40…Nh4+ 41.gxh4 [else, Re8# preceded by Qh3+ if necessary]

41…Qg4+

(1) 42.Kg1 [or Kh1] Re1#

(2) 42.Kf1 Qh3+ 43.Kg1 Re1#

40 ...Nh4+
The move cuts off f3 from the king and white is faced with either taking the knight or moving the king to the back rank. But the back rank won't do because (if 41. Kg1 or Kh1) then 41 ...Re1#.

On the other hand 41. Kf1 Qh3+ then white has a choice - whether the black rook delivers checkmate or the black queen: a) 42. Qg2 Qxg2# or b) 42. Kg1 Re1#

So all of this leaves only:
41. gxh4 Qg4+
White can't move the king to h1 because ...Re1#
42. Kf1 Qh3+
43. Qg2 (forced or else mate with the rook again)
43 ...Qd3+
44. Kg1 Re1+
45. Qf1 Qxf1#

There are lots to be said about why black can pull of the tactical fireworks (e.g., the how the white knight and rook are cut off from defence of the king), but it is really the black rook on the open e-file that is the killer here.

Probably white played it out to 43 ...Qd3+ and then resigned. It should be noted that black might miss the queen capturing the queen - it is always possible that black takes the white queen on f1 with the rook instead. (Incidentally that was the move that I thought of first until I realized of course that mate with 45 ...Qxf1 is a much better ending to the story). But even in that case (i.e., 45 ...Rxf1+), it is quite hopeless for white since black can scoop up the knight and also easily stop any of white's pawn from queening. (For example, 45 ...Rxf1+; 46. Kg2 Qxc4; 47. d7 (or 47. b6 Qxa6) Rd1; 48. Rxa7 Qg4+ and then ...Rxd7. Alternatively, 47. Rxa7 Rd1 and neither 48. d7 or 48. b6 will work. I suppose it is possible that white played past 43 ...Qd3 in the very slim hope that black might take the queen with the rook but again black will win with best play past that point as far as I can tell.)

In any case, here again is the best sequence without comment:

What makes it easy is a forced line seems paramount.

This is an easy one:

40...Nh4+ 41. gxh4
other moves are even worse: 41. Kf1 Qh3+ 42. Kg1 Re1#; or 41. Kh1 Re1#

41... Qg4+ 42. Kf1
all other moves: mate in one

42...Qh3+ 43. Qg2 Qd3+ 44. Kg1 Re1+ 45. Qf1 Qxf1#

In the meantime I am missing the weekends, were I complained not long ago to waste my time.

Once again.... Thursday? This is not a very varied solution and not hard to spot.

<40....?>

White: a4, b5, d6, f2, g3, h2, Nc4, Ra6, Qd5, Kg2

Black: a7, c5, f7, g6, h7, Nf5, Re8, Qd7, Kg7

Material: +♙

Characteristics:

Strategy
1) White has a very healthy passed pawn extra.

2) Both White and Black have quite some space, but Black has a loose position and doesn't seem to be able to use his space.

3) Both White and Black have active pieces: all pieces, except for Ra6.

4) White's Ra6 is the only inactive piece otb, but it's not bad after a potential Rc6.

5) White controls the passed pawn d-file, but it´s blocked. Black's control of the open e-file is extremely crucial in this position.

6) White´s pawn chain is pretty secure, as the b-pawn (the extra pawn) supports a6. Only d6 is an isolated pawn, but dangerous and well defended. The King side is safe. Black's pawn chain on the queen side is rubbish having lost the b-pawn. Both a7 and c5 are very weak. The king side is identical to White's.

Tactics:
1) White can check with Qd4†, Qe5† and Qxf7†: important here is the power of the centralized Queen. Black can check with Ne3†, a royal fork, and Nh4†.

2) White's King seems safe but has relatively weak light squares around him. Black's King has the same problem around the black squares, but those aren't as weak as White's light squares.

3) White nor Black has loose pieces. All pieces are adequately defended. Pc5 is in though, and there's little Black can do.

4) White's Pd6 is pinned to Qd7, but it is not going to move for any reason. Black endures no pins.

5) White nor Black have potential skewers.

6) White nor Black have potential discoveries.

Candidates: Ne3†, Nh4†

-ML-
The position's characteristics shows that White is actually more active and has a pawn more, so Black wants to attack the only weakness White has and use his own strength: light squares around the White King, and the Rook's e-file. This works perfectly with the clearance sac, actually leading to mate.

<40....Nh4†!> we enter, once again (see yesterday), a completely forced sequence here. All variations lead to mate.

/A\
<41.Kg1/Kh1 Re1‡ 0-1>

/B\
<41.Kf1 Qh3† 42.Kg1 Re1‡ 0-1>

Candidate Moves: ♘h4+

...I almost got it. I missed the ♕g2 interposition.

After Nh4+, if gxh4, Qg4+ really stands out!
And then you notice if Kf1, Qh3+, Qg2 followed by Re1+, Kxe1 and Qxg2, and you sort of have an advantage (am I missing something here?)