johnlspouge: Saturday (Very Difficult): White to play and win.
Material: Even. White has mating threats against Kd8. There are no immediate tactical strokes, so a strategic plan is required. The Black K is subject to mate threats when on c8 (temporarily, because of Rb7), d8, e8, or f8. The Black Pg7 takes away a flight square from Kd8 and cannot advance without immediate loss hxg6 and creation of a White passed Pg6. The White Rg3 has access to the invasion point f8 to the right of Kd8, but on the 3-rd rank, because of Pb4, Rg3 has no access to the left invasion point on the 8-th rank (e.g., a8) that would force an immediate win. White must leverage mate threats into tempi to restrain the Black Pb4, to advance Pe6, and to capture Pg7 and/or Ph6.
Candidates (44.): Rf3, Rg4
This counterintuitive move renders the Black Rb7 immobile. White aims to move his R to the 7-th rank, forcing the Black K to the b- or g-file to escape the mate threats. As the variations below show, if the K escapes to the b-file, the Black K-side Ps fall; and if to the g-file, the White R can cut the Black K off long enough for the White K to control e7 and force d8=Q without the aid of the White R, which can then sacrifice itself with impunity for the Pb4.
If Black does not play 44…b3 (e.g., 44…Ke8), Rg4-c4-c5 gains access to the a-file and then the 7-th rank anyway, and play proceeds as below.
44…b3 45.Ra4 (threatening 46.Ra8#)
45…Rb1 [Kc1 46.Ra8+ Rb8 47.d7+ Kc7 [or Kb7] 48.Rxb8 Kxb8 49.d8=Q+]
46.Ra7 (threatening 47.Rxg7 48.Rg8#)
The Black Pg7 must fall. To meet the threat 47.Rxg7, Black can counterattack:
(1) 46…b2 47.Rxg7 Kc8 48.Rg8+ Kb7 49.Rxb8+ Kxb8 50.d7 b1=Q [Kc7 is no better]
51.d8=Q+ K moves 52.Qd5
White defends both Pg2 and Ph5 and cuts the Black K off from the K-side. Black cannot prevent the White K from approaching Ph6. After Ph6 falls, White has a slow but certain win with Q+2Ps vs. Q.
Black can also frustrate the threat 47.Rxg7 directly:
(2) 46…Ke8 [Kc8 47.Rc7+ is an transposition into (1) with Pb3 instead of Pb2]
47.Rxg7 Kf8 48.Rf7+ Kg8 [Ke8 49.Rh7]
49.d7 (threatening 50.Rf1 51.Rxb1 52.d8=Q+)
Because the White K controls e7, the best choice for Black is a losing Q+P vs. R endgame.