Pawn endings: Emanuel Lasker and Reichhelm

white to play

I'm reading a chess book and here on pawn endings I find this example. The book says b1 is the winning move for white, but I don't understand why this is. Maybe because of distant opposition? Why b2 results in stalemate?

This line is proposed as winning for white: 1. Kb1 Kb7 2. Kc1 Kc7 3. Kd1 Kd7 4. Kc2 Kd8 5. Kc3 Kc7 6. Kd3

But doesn't 4. Kc2 loses the distant opposition for white?

If you think that you have an improvement over 4...Kd8, what would it be? Consider, for example:

After 1 Kb2 Ka8, how would White proceed?

Well, this line is proposed by the book, so I suppose it must be ok.

And I know Kb2 is bad for white. My book say it is (but doesn't say why). I wanna know if Kb2 is bad because it loses the distant opposition; if so, why 4. Kc2 is a winning move, but not 1. Kb2.

I suggest trying to experiment. Try to win after 1 Kb2 Ka8 and see what happens. I'll take Black if you like. Try to find a way for Black to survive after 1 Kb1 and see what happens. I'll take White if you like.

I think this is a "corresponding squares" situation, rather than being to do with distant opposition

I've had a long look at this, and it's all about corresponding squares. The concept of corresponding squares is described here:

http://www.captainchess.com/theory-corresponding-squares/

Although my solution is correct I don't see how this could be worked out over the board, since you are not allowed to make any notes, and I would not be clever enough to work this out without writing things down. So if anyone can shed any light on how they'd calculate this in practice then please let me know.

I started with a couple of observations:

(1) If the White King moves to c4, the Black King must move to b6, otherwise the White King gets to b5 and wins

(2) If the White King is 2 files or more to the right of the Black King, and it is White to move - then White wins. E.g. if the White King is on d3 and the Black King on b6 and White has the move then he can win be Ke3 and running to h4/h5 where he will be able to squeeze the Black King out and win the f5 pawn.

Therefore if the White King is on d3, the Black King must be on c7. The reason for this is that from d3 White is threatening Kc4, and from (1) above we know that Black must be able to reply to this with Kb6 - and since the King must be to the right of the b-file (see rule 2 above), the Black King must be on c7, since c6 is covered by a White pawn.

So we have 2 pairs of corresponding squares:

c4 - b6
d3 - c7

For each of these pairs of squares, the Kings are in mutual zugzwang - i.e. if White has the move then it's a draw, and if Black has the move then White is winning. This is true for all corresponding squares.

Now, here's where it turns into a logic puzzle for which I would need to keep notes...

To work out the corresponding square to c3:

from c3, the White King can reach d3 or c4. So to hold the game, the Black King must be able to reach (but not be on) b6 and c7. So the Black King must be on b7 (since Kc6 is illegal). So:

c3 - b7

Next try d2:

From d2 the White King can reach c3 and d3. So Black King must be able to reach b7 and c7, also must not be on the b-file (Rule 2 above). So c8

d2 - c8

Next, with similar reasoning:

c2 - b8

For b3, the White King can reach c2, c3 or c4 from our list. So the Black King must reach
b8, b7 and b6. There are 2 squares possible for this, a7 and c7. So:

b3 - a7 or c7

Continuing on like this:

b2 - a8 or c8
d1 - c7
c1 - b7
b1 - a7 or c7
a1 - b7 or b8
a2 - b7 or b8
a3 - b7 or b8

So in the original problem, White to play with Black's King on a7 - looking for a7 on the
right hand side of my list above, we have:

b1 - a7 or c7

So White should play 1 Kb1. If White plays 1 Ka2 then 1...Kb7 or 1...Kb8 draws. If White plays
1 Kb2 then Ka8 draws

Although it was quite satisfying to work this out, I wouldn't be able to do this calculation
over the board, since you're not allowed to take notes. Is there an easier way that I've missed?