Life In 19x19 :: This 'n' that

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This 'n' that http://www.lifein19x19.com/viewtopic.php?f=12&t=12327	Page 8 of 53

Author: Bill Spight [ Mon Dec 21, 2015 3:03 pm ]

Post subject: Re: This 'n' that

bayu wrote:

I've got a honest question about pratique.
When you say: this is no joseki anymore. How do you know? I'm not disputing whether the sequence in the game above is still joseki or not. I believe you that it is outdated. I'm wondering, how could I possibly have found out. Did a book or someone who knows tell you? Do you simply observe, that it doesn't show up anymore in pro games? Kogo still has the joseki in (and says GOOD VARIATION), it appears in the Go joseki app for Android (which doesn't seem to be based on Kogo) and I was very happy when I finally learnt it. What are the sources that tell you, that something is outdated?

The Suzuki-Kitani Small Joseki Dictionary (1966), does not even call it an old joseki, but simply says that it is favorable for White (in this case) because the corner is so large. I found out that it was an old joseki from its appearance in Gokyo Seimyo.

One problem seems to be that the outside stones need bolstering. For instance, I learned this as joseki.

Click Here To Show Diagram Code: [go]$$ $$ . . . . . . . . . . | $$ . . . . . . . . . . | $$ . . . . . . . . . . | $$ , . . . . . , . . . | $$ . . . . . . . . . . | $$ . . . . 9 . . . . . | $$ . . . . . . 3 5 . . | $$ . . . . 1 X 2 4 7 . | $$ . . . . . O X 8 6 . | $$ , . . . . O , . . . | $$ . . . . . . O . . . | $$ . . . . . . . . . . | $$ . . . . . . . . . . | $$ ---------------------[/go]

Inoue (1909) gives this sequence.

Click Here To Show Diagram Code: [go]$$ $$ . . . . . . . . . . | $$ . . . . . . . . . . | $$ . . . . . . . 9 . . | $$ , . . . . . , . . . | $$ . . . . . . . . . . | $$ . . . . . . . . . . | $$ . . . . . . 3 5 . . | $$ . . . . 1 X 2 4 7 . | $$ . . . . . O X 8 6 . | $$ , . . . . O , . . . | $$ . . . . . . O . . . | $$ . . . . . . . . . . | $$ . . . . . . . . . . | $$ ---------------------[/go]

Author: Bill Spight [ Tue Dec 22, 2015 9:59 am ]

Post subject: Re: This 'n' that

Back to the Doteki - Hoshiai game.

Both skydyr and bayu favored an approach to the bottom left corner. So did Doteki, but his approach was a bit unusual.

Click Here To Show Diagram Code: [go]$$Wcm32 Honinbo Doteki (W) - Hoshiai Hasseki $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . O . . . . . O . X . . . . . . . | $$ | . . . , O . . . . , . . . . X , X . . | $$ | . O O X O . . . . . . . . . . . . . . | $$ | . X O O X X . . . . . . . . . . . . . | $$ | . . X X . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . 3 . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . X X . . | $$ | . . 1 . . . . . . . . . . X X O O X . | $$ | . . . . . . . . . . . . . . O X O O . | $$ | . . X , 2 . . . . , . . . . O , . . . | $$ | . . . . . . . . . . . . . . . O . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

allows

, a big enclosure, but then :w34:

largely negates Black's influence on the left side.

The first time I saw that approach to the 3-4 was as a suggestion by Go Seigen in his 21st Century Go set, against the Chinese Fuseki. (Today the two space high approach is more popular.) It was a brilliant idea, but I suspect that Go had seen it in his studies when he was a kid.

Click Here To Show Diagram Code: [go]$$Wcm32 Honinbo Doteki (W) - Hoshiai Hasseki $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . O . . . . . O . X . . . . . . . | $$ | . . . , O . . . . , . . . . X , X . . | $$ | . O O X O . . . . . . . . . . . . . . | $$ | . X O O X X . . . . . . . . . . . . . | $$ | . . X X 8 . . . . . . . . . . . . . . | $$ | . . . . 7 . . . . . . . . . . 4 . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . O . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . X X . . | $$ | . . O . . . . . . . . . . X X O O X . | $$ | . . . . . 6 . 9 . . . . . . O X O O . | $$ | . . X , X . . . . , . . . . O , . . . | $$ | . . . . . . 5 . . . . . . . . O . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

stakes out the right side, but then :w36:

approaches the bottom left corner. How to respond is not all that obvious. :b37:

strikes me as a bit slow, as White already has a base on the left side. After taking his kikashi against the top left Black group, :w40:

is enterprising. (BTW, I have always liked such keimas towards the center.

