user30437: I did the following puzzle on chess.com:

And it shows solved with the following configuration:

I do not see why this is a solution. Black can take the Rook, and I don't see how to punish it.

user8860: It's won for White because Q+R coordinate to attack the exposed bK which Black can't defend for long.
I assume you started 1 Rg3 g5 2 Rxg5+ Kf7 3 Rxf6+.  Then Kxf6 4 Rg6+ and now:
4 ... Ke7 5 Qg7+ Rf7 (Kd8 6 Qg5+ Kc8 7 Qe7 and mates soon) 6 Qg5+ Ke8 (Ke6 7 Qg6+ Rf6 8 Qe8+ as in the next variation) 7 Rg8+ Kd7 (Rf8 8 Qg6+) 8 Qd8+ Kc6 9 Qe8+ Rd7 10 Rg7 1-0
4 ... Kf7 5 Rg7+ Ke6(f6) (or Ke8 6 Qg6+ Kd8 7 Qd6+ and mate) 6 Qg6+ Ke5 (Rf6 7 Qe8+ Kd6 8 Qe7+ Kc6 9 Qxf6+) 7 Re7+ Kf4 8 Qg3#
4 ... Ke5 5 Qg3+ f4 (Qf4 6 Re6+ Kxe6 7 Qxf4) 6 Qg5+ Ke4 (Rf5 7 Qe7#) 7 Qe7+ wins the Rook.  Black can try to counterattack with Kf3 8 Qxf8 Qe2 but 9 Rg3+ ends it (the pin prevents fxg3, and after Ke4 10 Qe8+ skewers the Queen).

user26335: After 2. Rxf6+ Black is fighting to survive and 2...Kxf6 is Black's only way to delay the final result (assuming best play).  White must respond with 3. Rxg6! which is the only move to maintain the advantage.
Black gets a very vulnerable King while White gets an active Queen and Rook with good coordination and a safer King.
After 3...Ke5 4. Qg3+ Black is forced to choose

between losing the Queen with 4...Qf4 5. Re6+, or
playing 4...f4 which is the human move (Engine wants to lose the Queen),

yet this f4-pawn push still loses to 5. Qg5+ as illustrated.
One commenter points out that the 6. Qxh5 is unclear.  I agree that 6. Qe7+ forking the King and Rook is the human move and the most practical move. I would play that as well.
For those interested, the less practical line 6. Qxh5 wins as well and leads to a situation where Black must either give up the Queen or submit to a forced mate.
[Title "Black is in Trouble"]
[FEN "5r2/6k1/1p2Rbp1/p2p1p1p/2q5/2P1R3/PP3PQ1/6K1 w - - 0 1"]

1. Rg3 Kf7 2. Rxf6+ Kxf6 3. Rxg6+! Ke5 4. Qg3+ f4 (4. Qh2+ {Also fine for White}) 5. Qg5+ Ke4 (5... Rf5? 6. Qe7#) 6. Qe7+ Kd3 7. Qxf8
(4...Qf4 5. Re6+ Kxe6 6. Qxf4)
(6. Qxh5 Kd3 7. Qd1+ Ke4 8. f3+ Kf5 9. Qb1+ {Black must give up the Queen or there's a forced mate in 10})


30437: <p>I did the following puzzle on chess.com:</p>
<p><a href="https://i.stack.imgur.com/SUxD5.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SUxD5.jpg" alt="initial configuration" /></a></p>
<p>And it shows solved with the following configuration:</p>
<p><a href="https://i.stack.imgur.com/gY8NY.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gY8NY.jpg" alt="final solution" /></a></p>
<p>I do not see why this is a solution. Black can take the Rook, and I don't see how to punish it.</p>
 8860: <p>It's won for White because Q+R coordinate to attack the exposed bK which Black can't defend for long.</p>
<p>I assume you started 1 Rg3 g5 2 Rxg5+ Kf7 3 Rxf6+.  Then Kxf6 4 Rg6+ and now:</p>
<p>4 ... Ke7 5 Qg7+ Rf7 (Kd8 6 Qg5+ Kc8 7 Qe7 and mates soon) 6 Qg5+ Ke8 (Ke6 7 Qg6+ Rf6 8 Qe8+ as in the next variation) 7 Rg8+ Kd7 (Rf8 8 Qg6+) 8 Qd8+ Kc6 9 Qe8+ Rd7 10 Rg7 1-0</p>
<p>4 ... Kf7 5 Rg7+ Ke6(f6) (or Ke8 6 Qg6+ Kd8 7 Qd6+ and mate) 6 Qg6+ Ke5 (Rf6 7 Qe8+ Kd6 8 Qe7+ Kc6 9 Qxf6+) 7 Re7+ Kf4 8 Qg3#</p>
<p>4 ... Ke5 5 Qg3+ f4 (Qf4 6 Re6+ Kxe6 7 Qxf4) 6 Qg5+ Ke4 (Rf5 7 Qe7#) 7 Qe7+ wins the Rook.  Black can try to counterattack with Kf3 8 Qxf8 Qe2 but 9 Rg3+ ends it (the pin prevents fxg3, and after Ke4 10 Qe8+ skewers the Queen).</p>
 26335: <p>After <strong>2. Rxf6+</strong> Black is fighting to survive and <strong>2...Kxf6</strong> is Black's only way to delay the final result (assuming best play).  White must respond with <strong>3. Rxg6!</strong> which is the only move to maintain the advantage.</p>
<p>Black gets a very vulnerable King while White gets an active Queen and Rook with good coordination and a safer King.</p>
<p>After <strong>3...Ke5 4. Qg3+</strong> Black is forced to choose</p>
<ul>
<li>between losing the Queen with 4...Qf4 5. Re6+, or</li>
<li>playing 4...f4 which is the human move (Engine wants to lose the Queen),</li>
</ul>
<p>yet this f4-pawn push still loses to 5. Qg5+ as illustrated.</p>
<p>One commenter points out that the 6. Qxh5 is unclear.  I agree that 6. Qe7+ forking the King and Rook is the human move and the most practical move. I would play that as well.</p>
<p>For those interested, the less practical line 6. Qxh5 wins as well and leads to a situation where Black must either give up the Queen or submit to a forced mate.</p>
<pre><code>[Title &quot;Black is in Trouble&quot;]
[FEN &quot;5r2/6k1/1p2Rbp1/p2p1p1p/2q5/2P1R3/PP3PQ1/6K1 w - - 0 1&quot;]

1. Rg3 Kf7 2. Rxf6+ Kxf6 3. Rxg6+! Ke5 4. Qg3+ f4 (4. Qh2+ {Also fine for White}) 5. Qg5+ Ke4 (5... Rf5? 6. Qe7#) 6. Qe7+ Kd3 7. Qxf8
(4...Qf4 5. Re6+ Kxe6 6. Qxf4)
(6. Qxh5 Kd3 7. Qd1+ Ke4 8. f3+ Kf5 9. Qb1+ {Black must give up the Queen or there's a forced mate in 10})
</code></pre>
