Endgames Without a King

Continuing with The Most Theoretical Endgames, there was one type of endgame I didn't mention, because it didn't fit as nicely into my schema as the others did. An example is shown in the diagram on the right. It's a 'four piece' endgame where the White King is facing three connected passed Pawns. The Black King is supposed to be occupied elsewhere on the board and isn't shown.

This type of endgame doesn't bother human players, who can easily work without the missing King. The engine, however, goes into a panic and starts displaying inaccurate results. As far as it's concerned, the King *must* be on the board.

Lone King vs. Three Connected Pawns

The Convekta version of 'Comprehensive Chess Endings' (CCE) by Averbakh goes a long way to solving this problem. It imagines four possible configurations where the Black King is on the board, but otherwise occupied. I've indicated these configurations by the letters (A) through (D) in the diagram. If you merge one of the configurations on the Queenside with the position to be studied on the Kingside, the resulting position is fully satisfactory to the engine.

Although the position might be satisfactory for an engine, it is not necessarily satisfactory for solving the endgame. Configuration (A), for example, allows the Black King to shuttle indefinitely between a8 and b7. This means the Black Pawns are never in zugzwang, a common endgame mechanism in all sorts of positions, especially when there are only Pawns.

Configuration (B) has the disadvantage that the Black King has no legal moves. This allows Black to sacrifice its Pawns, when the game ends in stalemate.

Configuration (C) works much better than (A) or (B). The position is in perfect equilibrium, where neither the White Pawns nor the Black King can move without disturbing the equilibrium and losing. I used (C) to study most of Averbakh's example positions with the aid of an engine. The engines are amazingly fast in analyzing these positions, sometimes calculating to depth 30 or more in a few seconds.

Configuration (C) falls down in positions where the start position of the White King is gradually shifted to the Queenside, i.e. to the e-file, then the d-file, and finally to the c-file. In those positions, the White King has the alternative of rushing to its a- and c-Pawns, thereby helping them to promote.

Although I didn't feed it to an engine, configuration (D) looks like a solution to that last glitch. Not only do the Black King and White Pawns prevent each other from moving, the Black King prevents its adversary from coming to the aid of its own Pawns.

I've seen other endgames where one or both Kings were supposed to be elsewhere. Now I have a trick for working with them also.