) Doteki won by 7 points.

Author: Bill Spight [ Wed Dec 23, 2015 5:30 pm ]

Post subject: Re: This 'n' that

Very ancient joseki and tricky play

Click Here To Show Diagram Code: [go]$$W Ancient joseki $$ -------------------- $$ | . . . . . . . . . . $$ | . . . . . . . . . . $$ | . . . . . 1 . . . . $$ | . . . X . . . . . , $$ | . . . . . 3 . . . . $$ | . . . . . . . . . . $$ | . . 2 . . . . . . . $$ | . . . . . . . . . . $$ | . . . . . . . . . . $$ | . . . , . . . . . ,[/go]

is a very ancient joseki. In fact, these are the opening moves of the oldest game record in my GoGoD database, dating to 196.

Click Here To Show Diagram Code: [go]$$W Belly attachment $$ -------------------- $$ | . . . . . . . . . . $$ | . . . . . . . . . . $$ | . . 4 3 . O . . . . $$ | . . 1 X . . . . . , $$ | . . 5 2 . O . . . . $$ | . . . . . . . . . . $$ | . . X . . . . . . . $$ | . . . . . . . . . . $$ | . . . . . . . . . . $$ | . . . , . . . . . ,[/go]

The belly attachment, :w1:

, is tricky. The nobi, :b2:

, is a solid reply.

After :w5:

, where does Black play?

(Answer after Christmas

)

Hint:

Author:	Bill Spight [ Mon Dec 28, 2015 8:57 am ]
Post subject:	Re: This 'n' that
Well, it's after Christmas. Happy Saturnalia! The answer is hidden for the benefit of later readers. Click Here To Show Diagram Code `[go]$$B Clamp $$ -------------------- $$ \| . . . . . . . . . . $$ \| . . . . . . . . . . $$ \| . . X O . O . . . . $$ \| . . O X . . . . . , $$ \| . 1 O X . O . . . . $$ \| . . . . . . . . . . $$ \| . . X . . . . . . . $$ \| . . . . . . . . . . $$ \| . . . . . . . . . . $$ \| . . . , . . . . . ,[/go]` The clamp is rather nice, isn't it? I would be happy to find this in a real game. Hayashi Gembi (1835) gives a few variations, as does Inoue Yasunobu (1909). Two of the sequences are virtually the same in each. (I have identified the variations in Inoue.) BTW, it's rather nice to find a joseki (through the White jump) that has lasted a couple of millenia. The large knight response is not played much these days, but I do not think that it has been deemed inferior. To have lasted so long, the joseki must be rather go-ish. The variations (after ):

Author: Bill Spight [ Thu Dec 31, 2015 10:01 pm ]

Post subject: Re: This 'n' that

On thoroughness

Over time I have come to appreciate thoroughness more and more. A good example is how Mr. K treats life and death. See http://mrkigo.sakura.ne.jp/ksikatuindex.html . He explores basic life and death problems in depth, so that the solver is prepared for anything. Also, I have found unexpected delights in things that I thought I already knew.

Here is an example from my own study not too long ago.

Click Here To Show Diagram Code: [go]$$W 2x4 in the corner $$ -------------- $$ | . . . . O X . $$ | . . . . O X . $$ | O O O O O X . $$ | X X X X X X . $$ | . . . . . . .[/go]

In this well known position Black to play can make ko or seki. White to play can live easily.

Click Here To Show Diagram Code: [go]$$W White lives $$ -------------- $$ | . 1 . . O X . $$ | . . . . O X . $$ | O O O O O X . $$ | X X X X X X . $$ | . . . . . . .[/go]

In the interest of thoroughness, I looked at other plays. How bad are they?

Click Here To Show Diagram Code: [go]$$W Status? $$ -------------- $$ | 1 . . . O X . $$ | . . . . O X . $$ | O O O O O X . $$ | X X X X X X . $$ | . . . . . . .[/go]

Suppose that White makes a crazy play on the 1-1. How bad is it? Can Black kill or make ko?

