1. ## Pawn endgame study by Ed. Lasker

Here is a nice little study that Edward Lasker made I believe. I can't find the book that reference it to him, so that is from memory. White to move and win. It is quite a challenging position for the player that has not seen this position before, and a set of 'corresponding squares'. In fact White must make about 10 precise moves where each move needs to be the only correct one, and if at one he fails, a draw results. Put your favorite box on this one, and it will probably be clueless (I would like to hear if that is not the case). Black can choose to follow many different move orders to confuse White, which makes it fun both to attack and defend this position.

I will give you the first two half-moves. 1.Kb1! is the only winning move. It seems that 1...Ka8 is the hardest nut to crack, although Black has other moves here. Can you continue the analysis?

 PGN Viewer

2. One has to find correspondence(hope it's a correct English term) squares, more a maths problem.
The pairs are:
c4-b6 (if black king is on a6 white can march to a king side)
d3-c7 (same reason)
c3-b7 (to access b6 and c7).
d2-c8 (that was the hardest to find. Black has to be near enough to king side and next to both b7 and c7)
c2-b8
b3-a7
b2-a8. That's where I stopped as it's enough to win

1.Kb1 Ka8 2.Kb2 Kb8 3.Kc2 Ka7 3.Kb3 Kb7 4.Kc3 Kb6 4.Kc4 is a sample

3. Further analysis. Other pairs:
d1-c7, c1-b7, b1-a7.
If white king is on a1,a2 or a3, black should keep their king on b7-b8. The moment white king moves to the b-file, black should put their king on a correspondent square (a7 or a8).
Which means after 1.Ka2 both 1...Kb7 and 1...Kb8 draw. After 1.Kb2 Ka8! is the only move.

4. Analysis from Igor in white:
Originally Posted by Igor_Goldenberg
Which means after 1.Ka2 both 1...Kb7 and 1...Kb8 draw. After 1.Kb2 Ka8! is the only move.
I agree with all your analysis. All your corresponding squares seem to check.

But if you just use the corresponding squares, it is not enough, because then the game would run 1.Kb1 Ka8 2.Kb2 Ka7 3.Kb1 Ka8 ... and eventually draw by repetition. I am not saying you suggest that. My point is that White needs to give up his opposition in the right way to regain it subsequently - how does he do that?

5. Originally Posted by Jesper Norgaard
Analysis from Igor in white:

I agree with all your analysis. All your corresponding squares seem to check.

But if you just use the corresponding squares, it is not enough, because then the game would run 1.Kb1 Ka8 2.Kb2 Ka7 3.Kb1 Ka8 ... and eventually draw by repetition. I am not saying you suggest that. My point is that White needs to give up his opposition in the right way to regain it subsequently - how does he do that?
Check my first post where I gave a sample. White has more corresponding squares and they move king toward c4. The idea is to play Kc4 when black cannot respond with Kb6. For example 1.Kb1 Ka8 2.Kb2 Ka7 3.Kb3 and so on.
If 3...Ka6 then 4.Kc2! Kb6 5.Kd2 Kc7 (forced, otherwise white runs to a king side) 6.Kd3 Kb6 7.Kc4.

I cannot cover every possible black move, but the algorithm is pretty simple.

6. Originally Posted by Jesper Norgaard
My point is that White needs to give up his opposition in the right way to regain it subsequently - how does he do that?
Opposition by itself is not important, it's just a tool to fight for access to key squares. In that example this tool is not required.

7. Originally Posted by Igor_Goldenberg
Check my first post where I gave a sample. White has more corresponding squares and they move king toward c4. The idea is to play Kc4 when black cannot respond with Kb6. For example 1.Kb1 Ka8 2.Kb2 Ka7 3.Kb3 and so on.
If 3...Ka6 then 4.Kc2! Kb6 5.Kd2 Kc7 (forced, otherwise white runs to a king side) 6.Kd3 Kb6 7.Kc4.

I cannot cover every possible black move, but the algorithm is pretty simple.
Yes that was the variation I was looking for

8. Originally Posted by Igor_Goldenberg
Opposition by itself is not important, it's just a tool to fight for access to key squares. In that example this tool is not required.
I agree that opposition is just a tool to fight for access to key squares, but all the same the notion of giving up the opposition in the appropriate moment to win it back with advantage, is useful. For instance the position
White:Ka4,f5
Black:Ka7,f6
White must play 1.Ka5! to gain opposition, and continue to hold it for a while to win. Instead 1.Kb5?,Kb7 is a draw. And even worse is of course 1.Kb4?,Kb6 where it is Black who will win the pawn, although this time it is still a draw. So the game will continue 1.Ka5 Kb7 2.Kb5 Kc7 3.Kc5 Kd7 4.Kd5 Ke7. Here White can continue to take side-opposition with 6.Kc5 but that will lead nowhere because of 6...Kd7. Instead he must give up opposition with 5.Kc6 the important thing here is that Black is not able to take it with 5...Ke6 because of the fxe6 threat. Instead he can maintain opposition with 5...Ke8 but that helps him little after 6.Kd6 Kd8 7.Ke6 Ke8 8.Kxf6 Kf8 9.Kg6 Kg8 10.f6 Kf8 11.f7 etc. and White wins even if he never regained the opposition.

