BeginnersEndgameExercise6/Solution

Black first  

captures the White stones and makes territory. Later, - is sente. (Remember that playing out sente plays gains nothing, on average.)

In the region between and (inclusive), Black has 14 points in territory and dead stones. In addition, Black has captured 1 White stone. White gets 1/3 point for in the ko. So the local count is 14 2/3.

White first  

is tesuji. If now , captures the Black dumpling.

After Black cannot respond at a because of damezumari.

To find the local count, let's look at the followers of this diagram.

Black follower  

After the rest is dame and protective plays.

In between the points (inclusive), White has 1 point of territory, while Black has captured 1 White stone. So the local score is 0.

White follower  

threatens , but Black throws in there, forcing White to protect through .

By comparison with the previous diagram, White has gained 1 point for taking . So the local count is -1 (one point for White).

Each play is gote, so the local count for the White first diagram is the average of 0 and -1, or -1/2.

The original plays are also gote, so the local count of the original position is the average of 14 2/3 and -1/2, or 7 1/12.

The miai value of a play is then the difference between 7 1/12 and -1/2, or 7 7/12. (That's also the difference between 14 2/3 and 7 1/12).