I'll leave this as a little problem. Enjoy!

Author:	Bill Spight [ Sat Jan 02, 2016 9:34 am ]
Post subject:	Re: This 'n' that
Answer hidden for those who might like it to be. Click Here To Show Diagram Code `[go]$$W White is alive $$ -------------- $$ \| O . . . O X . $$ \| . . . . O X . $$ \| O O O O O X . $$ \| X X X X X X . $$ \| . . . . . . .[/go]` Click Here To Show Diagram Code `[go]$$B Oshitsubushi $$ -------------- $$ \| O 1 2 . O X . $$ \| 3 4 . . O X . $$ \| O O O O O X . $$ \| X X X X X X . $$ \| . . . . . . .[/go]` Click Here To Show Diagram Code `[go]$$B White mistake $$ -------------- $$ \| O 1 2 . O X . $$ \| 3 . 4 5 O X . $$ \| O O O O O X . $$ \| X X X X X X . $$ \| . . . . . . .[/go]` Click Here To Show Diagram Code `[go]$$B Two eyes $$ -------------- $$ \| O 2 1 . O X . $$ \| . 3 4 . O X . $$ \| O O O O O X . $$ \| X X X X X X . $$ \| . . . . . . .[/go]` I find the answer charming. A hidden delight in a strange position. Here is another position where thoroughness produced delight for me. Click Here To Show Diagram Code `[go]$$B Deep sente $$ --------------------- $$ \| . 1 . . . X . . . . $$ \| O O . O O O X . X . $$ \| . X O O X X X . . . $$ \| . X X X . . . . . . $$ \| . . . . . . . . . . $$ \| . X . . . . . . . . $$ \| . . . . . . . . . .[/go]` The other two sente plays I saw at a glance. Looking further, I saw this one. This play adds no value in a real game. However, there may be other positions where the idea of this play bears fruit. Besides, it is a joy in itself. Happy New Year!

Author:	EdLee [ Sun Jan 03, 2016 7:41 am ]
Post subject:
Bill Spight wrote: there may be other positions where the idea of this play bears fruit. Hi Bill, Happy 2016. Click Here To Show Diagram Code `[go]$$B $$ --------------------- $$ \| . . . . . . . . . . $$ \| O O . . O . X . . . $$ \| O X O O X X . X . . $$ \| . X X X . . . . . . $$ \| . X . . . . . . . . $$ \| . . . . . . . . . .[/go]`

Author:	Bill Spight [ Sun Jan 03, 2016 9:39 am ]
Post subject:	Re:
EdLee wrote: Bill Spight wrote: there may be other positions where the idea of this play bears fruit. Hi Bill, Happy 2016. Click Here To Show Diagram Code `[go]$$B $$ --------------------- $$ \| . . . . . . . . . . $$ \| O O . . O . X . . . $$ \| O X O O X X . X . . $$ \| . X X X . . . . . . $$ \| . X . . . . . . . . $$ \| . . . . . . . . . .[/go]` Thanks, Ed!

Author: Bill Spight [ Sun Jan 03, 2016 6:57 pm ]

Post subject: Re: This 'n' that

No Pass Go

Way back when, when I was around shodan, I hit upon the idea of no pass go, which I called Nimgo, because the player without a move would lose. I thought that it might be useful for beginners, since it has a well defined game ending condition. That is not the case, at least not for straight no pass go, but I did not understand the game.

I had not heard of the group tax, but it is plain that in this corner, assuming that the enclosing White stones are alive,

Click Here To Show Diagram Code: [go]$$W Zero $$ ------------ $$ | C X C X O . $$ | X X X X O . $$ | O O O O O . $$ | . . . . . .[/go]

White cannot play on the marked eye points, and Black will not play there, unless forced. (In combinatorial game theory, the corner is worth 0. OC, I had no idea of CGT at that time.)

One advantage for beginners, I thought, was that life and death questions would naturally be settled by play. True enough.

Furthermore, I imagined that it would be plain that for you to spend four plays to capture a dead stone would cost nothing, because your opponent would also be making four plays inside his own territory or adding four dead stones to your territory. And also, it would be plain that playing a dead stone inside your opponent's territory to force him to capture it would not gain anything. That idea was a mistake. No pass go, at least straight no pass go, is not so simple, as we shall see.

