When Cortex first gave me this problem about 3 years ago, I thought I had solved it immediately (literally in 10 seconds) with a queen sacrifice:
The idea behind this move is that the queen is invulnerable- for example:
And white will mate in 4 moves total. However, it didn't take me more than a minute to realize that black didn't need to cooperate- he can just leave the white queen alone:
And white has a problem. If he plays 2.Qc1, black will mate white quickly (at move 5), and if he plays 2.Qa2, black has 2.....Qf4 after which I could never find a line that white didn't end up getting mated- everything for white after that is simply too slow to stop 3.....Qc7+. Finally, and most importantly, if white plays 2.fe7, black has the resource of 2...Qg5! 3.Qb2 a1(Q) 4.Qb4 Qe7!! 5.Qe7 Qb2+ and mate will follow by move 8 at the latest. However, this wasn't a wasted exercise- I realized that Qb1 probably plays a role at some point, but I needed to play it later- for example, the knight at f2 blocks a check from h2 white would have liked to play at move 4 in the sideline immediately above- this is important to remember. Also, from a clue Cortex had given me that the puzzle was "geometrical", I immediately realized that there were repetitive motifs involved, and looking at the position I saw really quickly the idea of 1.Qh2+, followed by the maneuver up the two diagonals h1/a1 and h2/b8. This allows an easy repetition draw and thus easy to spot, so I started there since it allowed me to get rid of that knight at f2:
Here, if black tries 2....Bd6, white takes at d6 and then buries black with 3.Kc7!!. I saw this easily at this point, and it plays a big role in the solution much later in the problem when it is a much harder move to find if you didn't see it at this stage. So keep Kc7 in mind. Continuing:
Now, because of the control of the c6 square by the knight at d8, white can't profit by continuing to march the queen up the two diagonals to close in on the black king, and he has traded the bishop for the knight that was on f2. In addition, white doesn't have time to play 8.fe7 since black has his own 8.....b1(Q)+ coming to win the position. So, it was pretty clear to me that my original idea of Qb1 had to be played to stop this, and the only way to do this was to return the white queen to h1, but the white queen has to return to the h1 square with check in order to prevent black from playing b1(Q) in reply- so we reverse the march up the board to go back down the ladder:
So, we have returned to the starting position, but minus white's bishop and black's knight that was on f2- the real key here is the missing knight, however. So, at this point, there really is nothing more to explore in this position but the offer of the queen at b1 that allows white to threaten fxe7. So...
So, now the question is- how does black best defend this? Even though I had already found the problems with Bxf6 earlier in the line, Cortex had given me an important clue to this by encouraging me to explore the variation at this point where black plays 14...Bxf6- it actually reinforced the idea of Kc7 by white at some point- so I leave as an exercise the following variation: 14....Bf6!? 15.Qb2! Bd4+ 16.Qd4 c1(Q)! 17.Qd6+! Ka8 18.Kc7!!. You can prepare yourself for what comes later by thinking about this position, though it will have changed in a major way by the time white is forced to play Kc7. In any case, the most testing defense black has is the same one I outlined right at the beginning- queening a pawn at c1:
15.fe7! Qg5! (everthing else loses quickly!)
16.Qb2! a1(Q)! (all else loses)
And from here all the moves for white at move 17 lose except for....
I mentioned that I thought of this move in the 1.Qb1?? line, but it wasn't possible at the key point because black had a knight at f2 that physically blocked access to the h2 square at the critical point- that was the point of white's first 7 moves- to get rid of that knight just so white can play 17.Qh2 here. Black is forced to play...
And from here it got really hard. I had these first 17 moves within an hour of Cortex giving me the problem. He had confirmed that I was on the right path however, and he had already given me the really important hint that allowed me to play a key move later on with confidence that I could do so and ignore all the rest of the ideas- a not unimportant clue for a problem of this complexity. So, what do we have here after 17 moves for each side? Black has two queens for white's one queen, and black has the key square of d8 guarded by the queen at g5. In addition, black's other queen is just waiting for the time to play a check on the white king. Indeed, this threat by black makes white's next move fairly easy to spot since there can't be anything else- it has to be a check and so.....
