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h3+ Kg5 h4+ Kg4 Rf8! Rb2+ Rf2! Rxb7 Rf4#
1.h3+ Kg5 2.h4+ Kрg4 (2. … Kf6/h6 3. Rf8/h8+ etc.) 3. Rf8 (threatening 4. Rf4#) 3. … Rb2+ 4. Rf2 Rxf2+ (4. … a2 5. b8Q Rxb8 6. Rf4#; 4. … a2 5. b8Q a1Q 6. Qf4#; 4. … a2 5. b8Q Rxf2+ 6. Kxf2 a1Q 7. Qc8#) 5.Kxf2 a2 6. b8Q a1Q 7. Qc8#
I got this one in about 20 seconds, but sort of by accident. I wanted to drive black’s king onto the g-file or to h6 so that I could check with the rook on the back rank and queen the pawn, but quickly discovered a deadly mate threat:
1. h3 Kg5 (Kf5 2.Rf8 Kany 3.b8Q)
2. h4 Kg4
Again, Kf5/f6 and Kh6 lose to Rf8 or Rh8 with check followed by b8Q. Continuing:
3. Rf8!
Threatens Rf4# and b8Q. There is only one critical line- where black checks from b2:
3. …..Rb2
4. Rf2!
The only move- otherwise black takes the pawn at b7 if white moves the king. With Rf2, white is again threatening b8Q, but now followed by Rf4#. Black has no good options:
4. …..Rf2
5. Kf2 a2 (what else?)
6. b8Q a1Q (else white covers a1)
7. Qc8#
Or…
4. …..g5 (covers f4)
5. Rb2! ab2
6. b8Q and the new queen covers black’s queening square.
Lastly….
4. …..a2
5. Rb2 a1Q
6. b8Q and black has nothing here- the attack on his king will come long before his queen can get into a position to harass white’s king.
1. h3+ Kg5
2. h4+ Kg4
3. Rf8 (threatening mate on f4)
3. … Rb2+
4. Rf2 Rxf2+
5. Kxf2 a2
6. b8Q a1Q
7. Qf4#
If this is the right solution then I solved it within one minute
1h3 kg5 2h4 kg4 3Rf8 rb2 4rf2 win
1.h3! Kg5 (If 1…Kf5, 2.Rf8+ followed by b8=Q wining)
2.h4+!Kg4 (If 2…Kf5/6 3. Rf8+ and b8=Q+ as above or if 2..Kh6 3.Rh8+ and then b8=Q)
3.Rf8!Rb4(white threatened Rf4# and Black had no time for a2).
4.b8=Q winning easily
If 3..Rb2+
4.Rf2! Rxf2
5.Kxf2.a2
6.b8=Q.a1=Q
7.Qc7#!
Harry
Well, the white don’t have any choice but move his rook to stop the a-pawn. If 1.Ra8 black assegures at last the drawn checking the king on 1st and 2nd rank or if white king decides move from kingside defense, the black king is able to get h-pawn since after ..Rxb7; Rxa6 Rb2 white must try conter attack on g-pawn but cannot win. Well, I consulted my old book, Técnicas de Finais, by Max Euwe & David Hooper, portuguese version (portuguese of Brazil), learning the lessons about rook against two pawns. That lessons inspirates me try this line:
1. Rg8 Rxb7
2.Rxg6+ Kf5
3.Ra6 Rb7+
4.Kf3 a7
5.h4 and if black tries promote a-pawn aproch his king to a file , white could tries get hpawn, sacrifice his rook for the a-pawn and advances his g and h pawn. I didnt decide yet if is drawn or not but I found this line that is a hope for white
It’s ok, h3-h4 and mate in few moves, ok, forget my drawn-line posted before