It was only much later, when I began to study CGT, that I began to understand no pass go. It shows its strangeness pretty quickly.

For instance, what is this marked area worth? (Outer stones alive, by convention.)

Click Here To Show Diagram Code: [go]$$B $$ ------------------ $$ | C X C X O C O X . $$ | X X X X O O O X . $$ | C C X O O C O X . $$ | X X X O C O X X . $$ | C C X O O C O X . $$ | X X X O C O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

It may look like Black is one point ahead here, with four moves (points) to three. However:

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------------ $$ | 5 X 7 X O 4 O X . $$ | X X X X O O O X . $$ | 1 . X O O 6 O X . $$ | X X X O 8 O X X . $$ | 3 2 X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

After

the local result is 0, with only the "group tax" points remaining.

Likewise:

Click Here To Show Diagram Code: [go]$$W White first $$ ------------------ $$ | 6 X 8 X O 3 O X . $$ | X X X X O O O X . $$ | 4 1 X O O 5 O X . $$ | X X X O 7 O X X . $$ | 2 . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

After

the result is also 0. Which means that the original position is worth 0.

And what that means is that the two two point eyes for Black are worth the same as the three one point eyes for White. How can that be? The variations show why. In each one White gets one play inside one of Black's two point eyes. Black gets four plays there, but White gets one, for a net of three plays for Black.

And what that means is that each two point eye is worth 1.5 points (moves).

And that means that, as the end of the no pass go game approaches -- and maybe earlier --, it is generally worthwhile to play inside your opponent's territory. Just as some beginners believe about regular go. :lol:

Author:	Jhyn [ Sun Jan 03, 2016 9:15 pm ]
Post subject:	Re: This 'n' that
Bill Spight wrote: And that means that, as the end of the no pass go game approaches -- and maybe earlier --, it is generally worthwhile to play inside your opponent's territory. Just as some beginners believe about regular go. A semedori of sorts, then. It looks highly nontrivial to find a balance between playing inside your own territory to increase its value by preventing your opponent from playing there, and playing inside your opponent's territory to reduce its value. I wonder if this impacts significantly the strategy. Have you given it a try on 9x9?

Author: Bill Spight [ Mon Jan 04, 2016 8:44 am ]

Post subject: Re: This 'n' that

Jhyn wrote:

Bill Spight wrote:

A semedori of sorts, then. It looks highly nontrivial to find a balance between playing inside your own territory to increase its value by preventing your opponent from playing there, and playing inside your opponent's territory to reduce its value. I wonder if this impacts significantly the strategy. Have you given it a try on 9x9?

Hi, Jhyn.

How to play in apparently simple no pass go positions may indeed be non-trivial.

As for play on a larger scale, I am a tyro. I suspect that staking out large territories or near territories is much less valuable than in regular go. For instance:

Click Here To Show Diagram Code: [go]$$ 2 points $$ ------------------- $$ . X . X . X . O X . $$ . X X X X X X X X . $$ . . . . . . . . . .[/go]

This group has two points of territory.

Click Here To Show Diagram Code: [go]$$ 2 points $$ ------------------- $$ . X . X . X . O X . $$ . X X X X X . . X . $$ . . . . . X X X X . $$ . . . . . . . . . .[/go]

So does this group.

The extra two board points add nothing to Black's score.

Author: Bill Spight [ Mon Jan 04, 2016 8:00 pm ]

Post subject: Re: This 'n' that

No Pass Go (ii)

Something to note about the 1.5 point eye:

Click Here To Show Diagram Code: [go]$$B Black first $$ ----------------- $$ | . X . X 1 . X . $$ | X X X X X X X . $$ | . . . . . . . .[/go]

If Black plays first the result is a 1 point eye.

Click Here To Show Diagram Code: [go]$$W White first $$ ----------------- $$ | . X . X 1 . X . $$ | X X X X X X X . $$ | . . . . . . . .[/go]

If White plays first the result is a 2 point eye, one move to capture the White stone, one move to fill the eye.

Each player loses 0.5 point by playing inside the eye. (OC, that is better than filling a one point eye, which loses 1 point.) The fact that a play loses points indicates that the 1.5 point eye is territory. As we shall see, there are eyes that are not territory, at least not yet.