Of course, black can't take the queen at g2 since it allows 19.ed8(Q)#. All black can do is....
Since there is nothing else, I just brought the white queen back up the board, but I also already had the main idea in hand, I just needed to put the white queen on the correct square before playing the next key idea....
Now we have reached another juncture- the question is can white take at g5 with the 22nd move? I don't see any way for white to avoid repetitive checks here if white plays 22.Qxg5- it is important that doing so gives up control of b8 square to black, so white has to play the inaccurate line as 22.Qg5? Qb2 23.Kc5!, but then there is nowhere for the white king to hide. So, white must continue to bring the queen up the board....
I had already, moves earlier, planned to put the white queen on d6. The reasoning was quite clear to me- it protects the pawn at e7 which black has been threatening to take since his 15th move of Qg5. In any case, I was now ready to tackle the idea of......
I have asked myself every time I revisit this problem this question- could I have found this move in this position unless I had found it in earlier, but inaccurate lines? I don't know the answer. Here, the move looks crazy since black now has two queens on the board, and it gives black a free move to check the white king. However, after some trial and error I did over a couple of days, I convinced myself that this was indeed the right idea- there are fortunately only two moves black can make here- the two possible checks from a5, but they are, ultimately both the same as far as I have ever been able to demonstrate:
26........Qaa5 (Qga5 is left as an exercise)
Here, it is mate if white plays 27.Qb6. Continuing:
I spent a lot time on the alternatives here for black- this really is the only try, but I leave the details to the curious. At least Qgb5 forces white to find an underpromotion to guard the key squares of b7 and c6...
An easy move, I think, to find- all the other moves are short mates. Basically, at this point of the puzzle, I was just working with a bit of faith that I was on the right track- I really couldn't see how this would finish, but I decided I would track this beast until I got the bear or the bear got me. In this position (FEN after white's 28th move is k1KN1N2/3P1Ppp/p2QP3/qq5p/2pP4/8/8/8 b - - 0 1), we have reached the technically most difficult point in the puzzle- the main problem for me were the numerous side variations that had to be closed off before one could identify the main line of the puzzle. Black has here all the pawn moves, and several moves with the queens that look plausible. I spent most of the two weeks I worked on this puzzle at this point. I am not going to post all the details, but I will highlight the most important parts of the most important variations. So, let's start with the two most obvious moves with the queens- Qab6 and Qbb6:
This move threatens Qb7+ followed by mate, so white really has no choice but to play....
This isn't a hard move to find in my opinion- it is the only plausible move white has here even though it looks crazy in isolation- white threatens to create a new queen at d8, gives the king an escape square, and the knight move prevents an immediate Qb8. Black really has only Qxc6 to play here, but I leave all the other moves as exercises for the curious:
31.Kd8 Kb7 (if c3, white plays Ke7 anyway)
32.Ke7 Qc7 (pinning the d-pawn again)
And it should be obvious white is going to win this- the pawns on the 7th rank are just too powerful. So, even 28....Qbb6 loses in this variation. In addition, if black plays a move like h4, g6/g5, or h6 at move 28, one should be able to easily show that 29.Nh7 or even 29.e7 wins quite easily. The details are left as exercises, though if you have specific questions about them, I can easily answer them in the comments section since I have prolific analysis of them. So, let's return to the mainline above after 28.ed8(N) (see the diagram immediately above for the position). After a lot of work, I determined that black's only real try at a defense at move 28 is...
This move has several threats behind it- first it is two moves from a 3rd black queen- mainly this prevents white from winning with a move like 29.Nh7- that move is too slow now. Indeed, c3 prevents all the other ways of winning for white- for example [29.Qc6 Ka7!! 30.Qb5 Qb5 31.Nc6+ Qc6 32.Kd8 c2! and white is toast]. It took me quite a while to convince myself that white could continue at move 29 with....
