Corresponding Squares

When I was studying a position from Paul Keres, Practical Chess Endings, trans. John Littlewood (1974) this morning, it was familiar. I thought the winning idea was rooted in understanding corresponding squares, especially because Keres placed it in that section of the book (he uses the term "related squares"). Nonetheless, I struggled to calculate the solution without working out all critical cases of correspondence, and only through trial and error against Stockfish did I discover the correspondence between a5 (Black) and d2 (White), highlighted in red below.

White to move

I solved the exercise before reading Keres' solution and discussion of related squares, but could not avoid seeing the paragraph below the diagram: "White's task is made extremely difficult by the fact that his passed pawn is on the rook's file and that he has no manoeuvring space for his king to the left of this pawn" (28). Hence, the general idea was clear from the outset if I did not already know that.

Initially, I thought triangulation would allow me to reach the same position, but with Black to move, calculating 1.Ka3 Kb6 2.Kb2 (distant opposition). But if Black plays 1...Ka6, do I still go to b2? This scenario did not occur in my play against Stockfish. Keres notes, as I learned later, that when the Black king is on a6 or b6, White's king can move to c1, c2, or c3 because all three squares allow Kd2 should Black play Ka5.

My play against Stockfish on the iPad follows.

1.Ka3 Kb6 2.Kb2 Ka5 3.Kb3

Already, I have successfully reached the diagram position above with Black to move.

3...Kb6 4.Kc3 Ka5

White to move

An advantage of playing against the computer is that you can learn immediately when you have gone astray. If the engine suddenly shows an evaluation of 0.00, it is clear that an error has been made. Here, 5.Kd3 seems tempting because after 5...Kb4, White has the resource 6.a3. However, it takes little calculation to see that 7.Ke5 fails because Black has 7...Kb3 8.Kd5 Kb4 and suddenly Black is winning.

I may have tried it anyway, but I do recall that I tried 5.Kc2 and watched the evaluation hit 0.00. What then? Going back to the b-file clearly makes no progress. Hence, there is only one move and I would have found it sooner had I more thoroughly considered all sets of corresponding squares instead of trying to see everything through brute force calculation.

5.Kd2! Kb6 6.Ke3

6.Kd3 is as good. Both squares lead to e4.

6...Kc6 7.Ke4

Tablebases indicate that 7.a3 and 7.Kf4 are both equally good, but taking the opposition in such positions is practically routine.

Black to move

7...Kd6 8.Kf5

This simple outflanking maneuver is not superior to 8.a3. However, it is easier to understand. I am reminded of Vladimir Kramnik's words in the Foreword to the 5th edition of Dvoretsky's Endgame Manual (2020): "it is impossible to attain real endgame mastery by just working with a computer. An explanation of why an endgame is winning, ... described in words and in language that a person understands (as opposed to computer variations), is needed" (12).

8...Kd7 9.Ke5 Kc6 10.Ke6

Continuing with moves that alternate between taking the opposition and outflanking is relatively simple and something I've done hundreds of times. Nevertheless, I spent a fair amount of additional time with this exercise after completing it. 

From the starting position, Stockfish notes that it is checkmate in 26 moves. My first effort following the discovery of 5.Kd2 led to checkmate with two queens on the 28th move. I knew that I could do better.

Where else have I seen this position?

I thought that the diagram Keres presents was familiar, but do not find it in Dvoretsky's Endgame Manual. Dvoretsky does have a similar position as a diagram that reaches a nearly identical position after three plies. it is a variation stemming from inaccurate play from a composition by Franz Sackmann (1913).

Black to move

After 10...Kb7 11.Kb4 Ka6, the position in the diagram from Keres is reached one square up the board. The solution follows the same process. Dvoretsky discusses it in the section on mined squares, noting that the d4 square is mined.

Perhaps this derivative from Sackmann's study was sufficiently lodged in my memory to recognize it, especially if I played out Dvoretsky's analysis.

I did find the original position in Alex Fishbein, King and Pawn Endings (1993), where it is no. 126 and credited to George Walker. However, I bought this book with unrealized intentions and have spent very little time studying it. Walker, A New Treatise on Chess (1841) has the position, which Walker states came up in a game that he observed. It is interesting that in Walker, the position is reached with black to move, but White did not know how to play it. Nor did Walker find the correct way. According to Johann Berger, Theorie und Praxis der Endspiele (1890), it was Josef Kling who pointed out the correct manner of play.

From Walker, A New Treatise on Chess

Walker's line follows.

White to move

1.Ka3 Kb6 2.Kb2 Ka5 3.Kb3

Here, we have the diagram at the top of this post, but with Black to move. It is the position I had to produce via triangulation.

3...Ka6 4.Kc3 Ka5 5.a3?

Walker does not give this move the question mark, I do. My memory fades on whether this was one of my errors this morning. I know I certainly considered it.

5...Ka4 6.Kd3 Kxa3 7.Ke4

Can we call this an example if hope chess? Even if Black plays 7...Ka4, the best White gets is a draw.

Black to move

7...Kb3

Now, White must fight for a draw. To Walker's credit (and the player of the White pieces), the drawing method is demonstrated.

8.Kd3

"Best", according to Walker. These days, the move gets a box--only move.

8...Kb4 9.Kd2 Kxc4 10.Kc2 

White draws by holding the opposition, as noted by Walker.