Sub-page of EndgameProblem40

During the endgame, a good practical method is to compare endgame plays in pairs. Most of the time this makes determing the exact numerical value of plays unnecessary. This problem is related to Endgame Problem 40 and is also by Bill Spight and used by permission. It touches on a method called difference method, which may well be above the head for kyu players.

Eric's idea of pairwise comparison is a good one. One way to do that is with the difference game, as expounded in *Mathematical Go*, by Berlekamp and Wolfe. Sometimes the difference game will give a clear winner, but even if it does not, it can suggest which alternative is preferable.

Here is the difference game with a mirror position in the bottom right corner. In the left corner Black has played atari, in the right corner White has descended.

The right corner is settled. White has 4 points. White cannot afford to play at *a*. If Black plays first, he wins.

As people have pointed out, White can make ko.

But Black takes the ko first. The theory behind difference games does not contemplate kos. (Even if we use the right corner for ko threats, Black wins the ko.)

How big is the ko?

The diagram shows two alternatives in the left corner:

If Black fills (left diagram), the score is +1 (for Black). If White wins the three-step ko, the score is -6 (right diagram).

The difference between winning and losing is 7 points, and there are 5 moves between these results, one for each step, and one for each player to win the ko. The value of each play in the ko is 1.4.

Let's stop play before playing the ko and assess the position.

In two moves Black can reach a position worth +1. Each move is worth 1.4, so this position is worth -1.8. Let's call it White's win.

Plainly, there will be some whole board positions in which one move will be correct, and others in which the other will be correct. That is normal between truly competitive alternatives.

But Black can win by 3 with one move, while White can only win by 1.8 in one move. That suggests that the atari is bigger than the descent.