From here, black has exactly two plausible options that don't lose as we have seen above- either Qb3 or to continue with c2. I am going to return to the idea of Qb3 at move 29 for black later in a comment to this story- this variation was a thorn in my side for a considerable time before I finally resolved it to my satisfaction, but it is a story in itself. Suffice it say at this point that it eventually, and essentially transposes into the main line of the puzzle, but with some issues that are still dangling loose as I write this. Black does best, though, to continue to push the c-pawn:
The only move that doesn't lose for white. Continuing:
30............Qc6 (if 30...Ka7? 31.Qb5! Qb5 32.Nc6+-)
31.Nc6 Qd5! (everything else loses, left as an exercise)
A second underpromotion, but again, not hard to find- all the alternatives are easily shown to be worse for white- white can't move the knight at c6, and has to find a way other than Kc7 to protect it- that leaves only the underpromotion. Continuing:
What else can black do here? The only alternative is the following illustrative line: [32....Qc6 33.Nc6 c1(Q) 34.Nd7!! Qc6! 35.Kd8 Kb7 36.f8(Q) Qc7 (if 36....Qe6? 37.Qb4+ starts a mate) 37.Ke8 h4 38.Qf3+ Ka7 39.Qe3! Ka8 40.Qe4+ Qb7 41.Qb7 Kb7 42.Kf7+-]. Believe me when I tell you- I examined this side varation to exhaustion- white wins all the lines after 32. ...Qc6 too. Continuing:
Again, this move is easy to find- white has nothing else even close to plausible. However, it was at this point in the puzzle that I first had a vision for what the ending might be- I could count the pieces here- if I was on the right track, and I believed I was- I knew that black could likely give up both queens for one of the knights and both of the remaining white pawns- I could literally see the line that follows over the next few moves. Like most anyone, I was doubtful that white could mate with the king and two knights- it was not the sort of thing I had any experience with outside the normal technical drawn ending most players are aware of, but I had hopes since the black king is already trapped in a corner. With 33.Nd7, white is threatening Nb6# and f8(Q)- black really has only...
Again, if Qc6+ 35.Kd8 will win as I show in the paragraph above. 34....Qe6 pins the d7 knight, removes one of the white pawns- is the only try- it also sets a devious trap.....
White has to move something, and the "obvious" 35.f8(Q) only draws as far as I have ever been able to demonstrate: [35.f8(Q)? Qc6+ 36.Kd8 Kb7 37.Qb4+ Qb5! 38.Qe4+ Qc6!=]. No, white has only the one legal knight move that protects the f7 pawn for a move. This allows black to get rid of that last white pawn, however.....
35........Qd6 (the only relevant line now)
Is this the only winning move? It is the only one I ever found here- white's moves are constrained, and what I found in the alternatives was that he either goes into this ending, or allows a repetition of moves. Continuing:
36........Qf8 (or get mated)
37.Nf8 Ka7 (or get mated quick)
At this point, I sorted of cheated- well, ok, cheated straight up. I had a suspicion that this was a forced win, but I wasn't sure, and I am terrible at endings with knights, and probably double terrible at two knight endings. I could see that white can keep the black king from escaping the corner, but I didn't want to spend a day or two working out the finish if all I had here was a forced draw. So I took this position after 37 moves- an 8 man position- and turned it into a 6 man position by removing the two black h-pawns, and also the each of the h-pawns along with the g-pawn and put those positions into an online Nalimov Tablebase. This allowed me to confirm that the position is winnable with all the black pawns on the kingside removed in 23 moves, and is winnable in 13-14 moves if one of the black kingside pawns remains along with the a-pawn. It is drawn if all the black pawns are removed, and it is winnable if the a-pawn is gone, and black has the other pawns on the kingside, but those are very, very long wins in this case- up to mate in 83 more moves. In any case, I considered it solved at this point, and you can put the described positions above with the various black pawns removed into the Nalimov Tablebase at move 37 if you want to see how black gets mated by move 51 in this puzzle. All in all, this was the most difficult puzzle I ever solved- at least solved by my standards, though some can quibble about my use of the Nalimov Tablebase to confirm if and how white mates black with the two knights and the king after the moves at 37. In any case, Alena showed how to mate from this point back in October.