Note: I am referring to territory in no pass go. Plainly it is not territory in the sense of regular go, but territory is an emergent concept of no pass go (with captures). Different forms of no pass go produce different concepts of territory.

Next, let us look at another kind of play in no pass go.

Click Here To Show Diagram Code: [go]$$B Dame $$ ------------------ $$ | . X . X O . O X . $$ | X X X X O O O X . $$ | . . X C O . O X . $$ | X X X O . O X X . $$ | . . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

To our previous position we have added a dame, :ec:

.

Suppose that White to play plays inside one of the 1.5 point eyes.

Click Here To Show Diagram Code: [go]$$W Play inside territory $$ ------------------ $$ | . X . X O . O X . $$ | X X X X O O O X . $$ | . 1 X 2 O . O X . $$ | X X X O . O X X . $$ | . . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

After

Black has 3.5 points and White has only 3. Black has gained 0.5 point in the exchange.

Click Here To Show Diagram Code: [go]$$W Fill the dame $$ ------------------ $$ | . X . X O . O X . $$ | X X X X O O O X . $$ | . . X 1 O . O X . $$ | X X X O . O X X . $$ | . . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

OC, White should simply fill the dame, leaving a net local score of 0. This is obvious from regular go. Likewise, Black to play should fill the dame.

The dame has a special name in CGT: STAR, which is written with an asterisk. The position in the "Dame" diagram is worth *. See http://senseis.xmp.net/?Star STAR is an infinitesimal. That is playing in it does not change the territorial count. In this case the net territory is 0, both before and after the dame is filled.

What about the next diagram?

Click Here To Show Diagram Code: [go]$$B Dame II $$ ------------------ $$ | . X . X O . O X . $$ | X X X X O O O X . $$ | W . X . O . O X . $$ | . X X O . O X X . $$ | X . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

How do we evaluate the eye with the :wc:

stone inside it?

Click Here To Show Diagram Code: [go]$$B Black atari $$ ------------------ $$ | . X . X O . O X . $$ | X X X X O O O X . $$ | W 1 X 2 O . O X . $$ | . X X O . O X X . $$ | X . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

makes the eye worth 2 points, and then White fills the dame, for a net local score of 0.

Click Here To Show Diagram Code: [go]$$B Black fills dame $$ ------------------ $$ | . X . X O . O X . $$ | X X X X O O O X . $$ | W 2 X 1 O . O X . $$ | 3 X X O . O X X . $$ | X . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

After Black fills the dame, White plays :w2:

. What is this eye worth? White has no move, so let :b3:

take the two White stones.

Click Here To Show Diagram Code: [go]$$B Black fills dame (ii) $$ ------------------ $$ | . X . X O . O X . $$ | X X X X O O O X . $$ | 4 . X X O . O X . $$ | X X X O . O X X . $$ | X . X O O . O X . $$ | X X X O . O O X . $$ | O O O O O O X X . $$ | X X X X X X X . . $$ | . . . . . . . . .[/go]

Now White plays inside the eye to produce a 2 point eye. So the eye with the two White stones in it is worth 2 points.

The net result is again 0. You can see that playing the dame and playing inside the eye are miai for both Black and White, leaving a 2 point eye.

Now we know that two dame are miai, and so are two STARs. In CGT we say that * + * = 0. The value of the eye with the White stone inside is 2 + *, also written 2*. It is not yet territory, because either player can play inside it without losing points.

Click Here To Show Diagram Code: [go]$$B Evaluate $$ ------------------ $$ | . X . X O . . O X . $$ | X X X X O O O O X . $$ | O . X O O . O O X . $$ | . X X O . O . O X . $$ | X . . X O O O X X . $$ | X X X X O . O X . . $$ | O O O O O O X X . . $$ | X X X X X X X . . . $$ | . . . . . . . . . .[/go]

As an exercise, you can evaluate this corner and decide where each player should play.

Author: Bill Spight [ Tue Jan 05, 2016 10:03 am ]

Post subject: Re: This 'n' that

No Pass Go (iii)

Well, I goofed. :oops:

The exercise is harder than I intended.

