Bill, you write faster than I can read. I really appreciate your effort. Very enjoyable quality read. It was great to use the oldest known game record for a joseki question. It shows that there must have been a sensible amount of training and knowledge around back then. Do you happen to know, why that game (or half of it) got recorded and survived till today?

I got lost with the tinies and minies. Is there a good reason for the difference? As I see it, a tinie for one player is a minie for the other and vice versa. I assume that they also appear in some sort of capture go.

_________________
If something sank it might be a treasure. And 2kyu advice is not necessarily Dan repertoire..

Thank you, bayu. I am glad you like all this. And I am writing a bit faster than I am comfortable with.



A question better addressed to John Fairbairn or Peter Shotwell. Since making my comment I have found out that that game record may have been a forgery. In any event, the book in which it appears was written a millenium later than the game was dated.



Tinies and minies are infinitesimals. A tiny is a White sente, and is a plus for Black; a miny is a Black sente, and is a plus for White. Here is a miny in straight No Pass Go.

Click Here To Show Diagram Code

[go]$$Bc Miny
$$ -----------
$$ | O X . X . |
$$ | O X X . X |
$$ | O O O X X |
$$ | . O . O X |
$$ | O . . O X |
$$ -----------[/go]

Each player has 1 point of territory, for a net count of 0. But there is a Black sente, or miny. Minies are less than 0, which means that White wins, regardless of who plays first.

Click Here To Show Diagram Code

[go]$$Bc Black first
$$ -----------
$$ | O X 3 X . |
$$ | O X X . X |
$$ | O O O X X |
$$ | . O 2 O X |
$$ | O 4 1 O X |
$$ -----------[/go]

is sente, leaving White with 1 point of territory after . The result is 0 with Black to play, so Black loses. If now and , Black can resign.

Click Here To Show Diagram Code

[go]$$Wc Miny
$$ -----------
$$ | O X 2 X . |
$$ | O X X . X |
$$ | O O O X X |
$$ | . O 3 O X |
$$ | O . 1 O X |
$$ -----------[/go]

is reverse sente, leaving White with 1 point of territory. The net score is 0, so Black to play loses, as above.



There are no known tinies and minies in regular go, except in chilled go, as explained on SL. I suppose that that holds for the Capture Game and Capture-N, as well.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I always feel embarrassed finding out that I can't even count to 3.
How do you count white as 1? I naively count it as 1.5: if white plays first, it's 2 points; if black plays first it's 1 point; take the average.

Maybe got something to do with Sente gains nothing. Black has the privilege to play the sente, so white's count is 1. But then:



Sente gains nothing. I also thought that sente loses nothing either. Is this not true for straight No Pass Go?

I hope these questions make sense. I'm not entirely sure;)



Thanks for explaining this. Maybe, Senseis could stand some improvement. The examples are given for chilled go, but regular go is not mentioned under tinies. It might be obvious when coming from a CGT perspective.

_________________
If something sank it might be a treasure. And 2kyu advice is not necessarily Dan repertoire..

Thanks for pointing that out.



Thank you for your questions. They are definitely sensible.

The saying that sente gains nothing refers to points. OC, playing a sente takes away the opponent's option of playing the reverse sente. That is worth something, but it is not something that is measured in points. The analogy that taking sente is like cashing a check is a good one. And sente gains nothing in No Pass Go as well.

A miny (a Black sente) is less than zero because White gets the last play in it, whether by playing the reverse sente or by answering the Black sente play. The difference between a miny and zero cannot be measured in points. This idea is new and strange to most people, who have no experience with infinitesimals.



There are a couple of things going on here. First, as you say, sente gains nothing. So let's look at the result after the Black sente.

Click Here To Show Diagram Code

[go]$$Bc Miny
$$ -----------
$$ | O X . X . |
$$ | O X X . X |
$$ | O O O X X |
$$ | . O O O X |
$$ | O . B O X |
$$ -----------[/go]

This position we can count.

The second thing that is going on is that territory does not mean the same in straight No Pass Go as it does in regular go. In regular go Black has three points of territory and White has four points of territory.

But in no pass go Black has only one point of territory because of the "group tax". Black has to keep two eyes to stay alive. So Black can only fill one eye, and has only one point.

As for White's territory, if White did not need the eye with the stone in it to live, it would be worth two points of territory. That is, White could take one move to capture the stone and another move to fill the eye. Because of the group tax White cannot afford to fill the eye and has only one move, i. e., one point.

And here is why a miny is an infinitesimal. Let's look at the position after White plays the reverse sente.

Click Here To Show Diagram Code

[go]$$Bc Miny
$$ -----------
$$ | O X . X . |
$$ | O X X . X |
$$ | O O O X X |
$$ | . O . O X |
$$ | O . O O X |
$$ -----------[/go]

In this case White has only one point, as well. Not only does the sente gain nothing, the reverse sente gains nothing, too!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Here is an easy problem. Like all my problems.

Click Here To Show Diagram Code

[go]$$B White to play
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X . . . . X
$$ X . . . O . . X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

Enjoy!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

OK. Easy problem. No particular point in hiding this.

