White is winning here. But how?

Question

White to move.

This puzzle is composed by myself!

Answer 1

This is a great problem ! White wins with several surprising sacrifices.

The solution must be:

1.Bc4!

It threatens

2.Bb5#

while avoiding

1...Qb4+

Then

If 1...dxc4 2.d7#
If 1...Qxc4 (the bQ is deflected from the c7-f4 diagonal) 2.d7 Kxd7 3.Qxf7 threatens 4.e8Q and 3...Qb4 is no perpetual check: 4.Ka6 Qc4 5.Ka7
1...Qxd6 doesn't help: 2.Bb5 Qd7 3.Bxd7 Kxd7 4.Qxf7

However

Black can take a rook with check with 1...Qxd2+

That's when it gets crazy:

2.b4! keeps giving material with check (2...dxc4 refutes other tries)
2...Qxb4 (if 2...dxc4 3.Rxd2 Nxd2 4.d7 Kxd7 5.Qxf7 Black won't survive long)
3.Ka6 Qxc4 (third consecutive capture with check!)
4.Kb7 (the main threat is 5.Kc7)

Now

if 4...Qb4 5.Kc8! Black is doomed
if 4...Kd7 5.Qxf7 Qb4 6.Ka6 Qc4 7.Ka7 and Black runs out of checks, while his king cannot escape the two White queens...

And we also counter

the switchback 4...Qf4, eyeing d6 and c7

with the ultra cool

5.Kc7! (not 5.Kc6? Nd4 - while the fantastic 5.Qxf7? Kxf7 6.d7 meets 6...Rg8) with the slow threat 6.Rb2 - 7.Rb8# (or 6.Qb1 - 7.Qb8#) that is hard to meet, e.g. 5...Nd2 6.Rb2 Rb1 7.Qxb1 (or 7.Qh8) Nxb1 8.Rb8# or 5...Qxf6 6.d7 Kxe7 7.d8Q#.

First I saw no defense for Black but @rhkoulen pointed

4...Qxa2, intending 5.Kc7/Kc8 Qxa4 controlling d7, or 5.d7 Kxd7 6.Qxf7 Qb3+ followed by Qxa4+, or 5.Qxf7 Kxf7 6.d7 Rg8 7.hg8 Rxg8 and White attack gets exhausted.

Yet White has one more crazy sacrifice

5.Qb1!!, giving one more piece with check just to deviate the bQ from a4.

Now :

5...Qxa4 6.Qb6 threatens 7.Qd8#, so 6...Qd7 7.Ka8 Qc8 (or 7...Qa4 8.Kb8) 8.Qb8 Qxb8 11.Kxb8 forces an exchange of queens and transposes in the winning variaton below.
And after 5...Qxb1 6.Kc8 the threat 7.d7# can only be postponed with 6...Qb8 (or 6...Qb7) 7.Kxb8 (that were the line with 5...Qxa4 merges), and then 7...Kd7 8.e8Q Kxe8 9.Kc7 Kf8 10.h7 Rg8 11.d7 will be followed by 12.d8Q# whatever Black does. Black can resist with 8...Kxd6, but then 9.h7 and White will have enough queens to win against Black's disorganized army.

White wins.

Answer 2

I'm going to type out the principal line and variations mostly for my own edification. Big props to @evargalo, as he has found some amazing moves. This is gonna be a long one, buckle up.
FEN: 4k3/4Pp1Q/3PpP1P/K1pp4/P4q1P/1B3p2/RP1RnPr1/5nr1 (if you want to play it out for yourself and/or prove me wrong)

The spoiler box below contains the main line. I am fairly certain it is correct. This puzzle is very beautiful and if you want the magic of solving it, don't click.

1. Bc4 $\hspace{0.85cm}$ Qxd2+
2. b4 $\hspace{1.03cm}$ Qxb4+
3. Ka6 $\hspace{0.85cm}$ Qxc4+
4. Kb7 $\hspace{0.82cm}$ Qxa2
5. Qb1 $\hspace{0.74cm}$ Qxa4
6. Qb6 $\hspace{0.73cm}$ Qd7+
7. Ka8 $\hspace{0.85cm}$ Qc8+
8. Qb8 $\hspace{0.75cm}$ Qxb8+
9. Kxb8 $\hspace{0.64cm}$ Kd7
It isn't mate, but it's certainly disastrous for black. I'd call white winning.
I love two things about this puzzle:
1. The forcing line has your opponent repeatedly checking you! Cool!
2. Stockfish 12 at d=20 (thanks Chess.com) has white losing until you put 6. Qb6 on the board, then it gives white a significant advantage. Brilliant!
Now, obviously, very few of blacks moves were forced so I'll go down the line justifying each one. If you find a better move for black, drop it in the comments!

