I wonder if the solutions are correct.

Two or three weeks ago, there was an endgame study where Black's bishop was on g8, his pawn on f7, white's pawn was on f6, and White won by sacrificing his knight on f8. I didn't understand why the Black would have to take the knight. I think the endgame is a draw if he didn't do so.

However, there is something I don't quite understand in last week's solution: How does White win if Black doesn't take the bishop on b3 but plays Rh2 instead?? Ra8 Rh7+, Kb8+ Kb6, a5+ Kc5 seems to be drawish to me...

Just wondering...

1.a7 Ba6 2.b7 Nb5 (idea 3... Nd6#) 3.Ng8+ Ke8 4.Nf6 Ke7 (4...Nxf6? 5.a8Q Nd7 6.Qxa6 and now there's no Nd6 mate because of the queen...)5.Ng8 ... a perpetual check

1.a7 Ne4 (idea 2...Nd6#) 2.Kb7 Bd5+ (2...Nd6+ 3.Kc6 Bb5+ 4.Kd5) 3.Ka6 isn't better, the king can escape...

I'm training endgames at the moment, reading dvoredskijs endgame manual- this study was really nice to train calculation and somehow really beautyful and not too difficult.

ups, last part was rubbish, 5...Bb5? 6.b8Q..

Ah, maybe it's good to continue this line: 5...Bb5 6.b8Q Ba6+ 7.Qb7 (7.Bb7 Nb6#) 7...Nb6+ 8.Kb8 Bxb7 with a draw

Good one i have to say..
I didn`t check it very well but i think Kb7 will do the job,threating to promote my c pawn,so his only move to prevent that is Nc5+..2-Kc6 threating to take his knight in addition to the promotion threat.His only move to stop both is to play Nxa6,notice i can`t promote my C pawn to Queen cuz 3-...Bb5 mate.
So C pawn promoting to Knight check..4-...Kf6(any other square and his Bishop is gone by fork).and after 5-b7 the position is clearly draw.

Aha, I see, this is really nice :)

1.a7 Ba6+ 2.b7 Ne4 3.g8N+ Ke8 4.Nf6 Ne4xf6 5.a8B with stalemate or if 5...Bb5 6.Kb7

truly beautyful :)

Felix, you might have missed the nicest variation:

1.a7 Ba6 2.b7 Ne4 with the idea of 3.g8N Ke8 4.Nf6 Nexf6 threatening Nd5 and Ne7

I hope the solution is correct, I spend some time and checked it a few times.

In the study of the last week: 2...Rh2! looks indeed like a good try (better than Rxb3). Obviously taking the rook is no good idea and also 3.Rb8 Rh7+ 4.Ka8 Ra7+! with stalemate or 3.Rc8 Rh7+ 4.Kb8 Rb7+!.

I guess 3.Rg8 is a move. 3...Rh7 4.Kb8 Kb6 5.Rg6+ Kc5 (5...Ka5 6.Rg5+ Kb6 7.a5+ etc.) 6.a5 .. and the a pawn runs

To Felix:

Thanks for the good idea 3.Rg8

I'm afraid I don't have enough time to analyze the options thoroughly, however, here are my ideas:

What about 3. ... Rh5, for instance: 4.Kb7 Rh7+, 5. Kc6 Rh6+, 6. Kc5 Th5+, 7. Kd4 Th4+, 8. Kd3 Rh3+, 9. Kc2 and maybe this leads to a won position for White (e.g. 9. ... Rc3+, 10.Kb2 Rc5, 11. Ra8+ Kb6 12. Bc2 (with the idea Be4).

Another idea was: 3. ... Rh7+, 4. Kb8 Rb7+, 5. Kc8 Ra7 (to prevent any movement of the pawn), 6. Rg5+ Kb6, 7. Rb5+ Kc6, 8. Kb8 probably wins.

However, this of course is not as cute as the "official" solution with the mate.

Mmh,
1.Kb7 Bd5+ 2.Kc8 (2.Ka7 Nb5#) 2...Nb5 (idea 3...Nd6#) 3.g8N+ (only move) 3...Ke8 4.Nf6+ Nxf6 5.b7 (5.a7 5...Nd7 6. a8X Nd6#; 5.Kb8 Nd7+ 6.Kc8 Nd6#) 5...Nd7 6.b8X Nd6#

Mmh,
1.Kb7 Bd5+ 2.Kc8 (2.Ka7 Nb5#) 2...Nb5 (idea 3...Nd6#) 3.g8N+ (only move) 3...Ke8 4.Nf6+ Nxf6 5.b7 (5.a7 5...Nd7 6. a8X Nd6#; 5.Kb8 Nd7+ 6.Kc8 Nd6#) 5...Nd7 6.b8X Nd6#

1.b7 Nb5 2.g8N Bxg8 and 3...Nd6#

1.g8X Bxg8 2.b7 (2.a7 Bd5 3.b6 Nb5 4.a8X Nd6#) 2...Nb5 3.X Nd6#

->1.a7 is the move :)

The other study o Rusinek with about the same theme (see the diagram on clicking on my name) seemms to be cooked in a nice way :
1.c4 Ne4 2.c5 Bc4 3.b8N Be6+ 4.Kb7 Na5+ 5.Ka6 Nc4 6.Kb7 Bc8 1-0
So Yochanan Afek has made the good choice.