Click Here To Show Diagram Code

[go]$$W Solution
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X a 3 b . X
$$ X 2 4 5 O . 1 X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

is the play, which was probably your first impulse. But it is a close call, since 2 is only slightly smaller, gaining or losing 1 point. gains 1 1/32 points. The reason it is so large is that threatens to play at "a", making a point of territory. In fact. "a" and "b" are miai. The end result is a net local score of 0.

Click Here To Show Diagram Code

[go]$$W Failure
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X 4 . . . X
$$ X 1 . . O 3 2 X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

Usually a difference of 1/32 point will not matter, but here it does. If White takes the 1 point play, Black will get a net local score of 1 point.

More on this little problem in the morning.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Difference games

I learned about difference games in the 1990s, from Berlekamp and Wolfe's Mathematical Go. Although the idea is mathematical, applying it does not require much in the way of math. The basic idea is that independent regions of the go board may be added and subtracted. Go tends to break up into independent regions in the endgame, and even before then regions may be only loosely coupled. To subtract one region from another one, change the colors of the stones in the first one and add the two together.

For instance, for this region:

Click Here To Show Diagram Code

[go]$$ A
$$ ------------
$$ . X C C O .
$$ . X X X O .
$$ . . . . O .
$$ . . . . . .[/go]

(The marked points are the points in play.)

this region is its negative.

Click Here To Show Diagram Code

[go]$$ -A
$$ ------------
$$ . O C C X .
$$ . O O O X .
$$ . . . . X .
$$ . . . . . .[/go]

OC, a region minus itself is zero. If you play the difference game out, the second player just plays mirror go.

A major use of difference games is to compare two plays. Let's take the little problem and subtract it from itself.

Click Here To Show Diagram Code

[go]$$W Zero
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X . . . . X
$$ X . . . O . . X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ O . . . X . . O O O
$$ O . X X O . . . . O
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

The bottom region is the negative of the top. In between is no man's land. This is a zero game, OC.

In the problem White has two plausible moves, which we wish to compare. We let White make one of those moves in top region, and let Black make the other move in the bottom region.

Click Here To Show Diagram Code

[go]$$W Difference game
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X . . . . X
$$ X . . . O . W X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ O B . . X . . O O O
$$ O . X X O . . . . O
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

If White's move is better than Black's then the combined position should be better for White than for Black (and vice versa). That means two things. First, White to play should win in the difference game. Second, Black to play should not win. Let's check that out.

Click Here To Show Diagram Code

[go]$$W White first
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X . 3 . . X
$$ X 2 4 5 O . W X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ O B . . X 6 1 O O O
$$ O . X X O 7 . . . O
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

White to play wins by one point. You can let Black try different moves, but White always wins. (Note that if White starts at 2 instead of 1, Black can play at 1 and tie by mirror go.)

Click Here To Show Diagram Code

[go]$$B Black first
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X a O O X c 5 . . X
$$ X 1 3 4 O d W X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ O B . . X 6 2 O O O
$$ O b X X O . . . . O
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

Again, if Black starts at 2 White can play at 1 and play mirror go. The result is jigo. Black does not win. (Note that "a" and "b" are miai, as are "c" and "d". In No Pass Go the player who moves to a jigo would win, but in regular go getting jigo is good enough to show that is a better play for White.

Now, let's play the difference game for the case when Black has the choice of moves.

Click Here To Show Diagram Code

[go]$$W Difference game 2
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X . . . . X
$$ X . . . O . B X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ O W . . X . . O O O
$$ O . X X O . . . . O
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$B Black first
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X . . . . X
$$ X 5 6 . O 7 B X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ O W 2 3 X . 1 O O O
$$ O . X X O . 4 . . O
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

Black wins by one point, as expected.

Click Here To Show Diagram Code

[go]$$W White first
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ X . O O X . . . . X
$$ X 4 5 . O . B X X X
$$ O O O O O O O O O O
$$ . . . . . . . . . .
$$ X X X X X X X X X X
$$ O W 1 2 X . 3 O O O
$$ O . X X O . . . . O
$$ O O O O O O O O O O
$$ . . . . . . . . . .[/go]

Again, if White starts at 3, Black at 4 is jigo. We can stop after , as the rest is miai. White wins by one point.

What does that mean? It means that we cannot say that one play is better than the other for Black. In some situations, one may be better, in different situations the other one may be better. Yes, the play has a slight advantage. If it makes a difference, it is more likely to be better than the other play. But sometimes the one point play will be better, even if it is smaller, on average.

Note that what allows White to win the difference game, going first, is the fact that is sente, after which White can play . This is a feature that you can take note of, even without setting up a difference game.

One advantage of comparing plays by difference games is that you do not have to calculate the size of the plays. And playing the difference game can be fairly easy, as in this case. As a local problem this is easy to read out, but in the context of the whole board, which play White should choose may not be so obvious. With one caveat, the difference game can say.

The caveat is ko. Why? Because we assumed that the regions were independent, and kos may break the independence of different regions. For instance, the number of possible ko threats may matter. And in this case we treated "a" and "b" in the original problem as miai, with one player getting one and the other player getting the other one. In a game with a ko, however, one player might win the ko while the other player gets both "a" and "b".

