How helpful is high-diving if you want to become a better competitive swimmer? They are separate competitive Olympic sports, which both happen to take place in a swimming pool. In chess, both game-playing and problem-solving are recognized by FM, IM and GM tiers of certification.
See the attached chessbase report of this year's world chess solving ...
There is a lot more to this, and this is really not a great tactics problem as much as it is a continuing the attack problem. The difference is that with a tactic problem, you can see the result all the way to the end. Other positions, you have just a feel for it, and you know that you are probably winning, but you have to play what you can calculate, you ...
I'm going to assume you meant ...Qc7 instead of ...Qc6, since the latter move blunders the queen.
After 1...gxf6 2.Qg4+, Black has three moves:
1) 2...Kf8 loses the queen to 3.Nd7+.
2) 2...Kh7 leads to mate after 3.Bd3+ f5 4.Bxf5+ exf5 5.Qxf5+ Kh8 (5...Kg7 6.Qxf7+
Kh8 7.Ng6#) 6.Qf6+ Kg8 7.Qxf7+ Kh8 8.Ng6#.
3) 2...Kh8 is Black's best move. After 3.Nxf7+ ...
The best example I could find in PDB is this one:
[Title "Theodor Steudel: Problemkiste 66, Dec 1989, no. 2339. Za6 in 12"]
[fen "8/1p1p1p2/1P1P1P1P/8/p7/p1p1p1p1/2P1P1P1/k1K w K - 0 1"]
1.h7 a2 2.h8=N a3 3.Ng6 fxg6 4.f7 g5 5.f8=N g4 6.Ne6 dxe6
7.d7 e5 8.d8=N e4 9.Nc6 bxc6 10.b7 c5 11.b8=N c4 12.Na6
(Far greater lengths are possible in ...
Thanks for the fun question, and welcome to CSE!
For starters, the king is obviously the piece to choose. I will answer this question assuming that the normal rules of chess are used and no special conditions are used other than what the question has.
The OP has clarified that both sides need a king and first/last rank pawns can do a double step.