I’m not sure if this is intended as a difficult problem, or as a beginner’s exercise, or as something in between, but here goes:
How about 1.c7 Rh8 2.Nd8 Rh6+ 3.Kc5 [3.Kb7?? Rh7 and sacs the rook] Rh5+ 4.Kd6 Rh6+ 5.Ke7 Rh7+ 6.Nf7 … ?
Or, if 2…Rh2 [intending to check perpetually from the rear] 3.c8=Q Rb2+ 4.Kc7 Rc2+ 5. Nc6, or if 3…Rc2 [hoping for 4.Qxc2 stalemate] 4.Nc6… ?
</SPOILER ALERT>
Or am I missing a defensive resource for black?
Bill Smythe
How about 1.c7 Rh8 2.Nd8 Rh6+ 3.Kc5 [3.Kb7?? Rh7 and sacs the rook] Rh5+ 4.Kd6 Rh6+ 5.Ke7 Rh7+ 6.Nf7 … ? <<
After 3. Kc5, Black can play … Rh1 (or … Rh2) with the idea of … Rc1 (or … Rc2) and exchanging the rook for the pawn.
Regards,
John
at work without a board but maybe 1. c7 Rh8 2. Nd8 Rh6+ 3. Kb5 Rh5+ 4. Kc4 Rh4+ king walks down to c1, back up d-file until … Rh6+ than white plays Ke7, Rh7+ and Nf7?
…scot…
Walks King to c2, not c1. Then Black has a try with Rh4, threatening c8=Q, Rc4! forcing stalemate, but Kd3 instead of Queening followed by your suggestion.
I don’t think either of these suggestions will work. You’ll end up with a position like this, where black can perpetually check white from the h-file:
However, your suggestion gave me a brilliant idea. Somehow get into a position like the following, with white to move:
Do not play c8=Q because of …Rc4+ leading to stalemate. Instead, promote to a rook! Then the stalemate possibility vanishes, and black must immediately play …Ra4 to avoid the “back-file mate” (sort of like a back-rank mate, only vertical). Black’s rook is stuck on the a-file indefinitely. Then maneuver the knight to a square like b3 or c2 where it mates.
I know, I know, I haven’t figured out the details. But I’m betting dollars to donuts that this underpromotion idea is the key. That’s what puzzle composers always like to do.
Bill Smythe
I’m not seeing the perpetual check in Mr. Doan’s example. Also, isn’t the knight on d8?
Alex Relyea
In the last diagram, after c8R, Ra4, White wins immediately after Kb3 (threatening the Rook and Mate on c1). This position is similar to the ending of the Saavedra Position.
It doesn’t really matter where the knight is. OK, technically it’s not a perpetual check, but in order to escape the horizontal checks from the h-file, white will have to move his king either to an advanced rank or way over to the kingside, where it is far from the scene of the crime and not helping in any way to threaten mate.
Bill Smythe
Aha! So at least I was right about the underpromotion.
Of course, if the black rook is on any rank other than the 4th, then there is no stalemate shot, so white can go ahead and promote to a queen, with an easy win.
Bill Smythe
Y’all have somehow teleported the Knight to the wrong square. 1 c7 Rh8 is forced (there’s no other move) 2 Nd8 Rh6 3 Kb5 (3 Kb7 Rh7, 3 Nc6 Rh8) Rh5 4 Kb4 (4 Kc6 Rh1 5 Kd7 Rh7, 5 Kd6 Rd1 and Rc1) Rh4 5 Kb3 Rh3 6 Kc2 Rh4! 7 Kd3! (7 c8=Q Rc4+!=, underpromoting doesn’t win) Rh3 8 Kd4 (can’t chase the Rook or the Rook gets to c file, so you have to run up the d file) Rh4 9 Kd5 Rh5 10 Kd6 Rh6 11 Ke7! Rh712 Nf7 wins.
Yes, this is what I was getting at. There is no immediate mate, but Q+N vs. R is an easy win.
Alex Relyea
Phooey. No underpromotion, then. The position looked so ripe for it, too.
Bill Smythe
Tablebase.
As Doan gave: 1 c7 Rh8 2 Nd8 Rh6 3 Kb5 Rh5 4 Kb4 Rh4 5 Kb3 Rh3 6 Kc2
Now both 6…Rh2 and 6…Rh4 lose in 20:
Check out k4it.de/?topic=egtb&lang=en
I’ve always been terrible at endgame studies.
But if anybody is looking for an Endgame Study Database, a guy named Harold van der Heijden sells one. Just google: endgame study database.
He’s working on an updated version for release in 2020, if the current version’s 85,619 problems can’t satisfy your cravings to solve endgame studies.
White to play & win
M. Liburkin, 2nd Prize, Shakhmaty v SSSR, 1931
So Bill Smythe wants something after Saavedra, but with a certain theme added? I have updated my signature. 
(I “solved” this online 25 years ago and complained that it was inelegant…but I had missed a variation!)
1 Nc1 Rd5+! (1…Rxb5 2 c7 Rc5 3 Nb3+) 2 Kc2 Rc5+! 3 Kd3 Rxb5 4 c7, then 4…Rb8! 5 c7xb8=B!±. 5 c7xb8=Q/R is stalemate. 5 Nb3+ Rxb3 6 Kc2 Rb4! 7 c8=Q Rc4! stalemates. Another (feeble) try is 4…Rd5+ 5 Kc4 Rd1 6 Nb3+.
Hmm, interesting. Does that cover everything, Bill B.?
Bill Smythe
No…Tom got the variation I missed & overlooked one variation I got.
After Saavedra! You asked for an underpromotion…did you think you only deserved one?
It’s not the most convincing example of underpromotion (at least for a problem), since the underpromotion gives an elementary win. (Usually you would expect there to be something more after that, not merely, can’t take Q or R but B is OK, done).
Your critique is EXACTLY THE SAME as mine was! But (like me some decades ago), you’re missing something: best defense after 1.Nc1 Rxb5 2.c7 and then…?