Each case must be decided on its merits. Difference games can help us to make close calls.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Difference games (ii)

I was quite surprised when David Wolfe told me that the White descent in the next diagram was not as good as the hane-and-connect.

Click Here To Show Diagram Code

[go]$$W Not so good
$$ -------------------
$$ . . . . . 2 1 . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .[/go]

The reason being, he explained, was that it allowed Black to play .

This surprised me for two reasons. One, in estimating territory here, the exchange of and , or vice versa, is standard. And correct. Two, Takagawa, in his Igo Reader series (囲碁読本), aimed at kyu players, had said that the descent was as good as the hane-and-connect, and I had often played it in positions like this, where I had follow-ups that I would not have had with the hane-and-connect.

The difference game makes David's point clear.

Click Here To Show Diagram Code

[go]$$W Difference game
$$ -------------------
$$ . . . . . . W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

For the difference game we set up the negative of the original position, then in the top we let White play the descent and in the bottom we let Black play the hane-and-connect.

Click Here To Show Diagram Code

[go]$$B Black first
$$ -------------------
$$ . . . . C 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

Black to play makes one point of territory (marked).

Click Here To Show Diagram Code

[go]$$W White first
$$ -------------------
$$ . . . . 2 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

White to play only gets jigo. The hane-and-connect is correct. Takagawa was wrong.

----

When I was a beginner one thing puzzled me -- among others, OC.

Click Here To Show Diagram Code

[go]$$B How to save?
$$ -------------------
$$ . . . . . . . . . .
$$ . . . . . . B O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .[/go]

How should Black save the stone?

Unlike a lot of beginners, I was not attracted to the descent

Click Here To Show Diagram Code

[go]$$B Not the descent?
$$ -------------------
$$ . . . . . . 1 2 . .
$$ . . . . . 3 B O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .[/go]

because it gave White in sente. In my Korschelt I saw that sometimes the pros played the solid connection, and sometimes they played the hanging connection, but there was no explanation for why they played the one or the other. Sometimes, for tactical reasons, they played the descent or the keima connection, but what about everyday bread and butter plays?

Can difference games can shed some light on this question? You bet.

First, let's compare the descent with the solid connection.

Click Here To Show Diagram Code

[go]$$B Descent vs. solid connection
$$ -------------------
$$ . . . . . . B . . .
$$ . . . . . . X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . W O X . .
$$ . . . . . . . . . .
$$ -------------------[/go]

Click Here To Show Diagram Code

[go]$$W White first
$$ -------------------
$$ . . . . . 4 B 3 C .
$$ . . . . 2 1 X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . W O X . .
$$ . . . . . . 7 5 6 .
$$ -------------------[/go]

White to play makes an extra point of territory (marked) and wins.

Click Here To Show Diagram Code

[go]$$B Black first
$$ -------------------
$$ . . . . . 5 B 4 . .
$$ . . . . 3 2 X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . W O X . .
$$ . . . . . 6 1 7 . .
$$ -------------------[/go]

Somewhat surprisingly, Black to play also wins. We cannot say that the solid connection is better.

Now let's try the descent vs. the hanging connection.

Click Here To Show Diagram Code

[go]$$B Descent vs. hanging connection
$$ -------------------
$$ . . . . . . B . . .
$$ . . . . . . X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . . O X . .
$$ . . . . . W . . . .
$$ -------------------[/go]

Click Here To Show Diagram Code

[go]$$W White first
$$ -------------------
$$ . . . . . 4 B 3 . .
$$ . . . . 2 1 X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . . O X . .
$$ . . . . . W 7 5 6 .
$$ -------------------[/go]

White to play wins, as with the solid connection.

Click Here To Show Diagram Code

[go]$$B Black first
$$ -------------------
$$ . . . . . 5 B 4 . .
$$ . . . . 3 2 X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . . O X . .
$$ . . . . . W 6 1 . .
$$ -------------------[/go]

White gets jigo.

But suppose that Black does not submit with and plays at 6, hoping that White will connect and then Black can go back and play at 5?

Click Here To Show Diagram Code

[go]$$Bm5 Black first, variation
$$ -------------------
$$ . . . 3 2 4 B O . .
$$ . . . . X O X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . 6 5 O X . .
$$ . . . 8 7 W 1 X . .
$$ -------------------[/go]

White simply ignores and captures the two Black stones. fills the ko at , but White still wins, with one more point of territory at .

So my beginner's intuition was correct, and the descent is normally inferior.

Finally, let's compare the solid connection and the hanging connection.

Click Here To Show Diagram Code

[go]$$B Solid connection vs. hanging connection
$$ -------------------
$$ . . . . . . . . . .
$$ . . . . . B X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . . O X . .
$$ . . . . . W . . . .
$$ -------------------[/go]

Click Here To Show Diagram Code

[go]$$W White first
$$ -------------------
$$ . . . . . 2 1 3 . .
$$ . . . . . B X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . C O X . .
$$ . . . . . W 5 4 . .
$$ -------------------[/go]

White makes one more point of territory (marked) and wins. No surprise.

