...Ra5, and concretely, about the property that one shouldn't try to win the rook by checks:
- is there a rule to recognize such positions, or is this an exceptional setup that one needs to know? The geometry in position after 4...Ra5 does look somewhat special to me, in that white King blocks the e1-h4 diagonal so that the rooks cannot be forked form e1.
- if this is a position one has to know, what are other other positions with a similar property? Among them, are there positions that have no quick win (so that white would need e. g. to allow the queen and the king to rejoin and set up some kind of 3rd or 2nd rank defense)?