The key here is to realize that the unprotected white rook moved to h6 cannot be taken because of the second rook check and mate on g6. Once that is noted the solution is very easy:
1. R*h6+ Kg7 (otherwise 1.…Rh7 2 R*h7#) 2. …Rg1+! and mates after (2…Ng5, R*g5)K*h6 4. Rg6#.
1 Rxh6+ Kg7
2 Rg1+ Ng5
3 Rxg5+ Kxh6
4 Rg6#
Rook check followed by rook check, preserving the White knight, which is vital for the checkmate position.
1. Rxh6+ Kg7
(1. .. Rh7 2. Rxh7#)
2. Rg1+ Ng5 (2. .. Kxh6 3. Rg6#)
3. Rxg5+ Kxh6 4. Rg6#
1.Rxh6 Kg7 2.Rg1 Ng5 3.RxNg5+ KxRh6 4.Rg6+mate
Gotta be Rh6:
1. Rh6 Kg7 (Rh7 2.Rh7#)
2. Rg1! Kh6
3. Rg6# an always pretty rook mate motif.
Rxh6+ Kg7
Rg1+ Kxh6
Rg6
The key here is to realize that the unprotected white rook moved to h6 cannot be taken because of the second rook check and mate on g6. Once that is noted the solution is very easy:
1. R*h6+ Kg7 (otherwise 1.…Rh7 2 R*h7#)
2. …Rg1+! and mates after (2…Ng5, R*g5)K*h6 4. Rg6#.
Psyche
If only my Knights could reach the f6 “arabian” square more often… the rest is simple.
1.Rh6+ Kg7 (1…Rh7, 2.Rxh7# classic Arabian mate)
2.Rg1+ Ng5 (2…Kxh6, 3.Rg6# another classic Rook and Knight mate)
3.Rxg5+ Kxh6
4.Rg6#
A 1.Rxh6+ Rh7 2.Rxh7 mate.
B 1.Rxh6+ Kg7 2.Rg1+ Ng5 3.Rxg5+ Kxh6 4.Rg6 mate.
Kamalakanta
1.Rxh6+ Kg7
(1…Rh7 2.Rxh7#)
2.Rg1+ Ng5
(2…Kxh6 3.Rg6#)
3.Rxg5+ Kxh6 4.Rg6#