Puzzle 1 from endgame textbook

The correct move is 1. K5.4

It's very important for Red to keep his pawn on one of the
central files. After 1.P5.6? Black can force the pawn to
the 7th file (or the pawn gets lost) and win the game:
1.P5.6? A5+6 2.K5.4 (2.K5.6 C5.4 and Black wins the pawn)
2. ... C5.4 3.P6.7 (3.P6.5 C4.6 4.K4.5 C6.5 and the pawn
is lost again) 3. ... K4+1 4.K4+1 (after 4.P7+1 K4-1 Black's
cannon will be able to pick up the pawn, e.g. 5.K4+1 A6-5
6.K4.5 C4.7 7.K5-1 C7+2 8.P7+1 K4-1 9.P7.8 C7-1 10.P8+1 C7-1
11.K5+1 C7.2, Answer3) 4. ... C4.6 5.K4.5 C6.5 6.K5.4 E5-3
7.K4.5 A6-5 8.K5.4 K4.5 9.P7.6 A5+6 and Black wins, Answer2.

However, Red can hold the position by playing 1.K5.4, see e.g.
1.K5.4 E5-7 2.P5.4 (but not 2.P5.6? A5+6 3.K4.5 E7+5 4.K5.4 C5.4)
2. ... A5+4 3.P4.5 A4-5 4.P5.4 A5-4 5.K4.5 E7+5 6.K5.4 K4.5
7.K4.5 K5.6 8.K5.6 A4+5 9.P4.5 (Answer1)

Puzzle 2 from a game He Liansheng - Li Aidong

The correct move is 1.R6-1

Red has the deadly threats P7+1 followed by P7.6 checkmate
or K4.5 followed by K5.6 and R6+3 checkmate. However, the
immediate 1.P7+1 will be refuted by 1. ... H9+8
2.K4.5 R7.5 3.K5.6 R5-1 and 1.K4.5 doesn't work because
of 1. ... R7.5 2.K5.6 R5-1. Thus, Red has to protect
his central cannon first.

After 1.R6-1 Black has no efficient defense, e.g.
1.R6-1 H9-7 2.K4.5 H7-5 3.P7.6 R7+3 4.K5-1 H5+6
5.K5.4 H6+4 6.K4.5 (Answer1) Since Black is not allowed
to check perpetually, he can't defend against P6+1 checkmate.

In the game, Red played 1.C5+1? with similiar ideas. However,
after 1.C5+1? H9-7 2.K4.5 H7-5 Red's rook is under attack
and has to lose an important tempo. Red played 3.R6+2 R7+3
4.K5-1 H5+6 5.K5.4 R7.4 6.C5-4 R4-7 7.P7.6 P3+1 (Answer2)
and Black had a won position. 

Puzzle 3 from a game ?? - ?? played in 1979

The correct move is 1. ... E3-1

This is a complicated endgame and it's difficult to see
why Black's central elephant needs to be protected by the
other elephant on the baseline.

Black played 1. ... P2.1? and Red managed to win the game:
1. ... P2.1? 2.R5.2 R6+4
(2. ... R6+5 makes it easier for Red, since the Black
rook can't take the central file, e.g. 3.R2+2 K6-1 4.R2-1 E5+7
5.R2+2 K6+1 6.R2.5 R6-4 7.R5-1 K6-1 8.H6+8 R6-1 9.H8+6 R6.9
10.R5-3 K6+1 11.R5.4 R9.6 12.R4+2 checkmate, Answer3)
3.R2+2 K6-1 4.R2-1 R6.5
(After 4. ... E5+7 Red can win as in the line 2. ... R6+5)
5.A4-5 R5-3
(Now it's too late for 5. ... E3-1 because of 6.H6-4 R5-2
7.H4+3 K6+1 8.R2+2 followed by R2.4 checkmate)
6.H6+5 R5.7
(6. ... K6+1 7.R2+1 K6+1 8.H5-3 E5+7 9.R2-1)
7.H5-3 K6+1 8.R2+1 K6+1 9.R2-5 E5-7 10.R2.4 K6.5 11.R4.5 K5.6
12.A5+4 R7.6 13.R5+6 (Answer2) followed by R5.4 checkmate.

With 1. ... E3-1 Black could have hold the position, e.g.:
1. ... E3-1 2.R5.2 R6+4 3.R2+2 K6-1 4.R2-1 E1-3 (this wasn't
possible after 1. ... P2.1) 5.H6+5 K6.5! (Answer1) Red can't
make any progress, but Black has to defend this tricky position
very carefully.