Material: Down N for B+P. The Black Kh7 has 1 legal move, to h8. White has a c-file battery, Rc1 and Rc7, with Rc7 on the same rank as Nd7, Bg7, and Kh7. The Black Nd7 might be embarrassed if it has to retreat. Other than diffuse pressure against the Black position, however, the only immediately obvious strategic theme is a heavy piece invasion. The White Nf3 and Ba3 are relatively inactive. The White Qb5 is threatened by Ba3, requiring an immediate response.

Candidates (31.): Qxd7, Qc6

31.Qxd7 Rxd7 32.Rxd7, threatening

33.Rcc7 34.Rxg7+ winning 2Ns for R.

The threat also includes winning Pe5, so the resulting passed Pe4 would give White the initiative and a winning advantage.

Black can try to counterattack.

32…Qb3 [or Qa2] 33.Rcc7 (threatening 34.Rxg7+)

cedes White a winning attack. The heavy pieces Ra1 and Qb3 must now defend Kh7 from a swarm 2Rs+2Ns, which can avail itself of powerful attacking moves like Ne8 and Nh4-Nxg6. Black has no feasible defense, e.g.:

(1) 33…Qxa3 34.Rxg7+ Kh8 35.Nh4 then 36.Nxg6#

(2) 33…Rg8 34.Ne8

(threatening 35.Nxg7 or 35.Rxg7+ 35.Nh4 36.Nxg6#)

34…Bb5 35.Rxg7+ Rxg7 36.Nf6+ Kh8 37.Rc8+ Rg8

38.Rxg8+ Qxg8 39.Nxg8 Kxg8

and White has N for P. Other moves 32… by Qg8 make no difference, because doubled on the 7-th rank, the White Rs can force the Black Q to lose contact with e8 and h4, permitting the variations above.

Black can try to frustrate the White plan of doubling Rs on the 7-th.

32…Bc8 [or Rd8] 33.Re7

and the Rs will double. Finally, Black can try to prepare a defense to the doubled Rs and the pin on Bg7.

32…Kh8 33.Rcc7

To avoid the original threat (on move 31., losing B+N for R), Black can move Bg7.

(1) 33…Bf6 34.Nf7+ Kg7 [or Kh7]

35.N7xe4+ Kh8 [else, drop Qg8 to avoid mate]

36.Nf7+ Kg7 37.N7g5+ Kh8 38.Rh7+ Qxh7+ 39.Rxh7+ Kg8 40.Rxh6

and White is up 2Ns+P for B. A similar maneuver is successful against 33…Bf8.

(2) 33…Bf8 34.Nf7+ Kg7 [or Kh7] 35.N7xe4+ Kh8

36.Nf7+ Kg7 37.N7g5+ Kh8

38.Bxf8 (threatening 39.Bg7+ 40.Bf6+ 41.Bxg5+ 42.Rh7+ and mate)

38…Qxf8 [Rxf8 permits the finish above] 39.Rh7+ Kg8 40.Ne6

The bishop is well and truly pinned and there is no escape for the black king. 33..h5 (trying for wiggle room)
34 and 35 sees the knights close in for white and curtains for black (at any rate a lost queen).

Nowadays, I generally spend about 2.0-2.5 hours on a Sunday puzzle; 1.5-2.0 hours on a Saturday. I do not know if that is "LONG", but it is less than 6 months ago by a factor of about 2-3. Today, I had little doubt about the key move, which helps a lot. After that, I just grind out variations, often as fast as I can type (hence, my interest in a methodical way of listing the defenses). I do not use a board, and I now have myself trained to visualize and remember critical positions in the middle of variations, which speeds the process enormously. I do not have access to a computer today, so I cannot check my variations, but I believe I caught most of the tactical themes.

A common script for stuffing a King's Indian is to rip open one or two queenside files, grab one, and invade with the heavy pieces. This game is a good case in point. If you know this script, then you know to look for sacs like this.

I normally spend about 20 minutes trying to solve the hard problems. Once I get my "solution" then I usually start computing to check my solution and the problem, and this can take longer, but the computer is doing the work then.

So 20 minutes is about the most I would spend on the first pass - and I usually try to figure it out in ten or less. Most of the positions are late middlegames when 10 or 20 minutes is probably all that you could afford without running into time problems.