Formulating Bähr's rule

Everyone is familiar with the idea of calculation shortcuts from their math education, and most chess players are familiar with the "rule of the square", which allows you to determine visually whether an unescorted passed pawn will promote without the need for a complete calculation. There is a similar technique known as Bähr's rule, which can be used in situations with blocked rook pawns where one side has an extra outside passed pawn. It was first formulated in 1936 by Walter Bähr in his book Opposition und Kritische Felder im Bauernendspiel or Opposition and Critical Squares in Pawn Endgames.

Bähr's conception of the rule

Bähr's original formulation of the rule required that the attacking king stand directly next to its passed pawn and that the defending king stand directly opposite that pawn or the opposing king. In those cases the board could be divided in two with a diagonal line and a result determined like this:

The problem with Bähr's rule is that, unlike the simple rule of the square, it works only in a very precise situation. If such an endgame arises and the kings are not standing exactly where Bähr prescribes, then the rule is useless.

Dvoretsky's method

Recently while studying these endgames and trying to understand Bähr's rule myself, I realized that it has been formulated very differently by other endgame theoreticians. I believe it was helpful for my understanding of the idea to study all three versions, and it was also an interesting exploration of the relationship between these visual rules of thumb, which use board geometry, and the brute force calculation that determines the precise truth of a position.

In his famous Endgame Manual, Mark Dvoretsky offers a different formulation of the rule, which allows it to be applied more broadly. His method involves memorizing a "normal" or model position and then comparing each new situation to that model. The player will first compare its blocked pawns with those in the model position and make tempi adjustments, then likewise with the position of the passed pawn, and the same with the king position. In trying to work all of this out, it seemed to me that the calculations involved might just as well have been spent on a direct calculation of the position itself. Perhaps I haven't fully appreciated something, however, and since he is considered the greatest chess author and coach, I plan to return to it someday.

Müller and Lamprecht's formulation

A third version of the rule was formulated by Karsten Müller and Frank Lamprecht in their excellent book Secrets of Pawn Endings. Their method involves using the passed pawn and the bishop file to find an intersection point, and then to compare the distance between the defending king and the passed pawn with the distance between the defender's pawn and the intersection point. If the distance between the defending king and the passed pawn is shorter, then the game is drawn.

It's not clear to me yet how this version differs from Dvoretsky's in practical application and which would cover more situations, but in my opinion, Müller and Lamprecht strike a nice balance: they retain a fairly simple visual idea while still covering more situations than Bähr's original formulation.

In all of these versions, however, there are a few exceptions, so it's debatable whether or not players should bother to memorize it, considering that it covers a very specific endgame position. Personally, I think it's worth studying, because it can help you make a snap decision in a time scramble or a blitz game, and even with plenty of time it would be a useful way to double check your calculations. It's also possible that just by studying these three versions of the rule, even if you don't ever use it, you will internalize something important about this endgame that will improve your instincts. If you choose not to, however, just remember the underlying principle: when you're going to sacrifice that passed pawn, try to make the route as long as possible for the enemy king.