Click Here To Show Diagram Code

[go]$$B Black first
$$ -------------------
$$ . . . . . 5 2 6 . .
$$ . . . . . B X O . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . . . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . . 4 O X . .
$$ . . . . . W 3 1 . .
$$ -------------------[/go]

White gets jigo.

So the difference game says that the hanging connection is best.

Wait! Are the pros wrong to play the solid connection, unless there is a tactical reason?

Remember the caveat about kos. The hanging connection leaves possible ko threats which the solid connection does not. Does that matter? Well, every case must be judged on its merits.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Difference games (iii)

More on the second line hane.

The second line hane occurs frequently, as does the descent on the second line. Often the difference between playing one or the other depends upon tactics. But what about when there is no particular tactical reason to choose one or the other? Let's explore that question with difference games.

Click Here To Show Diagram Code

[go]$$c Difference game, hane vs. descent
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . W . . . . . . . . |
$$ | X X X X X X X X X X O O O O O O O O O |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | O O O O O O O O O O X X X X X X X X X |
$$ | . . . . . . . . . B . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Which is better for the player to the right, the hane or the descent?

On the top side we let White play the descent, on the bottom side we let Black play the hane.

The answer is not obvious, nor is the play in the difference game, which could get complicated. Fortunately, in difference games we do not have to find the best play to win or tie; good enough play will do.

Click Here To Show Diagram Code

[go]$$Bc Black first
$$ ---------------------------------------
$$ | . . . . . . . . . 7 5 6 . . . . . . . |
$$ | . . . . . . . . . 1 W . . . . . . . . |
$$ | X X X X X X X X X X O O O O O O O O O |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | O O O O O O O O O O X X X X X X X X X |
$$ | . . . . . . . . 2 B 9 . . . . . . . . |
$$ | . . . . . . . . 4 8 3 . . . . . . . . |
$$ ---------------------------------------[/go]

With Black to play is good enough to win by 3.

Click Here To Show Diagram Code

[go]$$Wc White to play
$$ ---------------------------------------
$$ | . . . . . . . . 8 7 9 . . . . . . . . |
$$ | . . . . . . . . . 2 W . . . . . . . . |
$$ | X X X X X X X X X X O O O O O O O O O |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | O O O O O O O O O O X X X X X X X X X |
$$ | . . . . . . . . 1 B 3 4 . . . . . . . |
$$ | . . . . . . . . . 5 0 6 . . . . . . . |
$$ ---------------------------------------[/go]

fills at

may be good enough. If , wins by 3.

What if Black plays the hanging connection instead. That does not look promising, but let's see.

Click Here To Show Diagram Code

[go]$$Wc White first, variation
$$ ---------------------------------------
$$ | . . . . . . 6 3 9 5 . . . . . . . . . |
$$ | . . . . . . . 4 . 0 W . . . . . . . . |
$$ | X X X X X X X X X X O O O O O O O O O |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | O O O O O O O O O O X X X X X X X X X |
$$ | . . . . . . . . 1 B . . . . . . . . . |
$$ | . . . . . . . . 7 8 2 . . . . . . . . |
$$ ---------------------------------------[/go]

White wins by 2.

The first player wins the difference game, which means that whether to play the hane or the descent depends upon what's on the rest of the board.

----

Some time ago I noted that the hane-and-connect on the second line was fairly usual play, but that the descent was normally sente. That being the case, if you foresee that your opponent can get the hane-and-connect, why not forestall that by playing the descent with sente? Furthermore, by symmetry, that is so for both players, so normally shouldn't one player or the other play the descent with sente? In that case, we should see the pros make a lot of double descents on the second line, but we don't. OC, tactics often matter, but still. . . .

However, if you play the hane you do not necessarily have to connect, even when tactics don't matter. What do difference games tell us about playing the hane with sente versus doing nothing?

Click Here To Show Diagram Code

[go]$$c Difference game, sente hane vs. no play
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | X X X X X X X X X X O O O O O O O O O |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | O O O O O O O O O O X X X X X X X X X |
$$ | . . . . . . . . W B . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Click Here To Show Diagram Code

[go]$$Bc Black first
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . 1 . . . . . . . . . |
$$ | X X X X X X X X X X O O O O O O O O O |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | O O O O O O O O O O X X X X X X X X X |
$$ | . . . . . . . . W B . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

With Black to play, the descent is good enough to win. If White responds on the top side with a descent, Black connects on the bottom side. OTOH, if White captures the Black stone on the bottom side with gote, Black plays the monkey jump on the top side.

Click Here To Show Diagram Code

[go]$$Wc White first
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . 1 . . . . . . . . |
$$ | X X X X X X X X X X O O O O O O O O O |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | O O O O O O O O O O X X X X X X X X X |
$$ | . . . . . . . . W B . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

With White to play, the descent is also good enough to win. If Black responds with a descent on the top side, White captures the Black stone on the bottom side. OTOH, if Black saves the Black stone, White plays the monkey jump.

The upshot of it is this. Sometimes you play the descent to prevent your opponent from playing the hane, sometimes you play the hane with sente to prevent your opponent from playing the descent, and sometimes you wait. I suppose that sometimes you play the hane-and-connect, but I think that with enough foresight you pick one of the sente options, or your opponent will.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Difference games (iv)

Comparing plays with difference games has some practical advantages by comparison with calculating the size of plays.

