I have been terribly remiss in my Sunday Chess Problem responsibilities. So how about a charming little amuse bouche from the greatest of all endgame composers: Alexey Troitzky. The position below was composed in 1898 and calls for white to play and win.
A natural first reaction would be to give check with 1. Rc2 or 1. Qh1 or something like that. You're welcome to give that a try, but I think you'll find that white quickly runs out of checks. So we need to find something a bit more subtle.
The only way to win is the shocking 1. Re6+!
For what purpose is white sacrificing the rook? There are actually two points to white's idea. The first is that, though it may seem hard to believe, the black rook has now been decoyed to a very bad square. The second point is that the f1-a6 diagonal is now open. White makes immediate use of that fact with 1. ... Rxe6 2. Qa6+:
The point begins to emerge. If the black king moves to the seventh rank, then his queen will be lost to a skewer. But black isn't finished yet. Play continues 2. ... Kd5 3. Qc4+. Now what does black do? If he plays 3. ... Ke5, he loses his queen to a new skewer, this time along the long diagonal. (White would play 4. Qc3+ and grab the black queen as soon the king moves.) So black must try 3. ... Kd6 4. Qc5+,
after which it becomes clear that the black rook is on a very bad square indeed. That leaves only 4. ... Kd7 5. Qa7+:
after which white grabs the black queen after all. The resulting position of K+Q vs. K+R is a forced win for the queen, though it is not at all easy to execute.
Cute! Troitzky was the master of this sort of thing. Very spare positions where one side is dominated by the other. Alas, this sort of thing is largely played out. Modern endgame studies are much more complex, but often harder to enjoy because of the difficult analysis required.
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Hey! Nice one! I like the old endgame stuff -- one of my all-time favorite chess books is Endgame Magic by Beasley & Whitworth.
Maybe this one is even in that book, can't recall. In fact, your presentation of chess problems kinda reminds me of theirs -- nice amount of diagrams & descriptions from which the reader can pick up on the author's enthusiasm for his subject.
Of course, as you point out, the KQ vs KR endings are quite a chore for anyone. Used to be computers couldn't even guarantee mating an opponent within 50 moves. So the practical-game approach to this position in a real-life game at, say, the US Amateur Team East tournament in, say, round 3 would be to play the obvious Qg2+, forcing the immediate KR vs KR ending which the players would immediately agree to draw. This spares the White player from a couple hours of torment, quite likely ending in a 50-move draw, followed by self-recrimination resulting in a blown opening in Round 4.
I love Endgame Magic! It's a terrific book, which I go back to from time to time. Another good one is Secrets of Spectacular Chess by Levitt and Friedgood. As it happens, I found this study in a book called Endgame Artillery, by Alex Angos. It's a book specifically about endgames featuring only queens and rooks (and pawns and kings of course), Jon Edwards recommended it to me, and I've recently started going through it. It's a lot of fun!
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I like such problems very much because of their natural feel.
"though it is not at all easy to execute"
Once I had the endgame on the board. It was much easier than I expected. Basically the correct strategy rests on two pillars.
1. Try to mate as if it were a K+Q vs. K endgame;
2. Look for tactics that net the rook.