Corresponding squares

I did a little programming project.  With the use of a chess tablebase (for the Gaviota engine) and some Python, I managed to generate a list of all pairs of squares in mutual zugzwang in the position in the original post.  Corresponding squares are supposed to be squares in mutual zugzwang.  So, here is the information I found.  Red numbers are squares the black king is on, and black numbers are squares the white king is on.  For example, g6 has a red 28 written on it, and g2 has a black 28 written on it, so if the black king is on g8 and the white king is on g2, the kings are in mutual zugzwang.  Note that many squares have multiple numbers on them.

With just a couple exceptions, these squares of mutual zugwang seem to all be distant opposition.  So it appears the problem cannot be solved with corresponding squares alone.  However, these mutual zugzwang squares do help somewhat, for instance if white plays 1. f2 then black draws with 1. ... f6 because f2 and f6 are mutual zugzwang squares (both with number 26).

Conceptually speaking as you mention this chess board shows distant opposition.  However more generally, corresponding squares do not need to be in oppostion in order to be corresponding.  For example, White g2 has its corresponding square for Black in f-file not on g-file.

No, g2 has numbers 27 and 28, so if black is on g8 or g6 when white is on g2, it is mutual zugzwang.  There is no mutual zugzwang with white on g2 and black on the f file.  So, unless corresponding squares can refer to squares not in mutual zugzwang...  can they?

About your last question, you know the answer is no.  About your first sentence, my comment is "no" either.  g2 should have its corresponding square on f-file not g-file.  I repeat;  corresponding squares do not need to be in opposition.

There is an easy way to check whether two squares are in mutual zugzwang in this position.  Set up the position in an endgame tablebase with white to move.  White to move should be a draw.  Then switch so it's black to move.  Black to move should be a win for white.  The numbers in the diagram above were generated by computer (as I said, I wrote a script) so it should be 100% accurate unless I made a mistake transcribing them from the script output into the diagram.

Look at this other example.  This was taken from the wikipedia article:

http://en.wikipedia.org/wiki/Corresponding_squares#An_example_with_triangulation

It turns out that the wiki article is apparently WRONG. There are no relevant corresponding squares (defined as mutual zugzwang) in this position.  (There are some squares with mutual zugzwang but they are far away from the current position of the kings, and not relevant).  The wiki article claims that c3 - e3 correspond, and c2-f4, and b2-f3, and b3-f3.  Not accurate!  If white is on b2 and black is on f3, it doesn't matter whether white is to move or black is to move - white wins regardless since he can move to b3 and triangulate.  So either this example is wrong, or there is MORE to corresponding squares than simple mutual zugzwang - maybe in some cases corresponding squares do NOT involve mutual zugzwang.

The wiki article is apparently RIGHT. This is a classic CS example.

The issue, as you can confirm with an endgame tablebase such as Nalimov, is that the claimed corresponding squares in that example are NOT cases of mutual zugzwang!  So what are they?  For example, if the white king is on c3 and the black king is on e3, if white is to move he wins, and if black is to move, white still wins.

I have already answered that question of yours -twice. I will not answer it for the third time. Go back and read what I've already written.

Hi Berder.  Your second previous to last post made think about the algorithm that defines a Corresponding Square (CS).  Which by the way I much appreciate, so thank you for that.  Your definition is;
IF White moves THEN draws ELSE White wins.  That is correct for the classic first example that follows:

Dvoretsky also shows that Mined Squares are a type of CS.  In this case the algorithm is:  IF White moves THEN it loses ELSE White wins.  This is shown in example 2:

There is yet another case of CS.  Triagulation.  IF White moves then wins ELSE White wins.  Yes, in other words, White wins regardless.  Presented in example 3:

In summary there are three cases of CS:

1.- IF White moves THEN it draws ELSE White Wins

2.- IF White moves THEN it loses ELSE White Wins

3.- IF White moves THEN it wins ELSE White Wins

You might notice that the only common element is "ELSE White Wins"  This prompts me to define Corresponding Squares in the following way:

With both Kings in Corresponding Squares you win if your opponent is to move.

In consequence it seems as well that your CS algorithm needs to be revised.

Interesting, LikeTheLake.  The third case - IF white moves THEN white wins ELSE white wins - covers the great majority of winning positions for white, not simply those where there are corresponding squares.

I tried something else:  What if two squares correspond when if white is to move, he takes more moves to win than if black is to move?  So in that case white would rather not move, even though he wins either way.

With this interpretation I revised my script to produce a diagram that agrees with the wiki article.  As before, black numbers are where the white king stands, and red numbers are the corresponding squares where the black king stands.  Note that in this diagram, b2 and b3 correspond with f3 via numbers 37 and 41 (as in the wiki article), c2 and f4 correspond via number 42 (as in the wiki article), and c3 and e3 correspond via number 39 (as in the wiki article).  There are a lot of other numbers but they can be ignored since they are far from where the kings are currently positioned (d2 and f3).

This topic would turn out of much profit if, to every cited concept, like "mined squares", "poisoned squares", "strong squares", "weak squares" etc, someone could provide a good definition, some examples, brief considerations on the importance of the so defined concept, typical situations from which they can arise, and so on. I know this is the moment of "brainstorming", but, later, someone could try to build the glossary, the table of examples. Pawn Endings seem to be a territory where the chances of systematization are much more at hand than anywhere else.

 This the chess mentor lesson title.

There's not really a concrete way to spot them.  You just need to calculate variations where zugzwangs happen and generalize: If he steps here, he steps here, zugzwang.  Then say these squares correspond.

It's worth noting that the 1st example on corresponding squares misses the enthralling starting moves of the original challenge. Here is what it should have been:

Comprehensive Chess Endings: Volume 4 (Pawn Endings) has the best, most thorough explanation of Corresponding Squares that I've seen (Chapter written by Mikhail Zinar). This particular study can be solved using the "Quadratic System" described therein. I would do a very bad job of trying to explain the quadratic system, but the resulting corresponding squares are as follows:

You can start with f7/f5, and g7/g5. Because these two corresponding squares are adjacent to each other for both sides, you use the quadratic system. That system tells us that f8 corresponds with f4, and g8 corresponds with g4. (f8/g8 and f4/g4 are unambiguous because the path to the queenside is of different length). The quadratic system also extends this pattern to the "Rear Squares" behind those four main squares. This means that f3/f7 and g3/g7 are also corresponding. Moving even further you get f2/f8 and g2/g8, and eventually f1/f7 and g1/g7.

He has another (more recent) video also

https://youtu.be/vXH5NJZfbFs

I have that set (mine is old now). Did they revise since the1980's?

I found the first couple of systems fairly easy to understand. The 3rd/4th more difficult (maybe too much in a short time). But once I got to the Two systems (5th system of 10), I pretty much gave up.

I found it impossible to use OTB (probably because I didn't learn all of them) because you can't have board diagrams and make sketches during a game.

I should've tried to use it in correspondence games (post card chess), but had too many games and too little time back then.

Nowadays, I don't think you can use the books in online daily (this site). Last I read, opening books were allowed.

Do you actually use the systems in your play? If so, I think that's an amazing accomplishment.

I can tell many GMs (including the Super GMs) do not use it OTB.