Justification for 1... Qxd2+. The immediate threat is Bb5#.
Could black capture the bishop?
1.Bc4 dxc4 2.d7#$\quad$[fails]
1.Bc4 Qxc4 2.d7+ Kxd7 3.Qxf7 Kd6 4.e8=Q$\quad$[Mate is clearly now inevitable. There are too many threats (e.g. Qb8 Qxe6 etc.)]
Why doesn't this work immediately for white? Well, if the queen hadn't taken the bishop, the black queen could perform a perpetual with 2...Qc7+. Here we see the purpose of Bc4: distract the black queen.
Could black cover the b5 square?
1.Bc4 Nc3 2.Rxd5 exd5 [defends mate] 3.bxc3 dxc4 [defends mate] 4.Qb1 Kd7 [defends mate] 5.Qb7+ Ke6 [prolongs] 6.e8=Q+ Kf5 [prolongs] 7.Qbd7+ Kg6 [prolongs] 8.Qexf7+ Kxh6 9.Qh6#$\quad$[fails]
It's not clear at first glance, but all roads lead to force mate. I will not type all of them out; this is just one reasonable one with black fighting as hard as possible. White uses the fact that they have many possible checkmates to gain tempi and free the queen.
Could black's king run away?
1.Bc4 Kd7 2.Qxf7 Qxd2+ 3.Ka6 Qa5+ 4.Kxa5 Kxd6 5.e8=Q Ke5 6.Qexe6+ Kf4 7.Qc7#$\quad$[fails]
This fails for black because blacks only defense to e8=Q is to perpetual check white. Without d7+ having been played, the black queen can only give spite checks.
The only defense for black is checking white.
The only available check is Qxd2+.

Justification for 2... Qxb4+.
The justification is very similar to the previous.
White is still threatening Bb5#. You could again try to cover b5 with the knight, run the king away, or capture the bishop, but now you lose your queen with any of those as well, and I suspect most options create a forced mate, but I don't care to check.
The only defense for black is checking white.
The only available check is Qxb4+.

Justification for 3... Qxc4+.
This justification depends on blacks possibilities.
Remember that the intent of the move 1.Bc4 was to divert the queen from the b2-h7 so that black could not deliver a perpetual in one move. If black does not check the white king here, white will do one of two things:
4.Bb5. If white did not play Nc3 or Nd4, black will lose their queen.
4.d7+ Kxd7 5.Qxf7 Kd6 6.e8=Q. If a check is not applied, white will make a queen for free and black's king is in serious trouble (most lines are forced mate I think).
The only defense for black is checking white.
The only available check that does not blunder the queen is Qxc4.

Justification for 4... Qxa2+. Black is about to be checkmated with Kc7 d7#.
Could black blockade the d7 square?
4.Kb7 Kd7 5.e8=Q+ Kxe8 6.Kc6$\quad$[Mate is now inevitable]
Pawns on the a and d files can promote, and the rook and queen can both enter through the open b file. Fun sub-puzzle for the reader: this is forced mate, but which move for black prolongs the game the longest?
Could black prevent the king from supporting d7?
Not that I see.
4.Kb7 Qb4+ 5.Kc8 Qb8+ 6.Kxb8$\quad$[Black has no way to check or divert the white king without giving away the queen and losing.]
Could black cover the d7 square?
With 2 tempi? Yes. 4.Kb7 Qxa2 5.Kc7 Qxa4$\quad$[now white's plans are foiled]
This is black's best attempt to stop the checkmate on d7, but evargalo found the wonderful novelty 5.Qb1! (fun chess trivia: ! does not mean good move, ! means only move which preserves victory/draw), which we'll discuss in the next block.