Click Here To Show Diagram Code: [go]$$B Evaluate $$ ------------------ $$ | . X . X O . . O X . $$ | X X X X O O O O X . $$ | W . X O O . O O X . $$ | . X X O . O . O X . $$ | X . . X O O O X X . $$ | X X X X O . O X . . $$ | O O O O O O X X . . $$ | X X X X X X X . . . $$ | . . . . . . . . . .[/go]

White has 3.5 points. I had intended for Black to have 3.5*, with an eye worth 1.5 and an eye worth 2*, but that is not so, because the eyes are not independent.

Click Here To Show Diagram Code: [go]$$B Black first $$ ------------------ $$ | . X . X O . . O X . $$ | X X X X O O O O X . $$ | W . X O O . O O X . $$ | 1 X X O . O . O X . $$ | X . . X O O O X X . $$ | X X X X O . O X . . $$ | O O O O O O X X . . $$ | X X X X X X X . . . $$ | . . . . . . . . . .[/go]

After

Black indeed has 3.5 points.

Click Here To Show Diagram Code: [go]$$W White first $$ ------------------ $$ | . X . X O . . O X . $$ | X X X X O O O O X . $$ | W . X O O . O O X . $$ | 1 X X O . O . O X . $$ | X . . X O O O X X . $$ | X X X X O . O X . . $$ | O O O O O O X X . . $$ | X X X X X X X . . . $$ | . . . . . . . . . .[/go]

After

Black has only 3 points, plus an infinitesimal, a White sente.

Click Here To Show Diagram Code: [go]$$W White sente $$ ------------------ $$ | . X . X O . . O X . $$ | X X X X O O O O X . $$ | O . X O O . O O X . $$ | W X X O . O . O X . $$ | X 1 2 X O O O X X . $$ | X X X X O . O X . . $$ | O O O O O O X X . . $$ | X X X X X X X . . . $$ | . . . . . . . . . .[/go]

is sente, threatening to take six Black stones. After :b2:

Black has 3 points.

Click Here To Show Diagram Code: [go]$$W White sente (ii) $$ ------------------ $$ | . X . X O . . O X . $$ | X X X X O O O O X . $$ | O 2 X O O . O O X . $$ | W X X O . O . O X . $$ | X 1 4 X O O O X X . $$ | X X X X O . O X . . $$ | O O O O O O X X . . $$ | X X X X X X X . . . $$ | . . . . . . . . . .[/go]

Black can try capturing the two White stones, but the throw-in at :wc:

forces Black again. The result is still 3 points for Black.

Click Here To Show Diagram Code: [go]$$W White first $$ ------------------ $$ | . X . X O . . O X . $$ | X X X X O O O O X . $$ | W . X O O . O O X . $$ | 1 X X O . O . O X . $$ | X 2 . X O O O X X . $$ | X X X X O . O X . . $$ | O O O O O O X X . . $$ | X X X X X X X . . . $$ | . . . . . . . . . .[/go]

So after

Black plays the infinitesimal reverse sente at 2. (Don't you love it? :cool:

) Black has 3 points. White has 3.5 points and wins.

BTW, the White sente infinitesimal is called a TINY; a Black sente infinitesimal is called a MINY. See http://senseis.xmp.net/?TiniesAndMinies

Black to play can move to 3.5 points for Black (0 net points), White to play can move to 3 for Black, plus an infinitesimal. So the count for Black is originally 3.25, with a gote that gains 0.25 points on average.

Before I realized my mistake, I wrote this:

We have seen eyes in which the first player loses points and eyes in which the points stay the same. Are there eyes in which the first play gains points? You bet.

You betcha, indeed! :lol:

Author: Bill Spight [ Wed Jan 06, 2016 12:00 pm ]

Post subject: Re: This 'n' that

I had not intended to write so much about No Pass Go, interesting though it may be. Or maybe not!

Anyway, here are some 7x7 no pass go problems that I composed some years ago. Might as well present them now. If not now, when?

Edit: I think that I should call these studies, since problem implies that the player to play can win. Sorry if I made you frustrated.

