# BQM 222

Keywords: Question, Theory, Game commentary

I ran into this beast in a the 4th game between Rin Kaiho and Sakata Eio for the 22nd Honinbo title (1967) ( SGF).

How should I play it? In the game White’s move at a carried a very large threat, so Black would block at b. It seems like white a dominates d, but how do black c and d compare?

Lin (W) - Sakata

This was the position in which the problem arose.

Here White (Lin) chose over a move at .

## Bill’s analysis of the infinitesimal

Bill: This is a fearsome infinitesimal in chilled go. How to play it?

For the answer, let’s set up a difference game.

Difference game

First we mirror the bottom of the board, changing the colors of the corresponding stones. This produces an even position. Then we let Black play the stone and White play the stone, to compare these two plays for Black. If one play is better than the other, this position will favor the player who made that play.

### Difference game for Black’s options

Difference game, Black first

First, Black takes profit with - . Then White pushes in with and Black connects with . After Black has one more point in the bottom than White has in the top. Black to play wins the difference game.

Difference game, White first

is sente. Then and are miai. The result is jigo. Jigo is good for the second player, Black.

Black to play can win the difference game, while White to play cannot. Thus the difference game favors Black and the hane, , is better, unless the ko situation tells against it.

### Difference game for White’s options

Now, how should White play it?

Difference game II (White’s options)

Here we set up the difference game to compare White’s options. Does White play the hane-tsugi in the bottom, or block off the corridor in the top?

Difference game II, Black plays first

pushes and plays hane-tsugi. Then pushes, but with the result is jigo.

Difference game II, White plays first

White plays the hane-tsugi at , and pushes, but finally White wins by 1 point.

White to play can win the difference game, while Black to play cannot. So the difference game favors White, and the hane-tsugi is correct unless the ko situation dictates otherwise.

### Conclusion

Thanks Bill! So it’s a { * | 0^2 Miny-6 } then? It looks less like a beast written like that :).

Bill: Looks that way. :-) It looked more fearsome before I realized that in the Difference game, White first diagram was the right local play. ;-)

PJT I hoped to convert the game value to MathJax, but am not confident what the ASCII notation stands for as I found nothing like it in ONAG or WW;
I now suspect that “0^2 Miny-6” stands for “{:-_6:}|0^2”, analogous to “{:-_2:}|0^3” on p.23 of MGE (though the opposite way round),
i.e. shorthand for “{{:-_6:}|0" || "0}”, where  -_6  is the conventional notation for miny-6, i.e.{6|0" || "0}
This would make the entire value  { ast" || "{:-_6:}|0^2 }  or  { ast | {{:-_6:}|0" || "0} }  but I am unfortunately not quite able to work it out for myself :(

## Bill’s comparison of Lin’s play with the hane-tsugi

Bill: The natural follow-up question is how Lin’s play compares with the hane-tsugi on the bottom. We can answer that question with a difference game, too.

Difference game, original board
Difference game, mirror board

Bill: First we set up the difference game with White making Lin’s play on the original board and Black playing the hane-tsugi on the mirror board.

Difference game, White plays first
Difference game, White plays first, mirrored

White starts with sente, threatening the middle Black group, and then plays the hane-tsugi on the original board. Now the bottoms of the two boards are miai, and are equal. Black has nothing better than , but wins the difference game by 1 point.

Difference game, Black plays first
Difference game, Black plays first, mirrored

at a.

Black starts with the hane, , which leaves the bottom on the first board as a STAR in chilled go. Now we are on familiar territory. The rest of the game plays itself. Each player advances in the corridors until . Black wins the difference game by 1 point.

Since each player, playing first, wins the difference game, which play is better depends upon the rest of the board.

BQM 222 last edited by PJTraill on February 1, 2019 - 00:05