First, you do not have to calculate the size of plays. In my little problem White did not have to calculate the size of the either the play that gains 1 point or the play that gains 1 1/32 point. True, neither is that difficult to calculate, but to calculate the value of a play may mean reading out a somewhat sizable game tree, and calculating the values of other plays in the tree. Reading out a difference game may also involve a sizable tree, but the effort involved is not necessarily more. Besides, once you reach a mirror go position you can stop reading that variation.

Second, unlike with calculating values, you do not need to find perfect play. Good enough play will do. We saw the practical advantage of that in the second line hane vs. descent example.

Third, difference games can distinguish between plays of the same size. There are even plays of the same size where the difference game prefers one to the other, regardless of who plays first. Commenting on such a play, one pro suggested adding half a point to the value of that play. OC, he was unaware of difference games.

Fourth, difference games can tell us when a smaller play is a viable option. In my little problem, the difference game tells Black that the play that gains 1 1/32 point may not be as good as the play that gains 1 point.

----

Difference games in straight no pass go

Back in the 90s the late John Rickard and I explored play in some large eyes in no pass go. One problem, OC, is that no pass go values are not familiar to regular go players, so that it was not always easy to eliminate certain inferior lines of play. Difference games could have helped, but I did not use them, and I don't think that John did, either. (We did not get together in person, but corresponded via email.)

Here is a simple example. We ignore the group tax, assuming that the eye is immortal.

Click Here To Show Diagram Code

[go]$$W White to play
$$ --------------
$$ . X a b . . X .
$$ . X X X X X X .
$$ . . . . . . . .[/go]

Should White play at "a" or "b"? Our regular go sense tells us "b", because it threatens to take away a potential eye. But life and death is not the issue. OC, a one point eye does not have a fractional value in no pass go, so that may matter. We might work out the values of the resulting positions. (In fact, didn't I do that last week? But what were those values? )

Let's do the difference game.

Click Here To Show Diagram Code

[go]$$W Difference game
$$ --------------
$$ . X . W . . X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B . . . O .
$$ ---------------[/go]

Now, let's say that all I am interested in is whether is good enough. It is the play that looks good to me, and if it is as good as , that's good enough for me.

That means that all I have to do is to look at the game where Black plays first. If I get "jigo" (a zero game) as White, then is at least as good as . (Remember, there are no ties in no pass go, so moving to a zero game wins.) They may be equivalent, but I don't care about that. is good enough.

Click Here To Show Diagram Code

[go]$$B Black first
$$ --------------
$$ . X . W 1 . X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B . 2 . O .
$$ ---------------[/go]

If , makes mirror go, and wins. Each player has 3 points, but we're not counting, right?

Well, if is so good, maybe Black should play there. Besides, it is obvious that no matter where Black plays in the top eye, wins.

Click Here To Show Diagram Code

[go]$$B Variation
$$ --------------
$$ . X . W . 2 X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B . 1 . O .
$$ ---------------[/go]

Now if , makes mirror go and wins. It looks like a horrible play, but it is good enough, and that is all we care about.

Click Here To Show Diagram Code

[go]$$B Variation 2
$$ --------------
$$ . X . W 2 . X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B . . 1 O .
$$ ---------------[/go]

again looks bad, but it avoids mirror go. Now looks good. (In fact you can check the variations where White plays somewhere else and Black plays there. The opponent's good play is my good play. )

Click Here To Show Diagram Code

[go]$$B Variation 2a
$$ --------------
$$ . X 4 W 2 5 X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B 3 . 1 O .
$$ ---------------[/go]

Click Here To Show Diagram Code

[go]$$B Variation 2a, cont.
$$ --------------
$$ . X . 6 7 X X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B X 8 X O .
$$ ---------------[/go]

Click Here To Show Diagram Code

[go]$$B Variation 2a, fin
$$ --------------
$$ . X . O X X X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O 9 . O . O .
$$ ---------------[/go]

After White is obviously one point ahead, and wins.

Click Here To Show Diagram Code

[go]$$B Variation 2b
$$ --------------
$$ . X 5 W 2 3 X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B 4 . 1 O .
$$ ---------------[/go]

Click Here To Show Diagram Code

[go]$$B Variation 2b, cont.
$$ --------------
$$ . X X 6 . X X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O . O . X O .
$$ ---------------[/go]

After White obviously wins.

So is good enough.

BTW, in case you are curious,

Click Here To Show Diagram Code

[go]$$B Better for White
$$ --------------
$$ . X . W . . X .
$$ . X X X X X X .
$$ . . . . . . . .
$$ . O O O O O O .
$$ . O B . . . O .
$$ ---------------[/go]

In this position White is 0.125 point better on average than Black.

----

Now, in straight no pass go we know that it is generally a good idea to play inside your opponent's eye towards the end of the game. But what about playing inside your own eye? Which plays are good?

Well, since, group tax aside, a one point eye is worth a full point, while larger eyes are worth a fraction of the number of board points they surround, making a one point eye seems like a good idea. OC, that may not always be so.

And, in fact, it is not so when making the first play inside your own seven-point linear eye. As this difference game illustrates.

