White to move. How should white proceed?

Yancey, Here is the solution for the second puzzle

V-2

1.h3+ Kh4

2.Kg2 Ba7

3.Rd4+ Bxd4

4.Nxd4 e1=N+

5.Kf1 Kxh3

6.Kxe1 Kg4

It's a win.

V-3

1.h3+ Kf5

2.Nd4+ Bxd4

3.Re7 Bc3

4.a6 e1=Q+

5.Rxe1 Bxe1

6.a7

It's a win.

As for the third puzzle only Rf8 allows to pin the white queen, but this pin doesn't make sense. For example

1.Rf8 Rd1

2.g8=Q Rd3+

3.Qb3 any move along the third rank allows mate

In the third puzzle, consider this line: 1.Re8 Rd1 2.g8(Q) Rd3 3.Qb3. Now, white can't mate in 5 total- it takes 6 moves. I am asking you to describe why this happens with 1.Re8, but 1.Rf8 prevents black from extending the line to six moves. You will other moves at move 1 for the white rook run into the same problem.

As for the other puzzle, you leave out the critical line where black queens the pawn, though I do realize you understood it was still won for white. This was the line Troitsky wanted one to find eventually.

Yancey, here is the solution for the second puzzle.

V-4

1.h3+ Kh4

2.Kg2 e1=Q

3.Rd4+ Bxd4

4.Nxd4 Qe4+

5.Nf3+ Qxf3+

6.Kxf3

It's a winning position for white.

As for the third puzzle

1.Re8 Rd1

2.g8=Q Rd3+

3.Qb3 Re3

4.Qxe3 Kb1

5.Qd3+ Ka1

6.Re1#

Other moves create the threat of stalemate.

The third puzzle, however, is still not quite complete in my opinion because you still haven't explained the precise reason 1.Rf8 leads to mate in 5 while 1.Re8 (or 1.Rc8, for example) can only lead to mate in 6 at best. In other words, you are missing the critical line in 1.Rf8. In 1.Re8 line, black checks along the 3rd rank, and when white blocks the check with the queen, black can refuse the capture, but keep the queen pinned- this leads to the mate in 6 that you give above, however, when white has played 1.Rf8, explain why this same maneuver can't ever lead to mate in 6, but leads to mate in 5 instead.

Well done- that is the line that explains why 1.Rf8 is the shortest mate. The first time I saw this problem, I found 1.Rf8 by chance- it was the second try at a solution for me, and at first, I didn't see this particular line where the queen mates at d1. Here are the two references: https://www.youtube.com/watch?v=-mAAnRJxN6Y

http://web.chessdailynews.com/a-chess-masterpiece-puzzle/

- A chess masterpiece puzzle

White to move. How can White save this game? Source: ChessToday.net Chess Daily News from Susan Polgar

In puzzle #2, what about black's reply of 1. ....Kh4? That is the main line of the puzzle, in my opinion. In addition, in the line you give, what about 3. .....Be5?