Why are these sacrifices good moves? What are the winning follow-ups? I was using a phone-app called Chessrating to get an estimation of my ELO rating. It gives me 16 chess positions and I have four attempts to find the best move. There were some positions that I did not understand the app's move choice. Why are those the best moves in the given positions? Is the app correct on all of them?

These are the 7 positions that I didn't understand, out of about 50 positions I played. They are all about sacrifices, except positions 4 and 7.


I realize that if the sacrifice is accepted, some attacking chances show up, but I can't find a winning follow-up, and in the end, to me it looks like the player is just down the sacrifice.
Of course, it is not obligatory to accept the sacrifice, and for that reason I don't see some of those sacrifices as winning. For the moves to be really winning, they must be winning in both cases (if either the sac is accepted or declined).




Position 1 - Black to play, why Rh2?

1r2k2r/p4pp1/2n1p3/q2pP1N1/3P1PB1/1Pb5/P1Q5/1K1R2R1 b - - 0 1


My thoughts: after Qxh2, maybe black could play Rxb3+, but I don't see a checkmate follow-up, so it looks like black will be down a Rook.



Position 2 - White to play, why Re8?

r1bq4/1p4kp/3p1n2/p4pB1/2pQ4/8/1P4PP/4RRK1 w - - 0 1


My thoughts: I think there is no problem with black accepting the rook sacrifice, by Qxe8. Of course black can then take the Knight on f6 with check, but I don't see a checkmate follow-up, so it seems to me that white would be down an exchange.



Position 3 - White to play, why Nd7?

2rq1rk1/1b2bppp/p1n5/1p1BN3/5B2/P7/1P3PPP/R2Q1RK1 w - - 0 1


My thoughts: This time, I realize that if black accepts the knight sacrifice, they will lose their Queen in a follow-up tactic. But black could just decline the sacrifice with Re8, and I don't see a winning follow-up for white.



Position 4 - White to play, why Bf6+?

r4rk1/p6p/2q5/1p2pnB1/3p2Q1/P2P4/1P5P/R4RK1 w - - 0 1


My thoughts: after black plays Kf7, I don't see any special follow-up for white, and this check doesn't seem to have anything special.



Position 5 - Black to play, why e5?

1nr5/2rbkppp/p3p3/Np6/2PRPP2/8/PKP1B1PP/3R4 b - - 0 1


My thoughts: to me it looks like white can just take with the pawn (fxe5) and I don't see nothing special going on.



Position 6 - White to play, why Qh6?

2r2rk1/pp2pp1p/2np2p1/q4P2/2PBP1b1/2N5/PP1Q2PP/R4RK1 w - - 0 1


My thoughts: after the queen goes to h6, if I was playing black I would immediately capture the bishop with the Knight, that is, Nxd4. Now I don't see any compensation for white.



Position 7 - Black to play, why Bxf3?

2rr2k1/p4p1p/2n3pb/2Ppp3/3Pb3/P1R1PNqP/4Q1P1/B3RB1K b - - 0 1


My thoughts: to me it just looks like a simple exchange, exchanging the bishop for the knight. Of course white should take back with the pawn, because taking back with the queen leads to losing the rook. Why is this exchange considered good/useful?



So why are those considered the best moves on those positions? Do you see a better move in each position? In my opinion, a sacrifice can only be considered good if independently of the opponent accepting or declining the sacrifice, there is a good outcome. I don't want to hope that my opponent will play poorly.

(I know this is a long question, should I split it in several (seven) questions instead?)
This is a very long question. And you can use Stockfish to check tactical solutions.

Solutions of all positions except the 5th ends with mate or huge material gain. In position 5, Black plays a positional sacrifice, to turn that cramped position into a nice attacking one.



Position 1 - Black to play, why Rh2?

[FEN "1r2k2r/p4pp1/2n1p3/q2pP1N1/3P1PB1/1Pb5/P1Q5/1K1R2R1 b - - 0 1"]

1... Rh2 2. Qxh2 Rxb3 3. Kc2 (3.Kc1 Qa3 4. Kc2 Nb4#) (3. axb3 Qa1 4. Kc2 Nb4#) Qxa2 4. Kc1 Qb1#




Position 2 - White to play, why Re8?

[FEN "r1bq4/1p4kp/3p1n2/p4pB1/2pQ4/8/1P4PP/4RRK1 w - - 0 1"]

1. Re8 Qe8 2. Qf6 Kg8 3. Bh6 Qf7 4. Qd8 Qf8 5. Qf8#




Position 3 - White to play, why Nd7?

[FEN "2rq1rk1/1b2bppp/p1n5/1p1BN3/5B2/P7/1P3PPP/R2Q1RK1 w - - 0 1"]


1. Nd7 Re8 2. Bxf7+ Kxf7 3. Qd5+ Kg6 4. g4 Bg5 ( 4... h6 5. Qf5# ) 5. Qf5+ Kh6
6. h4 g6 7. Bxg5+ Kg7 8. Bf6+ Kf7 9. Qd5+ Re6 10. Bxd8  




Position 4 - Black to play, why Bf6?

[FEN "r4rk1/p6p/2q5/1p2pnB1/3p2Q1/P2P4/1P5P/R4RK1 w - - 0 1"]


1. Bf6+ Kf7 2. Rxf5 Ke6 3. Rxe5+ Kd6 4. Re6+ Kc7 5. Qg7+ Kb6 6. Rxc6+ Kxc6 7.
Rc1+ Kb6 8. Qc7+ Ka6 9. Rc6#  




Position 5 - Black to play, why e5?

[FEN "1nr5/2rbkppp/p3p3/Np6/2PRPP2/8/PKP1B1PP/3R4 b - - 0 1"]


1... e5 $1 { Black has a cramped position. White rooks are dominating d-file,
bishop and knight pressing hard on queenside. Black can't move.

With ...e5,
sacrificing pawn, Black vacates e6 square for bishop and doubles  and isolates
White's pawns in the center. After fxe5, White's pawns are not threatening any
more.}  2. fxe5 Be6 { Now it's the Black is attacking the c4 pawn, and c2 pawn
with doubled rooks. Black turned a cramped position into attacking one with a
very clever pawn sacrifice, which ruined White's pawn formation.}  




Position 6 - Black to play, why Qh6?

[FEN "2r2rk1/pp2pp1p/2np2p1/q4P2/2PBP1b1/2N5/PP1Q2PP/R4RK1 w - - 0 1"]


1. Qh6 Nxd4 2. Nd5 Qd8 ( 2... Rfe8 3. f6 Ne6 4. fxe7 Qc5+ 5. Kh1 Qd4 6. Nf6+
Qxf6 7. Rxf6 ) 3. Nxe7+ Qxe7 4. f6 Qxf6 5. Rxf6  




Position 7 - Black to play, why Bxf3?

[FEN "2rr2k1/p4p1p/2n3pb/2Ppp3/3Pb3/P1R1PNqP/4Q1P1/B3RB1K b - - 0 1"]


1... Bxf3 2. gxf3 exd4 3. exd4 ( 3. Rcc1 Bxe3 4. Qg2 Qxg2+ 5. Kxg2 Bxc1 6. Rxc1
) 3... Nxd4 4. Qe7 ( 4. Qd1 Bf4 5. Re2 Nxe2 6. Qxe2 Re8 7. Qg2 Re1 8. Qxg3 Bxg3
9. Kg2 Rxa1 10. Kxg3 Rxf1 ) 4... Bf4 5. Re2 Nxe2 6. Qxe2 Rb8 7. Rc2 Rb1 8. Qg2
Qxg2+ 9. Kxg2 Rxa1