Click Here To Show Diagram Code

[go]$$B One pt. eye vs. two pt. eye
$$ ---------------------
$$ . X . . B . . . . X .
$$ . X X X X X X X X X .
$$ . . . . . . . . . . .
$$ . O O O O O O O O O .
$$ . O . W . . . . . O .
$$ ---------------------[/go]

Is as good as ? Let's play the difference game with Black playing first to find out.

Click Here To Show Diagram Code

[go]$$B Black first, main line
$$ ---------------------
$$ . X . 2 B . 6 5 . X .
$$ . X X X X X X X X X .
$$ . . . . . . . . . . .
$$ . O O O O O O O O O .
$$ . O . W . 4 1 3 . O .
$$ ---------------------[/go]

and are miai, as are and . At this point all the eyes have integer values, so the score is easy to count.

Black has 2 + 2 + 1 = 5 points. (The eyes with the single dead White stones are worth 2 points, one to capture the eye and one to fill it. )

White has 1 + 1 + 2 = 4 points. (The eye with the two Black stones is worth 2 points, as you can see after White captures the two stones and then Black plays inside the eye.)

Black wins.

So we cannot say that is at least as good at .

But perhaps is not at least as good as , either. That would be a bother, but let's check it.

Click Here To Show Diagram Code

[go]$$W White first, main line
$$ ---------------------
$$ . X . 4 B . 6 5 . X .
$$ . X X X X X X X X X .
$$ . . . . . . . . . . .
$$ . O O O O O O O O O .
$$ . O . W . 2 3 1 . O .
$$ ---------------------[/go]

is correct for the 5 point eye, as we (John and I) previously found out. Then holds that eye to two points with sente. makes mirror go, to win.

So is better than .

Click Here To Show Diagram Code

[go]$$B Black better
$$ ---------------------
$$ . X . . B . . . . X .
$$ . X X X X X X X X X .
$$ . . . . . . . . . . .
$$ . O O O O O O O O O .
$$ . O . W . . . . . O .
$$ ---------------------[/go]

The Black "territory" is worth 4.5 points. The White "territory" is worth 4*. Black is half a point ahead, give or take the equivalent of a dame.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Yet Another Difference Game

I had not intended to post another difference game now, but this seems kind of interesting. Straight no pass go, Black to play. Ignore the group tax.

Click Here To Show Diagram Code

[go]$$B Black to play
$$ . . . . . . . . .
$$ . . . . X X X . .
$$ . X X X X . X X .
$$ . X . a b c . X .
$$ . X X . X X X X .
$$ . . X X X . . . .
$$ . . . . . . . . .[/go]

Should Black play at "a", "b", or "c"? Or does it depend upon the rest of the board? ("c" and "a" are equivalent, OC, so forget "c". )

At first I thought that "a" was best, because the count after "a" is 4.5, while the count after "b" is 4. But if Black "a", White "c", the local score is 4, while after Black "b" the local value is infinitesimally greater than 4, by 2 UPs. So maybe there are times when "b" is right.

OC, we set up a difference game.

Click Here To Show Diagram Code

[go]$$B Difference game
$$ . . . . . . . . .
$$ . . . . X X X . .
$$ . X X X X . X X .
$$ . X . B . . . X .
$$ . X X . X X X X .
$$ . . X X X . . . .
$$ . . . . . . . . .
$$ . . O O O . . . .
$$ . O O . O O O O .
$$ . O . . W . . O .
$$ . O O O O . O O .
$$ . . . . O O O . .
$$ . . . . . . . . .[/go]

Enjoy!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

OK. Here is the solution.

Which is better depends on the rest of the board.

I.e., whoever plays first wins.

Click Here To Show Diagram Code

[go]$$B Black first
$$ . . . . . . . . .
$$ . . . . X X X . .
$$ . X X X X . X X .
$$ . X . B . 1 . X .
$$ . X X . X X X X .
$$ . . X X X . . . .
$$ . . . . . . . . .
$$ . . O O O . . . .
$$ . O O . O O O O .
$$ . O . 2 W 3 4 O .
$$ . O O O O . O O .
$$ . . . . O O O . .
$$ . . . . . . . . .[/go]

gives Black 5 pts. White can get no more than 4 pts., which is his score after . Black wins.

Click Here To Show Diagram Code

[go]$$W White first
$$ . . . . . . . . .
$$ . . . . X X X . .
$$ . X X X X 7 X X .
$$ . X . B 6 1 . X .
$$ . X X . X X X X .
$$ . . X X X . . . .
$$ . . . . . . . . .
$$ . . O O O . . . .
$$ . O O 3 O O O O .
$$ . O . 2 W 4 5 O .
$$ . O O O O . O O .
$$ . . . . O O O . .
$$ . . . . . . . . .[/go]

holds Black to 4 points. White replies to each Black invasion. After we have a zero game with Black to play, which is a win for White. That may not be obvious, so play may continue with and , which are miai.

