Because I am teaching the concept of breakthrough in pawn endgames to young chess players this week, I also am honing my own skills. Chess Endgames (1999) by Laszlo Polgar contains 4560 positions. The bulk of the book consists of 169 themed sections, each containing 24 problems. Yesterday, I started working through the two dozen classified as "breakthrough". The second led to my post "Breakthrough". The fifth problem created some frustration: I could not find a solution.
I study the diagram in a book, and when I believe that I have the correct ideas worked out, I set up the position and play against an engine.
White to move
The theme offers a clue, and so 1.c5.
Houdini 1.5 replied 1...bxc5. 1...dxc5 is obviously no good in the light of 2.d6 cxd6 3.Kxd6+-. (Correction noted: see comments below.)
2.Kb5 Kc8! 3.Kxa5 c6 4.dxc6 d5!!
White to move
White cannot win from here. After perhaps half an hour or trying different lines, including some that lose for White, I checked the answer in the back of Polgar's book. He gives 2...Kd7 for Black. My engines--Houdini 1.5, Hiarcs 12, Rybka 4--concur that White is winning after 2...Kd7. They all find 2...Kc8 to equalize, however.
The key variation given in the solution is instructive:
2...Kd7 3.a4! Kc8 (if 3...c6+ 4.dxc6+ Kc7 5.b3! losing a tempo) 4.Kxa5 Kb7 5.Kb5 and the solution carried it out to move 13, but the plan is simple. The a-pawn will advance to deflect the Black king from defense of c7. Once that pawn falls, the White king mops up the entire Black army.
Black to move
Here 5...c6+ fails. 6.dxc6+ Kc7 7.a5 c4 8.Kxc4 Kxc6 9.b4+-.
In the original problem, Black can play the game of tempos too. That strikes me as the beauty of 2...Kc8! Instead of the logical seizure of the opposition with 2...Kd7, Black makes a move that leaves both Kb7 and Kd7 as possibilities. If White tries 3.Kc6, Black has adequate space for maneuver to keep the rearward c-pawn secure.
I ran Rybka in Monte Carlo mode for four hours. Then, I ran infinite analysis for another four. This morning, I let Hiarcs 12 work for nearly two hours. I am convinced that there is no win here.
Books usually have errors. This one frustrated me yesterday. Even so, failure is instructive.
3 hours ago