Justification for 5... Qxa4.
The fact of the matter is that it's over.
I see only two moves for black here that don't immediately force a mate: Qxa4 and Qxb1. And quite wonderfully, they transpose to each other.
5.Qb1 Qxb1+ 6.Kc7 Qb8+ 7.Kxb8
5.Qb1 Qxa4 6.Qb6 Qd7 7.Ka8 Qc8+ 8.Qb8 Qxb8+ 9.Kxb8
Note that the only difference is that in 5...Qxa4, white does not have a pawn on a4. That's the only reason I picked it for the main line. These are essentially the same position.
Finally, why is this position winning?
Simply put, white is too close to queening and mate, and black has somehow managed to make all their pieces sad and useless.
Obviously, 9... Kd7 is forced. If not, white just 10.Kc7 and pushes the d-pawn to victory. So, here's an example line on that vein: 9.Kxb8 Kd7 10.e8=Q+ Kxe8 11.Kc7 Kf8 12.h7 Ke8 13.d7+ 14.Kf8 d8=R#
Stockfish says black can live with 10... Kxd6 and escape through his pawns, but living isn't winning. White will have too many queens for black to have a chance.

I hope you enjoyed this massive brain dump, and bravo @Hobacle! Wonderful puzzle (if I'm right lmao).

If you've resisted the urge to click on spoilers and want to do it yourself, I believe in you! If you get stuck, here's some hints that have really guided my thinking.

White can't play d7+ because the black queen will perpet with Qc7+. How can you prevent this?

What that white queen doin up there? Seriously though. Why would the puzzle crafter even bother adding the queen if only to be boxed in by some rooks boxed in by some knights?

Answer 3

I believe it's

1. d7+ Kxd7
2. Qxf7

and black can't avoid

e8=Q double-check mate, eventually.

The best black can do is

to throw in a few desperate checks:
2...Qb4+
3. Ka6 Qb5+
4. Kxb5 Nd4+
5. Rxd4, with e8=Q next to follow.

Answer 4

EDIT: This does not work.

1. Qd3

threatening

2. Qb5#

If Black responds with

1. … c4

then White plays

2. Bxc4

maintaining the threat. Note that from this position

neither 2. … dxc4 nor 2. … Qxc4

are viable for Black since White can simply play

3. d7# in the former case, and 3. Qxc4 in the latter case, maintaining the previous threats.

Instead,

if 1. … Qxd2+, then 2. Kb6

threatening the caveman-like tactic

3. Kc6/Kc7 followed by 4. d7#

Black must deal with the above threat as well as the more immediate one still posed by

the White queen on d3.

So,

if 2. … Qxd3, then 3. Kc7, whereas 2. … c4 is again met with 3. Bxc4.

---

Evargalo pointed out in the comments that I missed the line

1. … Qb4+ 2. Ka5 Nd4

I think one way out is

3. h7, threatening 4. h8=Q+, and after exchanging the promoted queen with one of the black rooks, to play the white queen to h7 and continue from there.

This tactic is possibly too slow, again as pointed out in the comments, but I'm not sure…

The suggestion in the comments to respond to 3. h7 with 3. … Nc6 can perhaps be countered with 4. Bxd5. Note that 4. … exd5 loses to 5. d7+ Kxd7 6. Qxd5+ and so on. So, Black must instead continue with 4. … Ng8+ 5. Ka7. Now, if 5. … exd5, then 6. Qb5+ Qxb5 axb5 opens up the a-file for the white rook on a2, and seems to allow White to promote at least one pawn to a queen, if not two, so I guess Black is losing in this position. Similarly, if instead we have 5. … Nd7, then 6. Qb5 forces Black to capture on b5, and after axb5 by White we are in a similar situation as before.

However, what does seem to be a dead draw is 3. … Qb8 instead of 3. … Nc6. There is not enough time to pivot the White queen back over at h7 to continue the attack after promoting at h8. So, Evargalo is essentially correct that this line is not fruitful.

Answer 5

I see that Evargalo already gave the key move, but since he did not post the entire variant (and I think 5 xa4 is actually better defense than xb1): my main variant.

Main variant: Bc4 Qxd2; B4 Qxb4; a6 Qxc4; Kb7 Qxa2; Qb1 Qxa4; Qb6 Qd7; Qc7 Qb5; Kc8 Qa6; Qb7 Qxb7; Kxb7 Kd7; e8Q Kxe8; Kc8 Kf8; h7 Rg6; h8 Rg8; d7 ?; d8

Answer 6

This answer is wrong.

1.Rxd5

threatening

d7#

looks strong to me. If

1. ...Qb4+

simply

2. Ka6

and black will run out of checks soon. If

1. ...exd5

then

2.d7+ Kxd7 3.Bxd5 Qd2+ 4.b4 Qxd5 5.Rd2 Qxd2 6.Qf5+ Kc7 7.Qe5+ Qd6 8.e8N+

and wins.