Puzzle 1 from an endgame textbook

The correct move is 1. ... C4+2

To understand this endgame it's important to know
that cannon and two advisors can hold against a
rook in many positions. Typically they form a "three
advisor formation" yielding a draw when the king 
has enough space to move. However, there are certain
positions where the rook can win against cannon
and two advisors.

1. ... C4.3? yields one of the positions where the
rook can win:

1. ... C4.3 2.P5+1 E7+5 (everything else is hopeless)
3.R5+3 C3+6 (or 3. ... K5.4 4.R5.7 C3.1 5.R7+2 and Red
wins the cannon) 4.R5.7 C3.5 5.R7+2 A5-4 6.R7-6 C5-1
7.R7+1 C5-1 8.R7+1 C5-1 9.R7+1 C5-1 10.R7+1 C5-1
11.K5-1 C5+6 12.R7.5 and the cannon is lost (Answer 1)
1. ... C4+2 is much better. If Red plays as in Answer 1:
2.P5+1 E7+5 3.R5+3 Black can play 3. ... K5.4 and has
reached the typical "three advisor formation" with 
sufficient space for the king. Black's king can idle
(K4+1 K4-1 K4+1 ...) If Red hinders the king from idling
(e.g. 4.R5-1 K4+1 5.R5.8 K4-1 6.R8+3 K4+1 7.K5-1) the
cannon can take the idling part: 7. ... C4+1 8.R8-3 C4-1
(Answer 2)
However, Red's alternatives are not better as long as
Black is careful.     

Puzzle 2 from a Chinese master game 

After 1. ... P7+1 2.R2.3 the position looks quite drawish but
this is the much better move compared to 1. ... H2+3?

After 1. ... H2+3? 2.P3+1 R6.7 3.C1.3 (but not 3.R2.7 because
of 3. ... H3+5 4.E7+5 C1+7 5.E5-7 R7+3) Black's horse is
in trouble because of the threat 4.R2.7, see for example:
a) 3. ... R7.4 4.R2.7 R4+2 5.C3.4 (Answer1) followed by C4+2
b) 3. ... R7.1 4.R2.7 R1+2 5.A5+6 H3-1 (5. ... R1+2 6.C3+6 C1.7
7.R7-1) 6.C3.7 K5.6 (or 6. ... A5+4 7.C++7 E5-3 8.C7+8 A4+5 
9.C7.9, Answer3) 7.C++7 E5-3 8.C7+8 K6+1 9.R7+3 (Answer2) with 
strong attack.
c) 3. ... R7.1 4.R2.7 H3-1 5.R7.8 R1-1 6.C7.9 H1-2 7.C9+5 (Answer4)
and Red has a piece more (7. ... E3+1 8.C3+6)
     

Puzzle 3 from a Chinese master game

1. ... H7+5 is the better move.

After 1. ... R8.7 Black will regain a piece after
2. C3+5 R7-3 3.E3+1 R7-3 and his attack is still dangerous,
but the position isn't clear, Red may be able to organize
a successful defence beginning with 4.R8+1 (Answer1)

1. ... H7+5 is much more convincing. The threats H5+6 and P5.6
are very dangerous, see some lines as examples:
a) 1. ... H7+5 2.R8+1 H5+6 3.R4-4 R5.2 Answer2, Black wins
b) 1. ... H7+5 2.R8+1 H5+6 3.R8.5 H6+7 4.K4.5 R8.5 5.K5.6 H7-5
6.R4-6 R5.3 7.E3+5 R3-2 Answer3, Black wins
c) 1. ... H7+5 2.C3+7 P5.6 Answer4, Black wins
d) 1. ... H7+5 2.C3.5 R5.7 3.C5-2 H5+6 4.R8+1 (4.C5+7 A5+6 5.E7+5 
R7.5) H6+7 5.R8.3 R8.7 6.R4-6 (6.C5+7 A5+6 7.R4-1 R7.5) 
R7+1 7.K4+1 R7.6 8.K4.5 R6-2 Answer5, Black wins