Click Here To Show Diagram Code

[go]$$W White mistake
$$ . . . . . . . . .
$$ . . . . X X X . .
$$ . X X X X . X X .
$$ . X . B . 1 . X .
$$ . X X . X X X X .
$$ . . X X X . . . .
$$ . . . . . . . . .
$$ . . O O O . . . .
$$ . O O 4 O O O O .
$$ . O . 2 W 3 . O .
$$ . O O O O . O O .
$$ . . . . O O O . .
$$ . . . . . . . . .[/go]

and may look like miai, but they are not. makes a zero game and wins.

----

Many, if not all of these No Pass Go positions have analogs in regular go. For instance, here is an analog for the White position after .

Click Here To Show Diagram Code

[go]$$B Not miai
$$ . . . . . . . . .
$$ . . X X X X X X .
$$ . . O O O X O . .
$$ . O O . O 1 O O .
$$ . O . . W 3 . O .
$$ . O O 2 O . O O .
$$ . . O X O O O . .
$$ . . X X X X X X .
$$ . . . . . . . . .[/go]

and may look like miai, but they are not.

See http://senseis.xmp.net/?DOWN

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Hi Bill, I was watching a Haylee game on youtube and this position came up in the endgame:

Click Here To Show Diagram Code

[go]$$B
$$ | . O . . . .
$$ | O . O O O O
$$ | . O X X X O
$$ | . O . , . X
$$ | X O X X . O
$$ | . X X . . .
$$ | . X . X X .
$$ | . O X . X .
$$ | . 1 O X . .
$$ | 2 a O , . .
$$ | . . . O O X
$$ | . . . O . O
$$ | . . . . O .
$$ +------------[/go]

Haylee was white. After she hovered over 'a' but then she said she picked and said "seems like it's one point better". I thought of this thread and wondered about "one point". How would you decide which move is bigger? Do you think the difference is one point?

My first thought on seeing this was that avoids the following ko.

Click Here To Show Diagram Code

[go]$$B
$$ | . O . . . .
$$ | O . O O O O
$$ | . O X X X O
$$ | . O . , . X
$$ | X O X X . O
$$ | . X X . . .
$$ | . X . X X .
$$ | 3 O X . X .
$$ | 8 1 O X . .
$$ | 5 2 O , . .
$$ | 6 7 . O O X
$$ | . . . O . O
$$ | . . . . O .
$$ +------------[/go]

elsewhere

is not necessarily played immediately, OC.

This is a difficult ko for White to play, as White puts more at stake than Black. The practical thing is usually to play at 7. (That's what the textbooks show. )

Click Here To Show Diagram Code

[go]$$B
$$ | . O . . . .
$$ | O . O O O O
$$ | . O X X X O
$$ | . O . , . X
$$ | X O X X . O
$$ | . X X . . .
$$ | . X . X X .
$$ | 3 O X . X .
$$ | 9 1 O X . .
$$ | 5 2 O , . .
$$ | 7 6 . O O X
$$ | 8 0 . O . O
$$ | . . . . O .
$$ +------------[/go]

As - is sente, we assume that Black will be able to play it sometime after . I expect that this is what Haylee envisioned, not just from general principles, but from an awareness of the ko threat situation in the actual game.

Let's compare that with the result after in the game and capturing one stone.

Click Here To Show Diagram Code

[go]$$B
$$ | . O . . . .
$$ | O . O O O O
$$ | . O X X X O
$$ | . O . , . X
$$ | X O X X . O
$$ | . X X . . .
$$ | . X . X X .
$$ | 3 O X . X .
$$ | . 1 O X . .
$$ | 2 5 O 7 . .
$$ | . 6 8 O O X
$$ | . . . O . O
$$ | . . . . O .
$$ +------------[/go]

elsewhere

- is sente. White has 7 pts. in the very corner instead of 6. I believe that this is what Haylee meant by "one point better". I suppose that does not gain anything to its right.

Since our current theme is difference games, what does the difference game tell us about these two positions?

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[go]$$B Difference game
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | X . X . X . . . . . . . . . O . O . O |
$$ | . X O X . . . . . . . . . . . O X O . |
$$ | W . O , . . . . . , . . . . . , X B . |
$$ | . . . O O X . . . . . . . O X X . . . |
$$ | . . . O . O . . . . . . . X . X . . . |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

Difference games do not work well with kos, but let's assume that Black avoids the ko in the bottom right corner.

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[go]$$W White first
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | X . X . X . . . . . . . . . O . O . O |
$$ | . X O X . . . . . . . . . . . O X O 9 |
$$ | W 2 O 4 . . . . . , . . . . . , X B 1 |
$$ | . 3 5 O O X . . . . . . . O X X . 6 7 |
$$ | . . . O . O . . . . . . . X . X . 0 8 |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

does not immediately answer , and looks right? Is necessary? Fortunately, with a difference game we do not have to answer that question. is good enough to win.

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[go]$$B Black first
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | X . X . X . . . . . . . . . O . O . O |
$$ | . X O X . . . . . . . . . . . O X O . |
$$ | W 1 O , . . . . . , . . . . . , X B 3 |
$$ | . 2 . O O X . . . . . . . O X X . . . |
$$ | . . . O . O . . . . . . . X . X . . . |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

may not be the best play, but it is good enough to win.