In the present pawn ending the import opposition is Kc4 vs. Kb6. That means that it is different than the normal opposition without pawns in the way of for example Kc4 vs. Kc6. This opposition could be called the Knight's opposition (because c4 and b6 is at Knight's move distance). If Black plays Kc7 we want to maintain or win the Knight's opposition with Kd3 (opposite color square, one file more to the right).
If we look at your winning continuation 1.Kb1 Ka8 2.Kb2 Ka7 3.Kb3! Ka6 (Ka8 was no longer possible because of 4.Kc4 and 5.Kb5) White can no longer maintain the Knight's opposition and win the game, he must give it up right now with 4.Kc2! (4.Kc3 Kb7! 5.Kc2 Kb8 or 5.Kc4 Kb6 would be a draw) because it obliges Black to the unfavorable squares b6-c7 if he wants to maintain opposition, so 4...Kb6 5.Kd2! Kc7 (Kc6 is not possible) 6.Kd3 and with that move White regains the opposition and wins easily. He either wins with Kc4 and Kb5, or runs to the h-file to break in (6...Kb7 7.Ke3). White gave up opposition with 4.Kc2! to regain it with 6.Kd3.

In resume I think opposition (Knight's opposition) is very important in this exercise, as well as understanding when to give it up and why. I know Fritz or Rybka solves these without knowing opposition, but for a human the understanding of opposition makes you able to reason more clearly, and recognize the patterns.

9. Originally Posted by Igor_Goldenberg
If 3...Ka6 then 4.Kc2! Kb6 5.Kd2 Kc7 (forced, otherwise white runs to a king side) 6.Kd3 Kb6 7.Kc4
I would argue 7.Kc4 is inaccurate. It will maintain the opposition, but gains nothing after 7...Ka6 where the right path is 8.Kd3 Kb6 9.Ke3! and winning in the h-file. So the accurate move order would be 6.Kd3 Kb6 7.Ke3! White still wins but loses 2 tempi with 7.Kc4

10. Sorry for examining your answers so rigorously, Igor, but I'm sure that even if you didn't learn anything (you're a big boy now) perhaps a lot of other players learned something!

11. Originally Posted by Jesper Norgaard
I agree that opposition is just a tool to fight for access to key squares, but all the same the notion of giving up the opposition in the appropriate moment to win it back with advantage, is useful.
IG is right though. In 1988, I was in a chess seminar in the then USSR, and Yuri "Mr Endgame" Averbakh himself said that the goal was to penetrate (to certain key squares) and the opposition should be used only if it's a means to this end. His example was:

 PGN Viewer

Or else: the opposition is a subset of corresponding squares in many positions.

12. Originally Posted by Jesper Norgaard
I would argue 7.Kc4 is inaccurate. It will maintain the opposition, but gains nothing after 7...Ka6 where the right path is 8.Kd3 Kb6 9.Ke3! and winning in the h-file. So the accurate move order would be 6.Kd3 Kb6 7.Ke3! White still wins but loses 2 tempi with 7.Kc4
You are right, of course. I was just illustrating the concept of correspondence squares. In that position 7.Kc4 is completely unnecessary.

13. Originally Posted by Jono
IG is right though. In 1988, I was in a chess seminar in the then USSR, and Yuri "Mr Endgame" Averbakh himself said that the goal was to penetrate (to certain key squares) and the opposition should be used only if it's a means to this end.
In fact we are probably just entering a philosophical area. The fact that seemingly no computer chess programs have been programmed to use "the opposition" is a clue - opposition is not necessary to find the best moves, period! The computers just calculate to the end.

But the human brain that works better with ideas that are universally true over many variations, than with concrete calculation, is often helped by "perceiving opposition" in many positions. A human has difficulties maintaining a variation tree with many overlapping positions in the head. The opposition as an idea, a vision, can help.

Whether "Knight's opposition" as I propose for this study makes sense to all humans or not, is individual. It seems IG disagrees. Anyhow I think I have no objections to Jono's and IG's and Rybka's ways of handling things - it leads to the same solutions of course. The right solutions!