An interesting chess puzzle ;)

Originally posted by Mephisto2
I don't have all variations yet, but I believe yes,white can draw with 1.Bg7 Rh7 (otherwise Bxd4+ and the fork on c6) 2.Kg4 Kxa7 (to be analysed what happens after other moves) 3.Kh5 Nf5 (Ne6? but I think that too will draw) 4.Bxb2 Rxh6+ 5.Kg5 Rh2 6.Be5 Rf2 7.Bf4 Nd4 (only move to escape from Be3+) 8.Be3 and black cannot get out of the pin without losing ...[text shortened]... Kc5 12.Kd3 Rd6 (or other move on d-file) 13.Kc3 etc...

Anyone helps for the other variations?

That was the original idea of the composer. There were two attempts to find
a lapse in this study. The first one was:
3...Ne6 4.Bxb2 Rb7 and Black should win. However this turned out to be a draw after all 5.Ba3 Rb3 6.h7! Rxa3 8.Kg4 Ra4+ 9.Kg3=
The second one however showed a clear win for Black through an unexpected move. What was it?

Originally posted by ilywrin
That was the original idea of the composer. There were two attempts to find
a lapse in this study. The first one was:
3...Ne6 4.Bxb2 Rb7 and Black should win. However this turned out to be a draw after all 5.Ba3 Rb3 6.h7! Rxa3 8.Kg4 Ra4+ 9.Kg3=
The second one however showed a clear win for Black through an unexpected move. What was it?

well, I wasn't sure about what happened when black avoids to go into the black-squared diagonal and played 1.Bg7 Rh7 2.Kg4 Ka7! (instaead of Kxa7). Could that be it? 3

Indeed 2...Kb7!! 3.Kh5 Nf5 4.Kg6 Nxg7 5. Kxh7 Ne8! 6.Nb5 Kb6 7.Kg8 Kxb5 8.h7 Nf6+ 9.Kg7 Ng4+ 10.Kg8 Nh6+ and wins 😉
The author is Villeneuve-Esclapon and the study is now altered as to begin from the position after 2...Kxa7. Starting 1.Kh5, etc.
Okay now something easy yet curious:
White to win:

Originally posted by BigDoggProblem
You gave up too soon! It appears 4.Ke1 is indeed forced.

If 4.Kf1?? Ke3! 5.Ke1 f2+! 6.Kf1 Kf3! wins for Black. White is forced to move his h-pawn.

5. ...h5 seems to be the only move winning.

Okay now something easy yet curious:
White to win:
[fen]6nk/2p4p/3B1P1P/5KpP/8/8/8/8 [/fen]
[/b]

Still too hard for me. I only see that the knight inevitabely takes the f-pawn and after Kg8 the position is drawn.

Originally posted by jfkjmh
5. ...h5 seems to be the only move winning.

And how does Black win exactly?

Originally posted by ilywrin
And how does Black win exactly?

6.Kd1 f2
6.h4 Kf4
6.Kf1 h4 7.Ke1 f2+ 8.Kf1 Kf3 and the difference to the line mentioned by BigDoggProblem is that White can no longer play h4. Black wins.

Originally posted by jfkjmh
6.Kd1 f2
6.h4 Kf4
6.Kf1 h4 7.Ke1 f2+ 8.Kf1 Kf3 and the difference to the line mentioned by BigDoggProblem is that White can no longer play h4. Black wins.

This seem correct. Thanks for pointing it out 😉

Originally posted by propawnkiller
Still too hard for me. I only see that the knight inevitabely takes the f-pawn and after Kg8 the position is drawn.

Sometimes you have to look into the past in order to see the future 😉

Originally posted by jfkjmh
6.Kd1 f2
6.h4 Kf4
6.Kf1 h4 7.Ke1 f2+ 8.Kf1 Kf3 and the difference to the line mentioned by BigDoggProblem is that White can no longer play h4. Black wins.

Good work. I had the main idea, but (as usual in endgame studies!) it needed refinement.

Originally posted by ilywrin
Indeed 2...Kb7!! 3.Kh5 Nf5 4.Kg6 Nxg7 5. Kxh7 Ne8! 6.Nb5 Kb6 7.Kg8 Kxb5 8.h7 Nf6+ 9.Kg7 Ng4+ 10.Kg8 Nh6+ and wins 😉
The author is Villeneuve-Esclapon and the study is now altered as to begin from the position after 2...Kxa7. Starting 1.Kh5, etc.
Okay now something easy yet curious:
White to win:
[fen]6nk/2p4p/3B1P1P/5KpP/8/8/8/8 w - - 0 1[/fen]

Black's last move was 0...g7-g5. 1.hxg6 e.p. wins.

Very much so 😉 A study by Troitsky (one of the first to use retrograde analysis in studies).
Okay a new one:
White to win

I think 1. Nc7. If 1.. Kxc7 then pawn promotes. So 1.. Nf6 2. Nd5+ and White can promote.

Originally posted by crazyblue
I think 1. Nc7. If 1.. Kxc7 then pawn promotes. So 1.. Nf6 2. Nd5+ and White can promote.

What about 2...Nxd5+?

oh dang, missed the check. its late 😳

edit: Here is my next try. 1. Ka3!
White cannot move the knight, because of the problem in my solution above (the check after sacing the knight). So where can White move the king?
1. Kc4....Black will give check followed by Ng6 and pawn win.
1. Kc3 will lead to the same problem in my previous post. 1... Kc5 2. Nd6 Nf6 3. Ne4+ Nxe4+
1. Ka4/b3 will give Black a check allowing his knight to move to e4 and free the way for his king.

After 1. Ka3 Blacks problem is, that it cannot move the knight, otherwise white plays Nf6 followed by promotion. So 1... Kc6 (of course not c5 because of the fork motive above). 2. Ka2. And now Black is doomed. Still cannot move the knight and any king move will also finish him. If the king chooses a White square he will get check right away. If he chooses 2... Kb6 3. Nc7 Nf6 4. Nd5+. On 2... Kc5 follows 3. Nd6 Nf6 4. Ne4+. Very nice indeed (I hope it's correct this time 🙄 )