First, the eighth row is very weak, so, if white could take the black Rook on a8 with check, white would win by checkmate. On the other hand, The queen in column b is the only piece that defend the b3 square, thus:
Kg3 (attaking the Rook on a8 with the Queen) Qe1+
Black can't take the Queen on h1: 1. ... Qxh1 2. Bb3+ Kh8 3. Nf7+ Kg8 4. Nd8+ Kh8 5. Rf8# Or 3. Ng6+ hxg6 4. Rh4#
Kh3 (No more checks with senses) The game is lose for black. If black move the Rook, for example:
2.. ... Re8 (attaking the Knight)
3.. Qd5+ Kh8
4.. Nf7+ Kg8
5.. Nh6+ (double check) Kh8
6.. Qg8+ Rxg8
7.. Nf7# (Smothered mate)
1. Kg3 Rd8 -> What now?
Also:
1. Kg3 Qg1+ -> What now?
And:
1. Kg3 h6 -> What now?
Finally:
1. Kg3 h5 -> What now? :)
Sorry for bad formatting before editing the comment!
Here I go:
After
Interesting is also:
After
The last sequence (1. Kg3 h5) is accepted, but please check your notation again (4. Kg3 is not possible).
By the way, just as a hint: chess software is not allowed to solve my problems (apart from some exceptions where I explicitly inform the competitors). I am sure you are aware of that, are you? :-)
I mention that because your variations are very accurate without any small typical human errors ... I didn't say you are using chess software - just wanted to state the rules again, just in case ... :)
Edit: Sleeping time for me: I will check all remaining answers tomorrow. Good night for now. :)
4.. Qg2 Rd8 -> What now?
5.. Rxd4!! This move took me a long time to see it
If Rxd4, Qa8#
If Qxd4, a8=Q winning the game.
3.. Qxh2 Qa3 -> What now?
4.. Qg2 (attacking the Rook again) Rd8
5.. a8=Q Qd6
6.. Bb2+ Kh8
7.. Nf7+ Kg7
8.. Nxd6+ Kh8
9.. Qd8#
4.. Nxg6 Qg1+ -> What now?
5.. Qg2 Qe1+
6.. Kh3 Bxa7
7.. Rf8!!
I'm very sorry if I made you doubt. Chess is my favorite sport and I always try to do my best.
Don't worry! I had to ask that. It seems you are very talented & engaged solving my chess problems. :)
I will put an official solution now with a few selected variations.
I hope you had solving this chess riddle which cost me quite some time to compose. :)
tnx for resteem