johnlspouge: Saturday (Very Difficult): White to play and win.
Material: B for N. White has a battery Re1 and Re2 on the e-file against Pe6. His Qc2 is active, attacking the light squares near the Black Kg8; Ng4 and Bh4 attack the dark squares. The Black Pg7 must protect against the move Nf6, forking Qd7 and Kg8, so Bh4 may occupy f6 with impunity. The Black Nf8 must protect the light squares around Kg8, to prevent the moves Qg6 and then Nf6+. Only the White Bf1 needs to be activated.
Candidates (30.): Rxe6 (deflecting Nf8)
30.Rxe6 Nxe6 [else, drop a P and permit invasion by the Rs]
Candidates (31.): Qg6, Rxe6
31.Qg6 (threatening 32.Nf6+ then 33.Nxd7+ or 33.Qh7#)
To meet the threats, Black must move Kg8, because moves with Ne6 or Qd7 are ineffective.
(1) 31…Nf8 32.Re8 Qxe8 33.Nxh6+ Kh8 34.Qxe8 (threatening 35.Qxf8+)
34…Nb6 [gxh6 35.Qxf8+ Kh7 36.Qf7+ Kh8 37.Bxf6#] 35.Bd8 Kh7 36.Qxf8
36…Kxh6 [gxh6 Qf7+ and mate soon] 37.Qxd6+ K moves 38.Bxb6
leaving White with Q+B+3Ps for 2Rs.
(2) 31…Qf7 32.Nh6+ winning Qf7.
(3) 31…Q other feasible move 32.Rxe6,
threatening 33.Re8# and if Nc8 moves, 33.Re7 and mate soon, so there is no feasible defense.
The Kg8 has 2 legal moves.
(4) 31…Kh8 32.Rxe6
threatening 33.Re8#, 33.Nf6 gxf6 [else, 34.Qh7#] 34.Bxf6+ Qg7 35.Qxg7#, and 33.Re7
There is no feasible defense.
(5) 31…Kf8 32.Nxh6 (threatening 33.Qh7 34.Qh8#)
Black cannot accept the sacrifice, which takes the critical defensive square f7 away from Qd7:
32…gxh6 33.Rxe6 (threatening 34.Rf6+ and mate soon)
33…Ra7 34.Rf6+ Qf7 35.Rxf7+ Rxf7 36.Qxh6+
White emerges with Q+2Ps and the initiative against an open K for 2Rs, so Black must refuse the sacrifice. To prevent the threat in 32.Nxh6, Black cannot prevent 33.Qh7 with 32…Ng5 33.Bxg5, so he must prepare to play 33…gxh6.
32…Ra7 [or Nb6] 33.Rxe6 (threatening 34.Qf4+)
Because 33…gxh6 is a transposition of a previous losing line, and because of 33…Ne7 34.Qf7#, Black can resign.
The key to the solution is the follow-up 32.Nxh6. The position demands denial of 2 key defensive squares, f7 to Qd7, and e7 to Nc8. White must delay capture of Ne6 in favor of 32.Nxh6, which contains enough venom to accomplish the 2 objectives.