The critical endgame position in a recent online correspondence game illustrates how a handful of simple ideas are the keys that unlock the truth.
Black to move
This position came about after White thrust his h-pawn forward with hopes of creating a structural weakness in Black's pawn structure. White's plan was to exploit the resulting weakness with his rook while his king guarded the open file. Black responded to the h-pawn thrust with an offer to exchange rooks. White believed that he had a winning pawn endgame, and so traded rooks.
Was White correct in this assessment?
White's h-pawn is en prise. Capturing this pawn gives Black a passed pawn on the h-file. White invites this capture because his king is in the square of the pawn.
The square of the pawn is a simple idea that should form part of the instruction for every beginning player. Imagine a square made up of some of the squares on the chessboard. The first two corners are the pawn's square and its promotion square. Two squares an equal distance to the side form the other two corners. If the defending king is in this imaginary square, he stops the pawn.
The Black king is outside the square. However, if it is Black's move, he steps into the square and stops the pawn. If it is White's move, the Black king will always remain one square outside as the pawn runs and the king attempts a futile chase.
Note that if the White pawn were on b2, the square would be the same. The king always move one square. A pawn may move two if it has not yet moved.
In the game, not only does the White king have the ability to stop a passed h-pawn, he can penetrate into Black's position along the same. Black's king, however, can prevent decisive penetration there.
The game continued 32...gxh4 33.Ke2 Ke6 34.Kf1 f5 35.Kg2 Kf6 36.Kh3 Kg5
White to move
The Black king has stopped White's kingside penetration. Alas, Black has a problem on the queenside that also requires the attention of his king. White has a pawn majority. Even before the beginning chess player learns about the square of the pawn, he or she should learn about majorities. A pawn majority offers conditions for creating a passed pawn by force.
I teach the concept of majorities to young children by having them play Pawn Wars (first pawn to promote wins) from the following position.
Both sides have pawn majorities. Who will promote first? With best play, the player who moves first will promote first.
The game continued 37.c4 bxc4 38.bxc4
Is the Black king within the square of White's passed c-pawn? Yes. However, chasing the c-pawn gives up the kingside pawns. White will create another passed pawn. My opponent opted not to chase my pawn. Instead, he sought to create his own passed pawn. This quest was successful.
38...fxe4 39.c5 Kf4 40.fxe4 Kxe4 41.c6 Kf3 42.c7 Kxf2 43.c8Q e4
From here, the queen easily mops up Black's pawns. The most important ones are the a-pawn and the e-pawn. White could capture the a-pawn, exchange the queen for the e-pawn, and then run the b-pawn to promotion.
Back to the original position. What if Black chose a different move, say 32...Ke6. White may still create a passed pawn via 33.c4 bxc4 34.bxc4 Kd6 35.c5+ Kc6
White to move
The White king must remain close, ready to step to e2 to remain in the square if Black ever captures the h-pawn. Meanwhile, Black threatens 36...a5! 37.bxa5 Kxc5 when the Black king is in the square of White's a-pawn, will capture it, and will then return to action on the kingside where knowledge of opposition and triangulation, as well as the gxh4 threat will win the f-pawns, then the e-pawn.
Instead, if Black does not capture the h-pawn, White must resist the queenside pawn break. Instead, the players create fortresses with limited points of penetration. White cannot force issues on the kingside. Black cannot force issues on the queenside. White's king must remain within the square of the potential passed pawn on the h-file, but Black need not create it. Black's king must remain with the square of the potential passed c-pawn, but White need not create it.
I played the position against Rybka 4. The truth of the position prior to 32...gxh4?? is that the game should be a draw. Perhaps White's decision to trade rooks was an error, or perhaps that exchange was necessary to avoid a losing position. White was wrong to believe that he exchanged into a winning pawn endgame even though he managed to win it.