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I actually remembered to check the material today. Black is up the exchange, so a simplifying combination might work as well as Mate or Ruinous Loss of Material.

There appear to be various and sundry ways of doing this.

Checking for forcing moves, I didn't try the queen sacrifice first since it wasn't Monday. However, 39...Rxg3+ quickly went nowhere, so I tried <39...Qxh3+>. After all, any move that gives you the chance of two discovered checks in a row is fun to look at, if nothing else.

So 40.Kxh3 hxg3+ 41.Kg2 gxf2+. Now 42.Kxf2 Rh2+ regains the queen while winning a pawn and simplifying. 42.Kf1 Rh1+ leads to the same line. 42.Kf3 just lets the pawn promote.

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42...Rf8+ just loses the g-pawn, while 42...g2 runs into 43.Qd7+ Kh6 43.Qxe6+ Rg6 (43...Kh7 44.Qf7+) 44.Bd2+. I haven't analyzed this to any great extent (let the Silicon Creatures have their fun), but offhand it looks to me that White has at least a perpetual check.

Even if 42...g2 were good enough to win, I'd rather force matters with 42...Rxf2+ and enjoy the simple life.

<39...Qxh3+ 40. Kxh3>

[40. Kg1 hxg3 41. Qd7+ (41. Qd3+ Qf5 42. Qd7+ Kg6 ; 41. Qc2+ Qf5 ) 41...Kg6 42. Qd3+ Qf5 ]

<40...hxg3+ 41. Kg2>

(Obviously 41...Qxh6+ 42. Kxh6 )

<41...Rh2+>

Black might as well showboat.

<42. Kg1>

(42. Kf1/Kf3 Rxf2+ )

<42...gxf2+ 43. Kxh2>

(43. Kf1 Rg1+ 44. Ke2 f1=Q+ 45. Ke3 Rh3+ 46. Ke4 Qf5#)

<43...f1=N+ 44. Kh1/Kh3 Nxd2>

39..............Qxh3+!
40.Kxh3 hxg3+

<if 41.Qxh6+ Kxh6 42.fxg3 Re8, Black will have a Rook for a Bishop>

41.Kg2 gxf2+
42.Kxf2 Rh2+
43.Kf3 Rxd2
44.Bxd2
And Black will have R+p for a Bishop

If on move 42, King does not take the pawn and moves to f1:

White may have been congratulating himself on constructing something of a fortress in the puzzle position, reposing confidence in a pin on the h-file: If Black tries taking the g-pawn, he will lose immediately to xh6#, while the bishop on c3 creates some threats of its own with an imminent d7†. This rather forces Black's hand. Fortunately for the second player, he's ahead by an exchange, so a simple exchange of material will work in his favor.

All of this suggests a look at the key move: <39. ...xh3†<>>, after which play might continue as follows:

<40. xh3...<>>

Effectively forced. White can't simply abandon the rook, even if Black didn't have a withering attack to follow.

<40. ...hxg3†
41. g2, h2†<>>

This is simplest and most forcing. White has three legal replies:

(1) 42. f3, xf2†
43. xf2, gxf2
44. xf2, g6

With an advantage of the exchange and a pawn, Black will bring home the endgame without difficulty.

(2) 42. f1, xf2†

This transposes to line (1).

(3) 42. g1, gxf2†† and either

(3.1) 43. xh2, f1=†!
44. h1/h3, xd2 or

(3.2) 43. f1, g1†
44. e2, f1=†† .

Second observation: white, while down in material, has threats of his own, control over the a1-h8 diagonal, attacking the rook on h6, potential check on the seventh rank and, if given time, threats along the b1-h7 diagonal.

Conclusion: nip it in the bud with a series of forcing moves starting with 39... Qxh3+ 40. Kxh3 (practically forced, else ) hxg3+ 41. Qxh6 Kxh6 If white tries 41. Kg2 then gxf2+ continues the forcing sequence and sets up a skewer along the second rank, . Couldn't find a forced mate though.

ENDS

41.Qxh6+ Kxh6; or

If Black can trade off a rook and the Queens, he will win. The straightforward <39...Qxh3+> accomplishes this after a short series of forced moves:

<39...Qxh3+ 40.Kxh3 hxg3+> This puts White in a quandary:

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White has two replies, 41.Qxh6+ & 41.Kg2, both losing:

(1) <41.Qxh6+ Kxh6 42.fxg3> and Black has a simple exchange-up won game

(2a) <41.Kg2 Rh2+ 42.Kf1 (or 42.Kf3)...Rxf2+> and Black wins since he's up an exchange plus a pawn

OK, let's run the Q-sac line.

39 ... Qxh3+ 40 Kxh3 hxg3+ 41 Kg2 Rh2+

a) 42 Kf1 g2+ 43 Ke2 g1+Q (44 Qd3+ Qg6)

b) 42 Kg1 gxf2+ 43 Kxh2 f1=N+ and Black has R+passer vs. B, and BK is trapped on the h-file.

Oh, not quite. WK is on the f-file.