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[go]$$B Black first, variation 1
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | X . X . X . . . . . . . . . O . O . O |
$$ | . X O X . . . . . . . . . . . O X O 8 |
$$ | W 1 O , . . . . . , . . . . . , X B 2 |
$$ | . 3 . O O X . . . . . . . O X X . 5 6 |
$$ | 9 4 . O . O . . . . . . . X . X . . 7 |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

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[go]$$B Black first, variation 2
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | X . X . X . . . . . . . . . O . O . O |
$$ | . X O X . . . . . . . . . . . O X O . |
$$ | W 1 O , . . . . . , . . . . . , X B 2 |
$$ | . 3 . O O X . . . . . . . O X X . . . |
$$ | . . 5 O . O . . . . . . . X . X . 4 . |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

It seems that Black to play wins the difference game by taking or threatening to take the stone. (Except perhaps when the players need an eye in the very corner to live.)

That means that under some circumstances, depending on the rest of the board, the second line atari is right.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Correction

Actually, we can remove the exception about needing the corner eye if we go back to the original comparison.

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[go]$$B Difference game
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | . O X . X . . . . . . . . . O . O X . |
$$ | . X O X . . . . . . . . . . . O X O . |
$$ | W . O , . . . . . , . . . . . , X B . |
$$ | . . . O O X . . . . . . . O X X . . . |
$$ | . . . O . O . . . . . . . X . X . . . |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

This is the original game to compare in the right and in the left. I set up the other difference game in part because normal play is to make the capture, and in part because I thought that doing so would make the exposition simpler and clearer. That's not so without the assumption that the stones on the bottom are immortal.

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[go]$$B Black first wins
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | 5 O X . X . . . . . . . . . O . O X 2 |
$$ | . X O X . . . . . . . . . . . O X O . |
$$ | W 1 O , . . . . . , . . . . . , X B 3 |
$$ | . 4 . O O X . . . . . . . O X X . . . |
$$ | . . . O . O . . . . . . . X . X . . . |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

Normally, to play first loses a ko threat. looks strange, but if White plays at 4 we transpose to this variation. is also strange, but it is good enough to win.

OC, White to play wins.

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[go]$$W White first wins
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . X . |
$$ | O . O O O O . . . . . . . X X X X . X |
$$ | . O X X X O . . . . . . . X O O O X . |
$$ | . O . , . X . . . , . . . O . , . X . |
$$ | X O X X . O . . . . . . . X . O O X O |
$$ | . X X . . . . . . . . . . . . . O O . |
$$ | . X . X X . . . . . . . . . O O . O . |
$$ | 4 O X . X . . . . . . . . . O . O X 1 |
$$ | . X O X . . . . . . . . . . . O X O 9 |
$$ | W 2 O 0 . . . . . , . . . . . , X B 5 |
$$ | . 3 a O O X . . . . . . . O X X . 6 7 |
$$ | . . . O . O . . . . . . . X . X . b 8 |
$$ | . . . . O . . . . . . . . . X . . . . |
$$ ---------------------------------------[/go]

at "a", at "b"

OC, we are assuming that Black avoids the ko on the right.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Error in Tesuji and Anti-Suji of Go?

This example comes from Sensei's Library ( http://senseis.xmp.net/?canceledit=Tesu ... o%2FErrata ), where Dieter (Knotwilg here) suggests a different play from the one Sakata gives.

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[go]$$B Connecting underneath
$$ --------------------
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . X X X X X . |
$$ . . . . . O O O X O |
$$ . . . . . . . . O . |
$$ . . . . . O O . . . |
$$ . . . . . . . X X X |
$$ . . . . . . O . . . |
$$ . . . . . . . O O . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |[/go]

Problem: How to connect the three isolated black stones to the corner?

Sakata's solution starts with in this diagram.

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[go]$$B Connecting underneath
$$ --------------------
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . X X X X X . |
$$ . . . . . O O O X O |
$$ . . . . . . . 1 O . |
$$ . . . . . O O . . . |
$$ . . . . . . . X X X |
$$ . . . . . . O . . . |
$$ . . . . . . . O O . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |[/go]

Dieter suggests this way.

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[go]$$B Connecting underneath
$$ --------------------
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . X X X X X . |
$$ . . . . . O O O X O |
$$ . . . . . . . . O 1 |
$$ . . . . . O O . . . |
$$ . . . . . . . X X X |
$$ . . . . . . O . . . |
$$ . . . . . . . O O . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |[/go]

Is Dieter right?

Edit: That is, is Dieter's play better?

Enjoy!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

_________________
La victoire est un hasard, la défaite une nécessité.

Same principle:

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[go]$$ B to play (a or b)
$$ --------------------
$$ . . . . . . . . . |
$$ . . . a O b . . . |
$$ . . . O X O . . . |
$$ . . . . X . . . . |
$$ . . . . . X . . . |
$$ . . . . . . . . . |
$$ . . . . . . . . . |
$$ . . . . . . . . . |[/go]

Whichever way B cuts, W should normally capture the cutting stone

Therefore if B wants the outside, cut at B
Therefore if B wants the corner, cut at A

However, if B knows W will defend the outside no matter what, cut at B

However, if B knows W will defend the corner no matter what, cut at A

Then W will be unable to capture the cutting stone