Study 1

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . O O . X . . | $$ | X X X X X X X | $$ | . X . X O . . | $$ | X X X X O O O | $$ | . O X O . O . | $$ | O O O O O O O | $$ | . . O X X . . | $$ ---------------[/go]

Study 2

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . . . . X . . | $$ | X X X X X X X | $$ | . X . X O . O | $$ | X X X X O O O | $$ | O O X O . O . | $$ | O O O O O O O | $$ | . . O X X . . | $$ ---------------[/go]

Study 3

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . O . . X . . | $$ | X X X X X X X | $$ | . X . X O X O | $$ | X X X X O O O | $$ | . O X O . O . | $$ | O O O O O O O | $$ | . . O X . . . | $$ ---------------[/go]

Enjoy!

Author: Bill Spight [ Fri Jan 08, 2016 8:10 pm ]

Post subject: Re: This 'n' that

I don't think there is much point in hiding this.

Study 1

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . O O . X . . | $$ | X X X X X X X | $$ | . X . X O . . | $$ | X X X X O O O | $$ | . O X O . O . | $$ | O O O O O O O | $$ | . . O X X . . | $$ ---------------[/go]

Black has 1.5 points of territory in the top right, and 2.5 points in the top left, for 4 points total.

White has 1.5 points in the bottom left, and 2.75 points in the bottom right, for 4.25 points total.

There are 3 dame, so the net value is -0.25* (for Black). White wins.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | 8 O O 9 X 7 . | $$ | X X X X X X X | $$ | . X . X O 1 2 | $$ | X X X X O O O | $$ | 3 O X O . O . | $$ | O O O O O O O | $$ | 6 . O B X 4 . | $$ ---------------[/go]

After the dame are filled (and even before) experienced players could score the game.

But we are not experienced, so let's play it out. :w4:

loses only 0.25 point, making the net score 0. Any other play loses. The next four plays lose 0.5 points each. :b9:

is the first play to lose 1 point on average.

Click Here To Show Diagram Code: [go]$$Wm10 $$ --------------- $$ | X 2 1 . X X . | $$ | X X X X X X X | $$ | . X . X O X O | $$ | X X X X O O O | $$ | X O X O . O . | $$ | O O O O O O O | $$ | O . O X . O . | $$ ---------------[/go]

After

it is obvious that White is one point ahead, and wins.

----

To evaluate the territory we have to evaluate the eyes with four board points and two stones. It is obvious, I hope, that if Black plays first in the top left eye it will have 2 points of territory, and if White plays first in the bottom right eye it will have 2.5 points of territory. If the other plays plays first we get an eye with three stones in it.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | O O O . X . . | $$ | X X X X X X X | $$ | . X . X O X O | $$ | X X X X O O O | $$ | X O X O . O . | $$ | O O O O O O O | $$ | O . O X . O . | $$ ---------------[/go]

White has no play in the top left, Black can move to this position in one net play.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | X O . X X . . | $$ | X X X X X X X | $$ | . X . X O X O | $$ | X X X X O O O | $$ | X O X O . O . | $$ | O O O O O O O | $$ | O . O X . O . | $$ ---------------[/go]

The top left is now worth 2 points of territory, so in the previous position it was worth 3. (If that is not clear, compare it with three one point eyes for White.)

Back to the original position.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . O O . X . . | $$ | X X X X X X X | $$ | . X . X O . . | $$ | X X X X O O O | $$ | . O X O . O . | $$ | O O O O O O O | $$ | . . O X X . . | $$ ---------------[/go]

In the top left eye Black can move to 2 points and White can move to 3 points (for Black). Each move loses 0.5 points, and the eye is worth 2.5 points.

In the bottom right eye White can move to -2.5 points and Black can move to -3 points. Each move loses 0.25 points. That is why White plays there after the dame are filled.

Author: Bill Spight [ Fri Jan 08, 2016 9:55 pm ]

Post subject: Re: This 'n' that

Study 2.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . . . . X . . | $$ | X X X X X X X | $$ | . X . X O . O | $$ | X X X X O O O | $$ | O O X O . O . | $$ | O O O O O O O | $$ | . . O X X . . | $$ ---------------[/go]

In the top left eye Black has 3 points of territory plus 1.5 points in the top right, for a total of 4.5.

In the bottom right eye White, as we know, has 2.75 points plus 1.5 points in the bottom left, for a total of 4.25.

There is one dame, or *.

The net value is 0.25*. Black wins.

OC, Black fills the dame first.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . 7 6 . X . 5 | $$ | X X X X X X X | $$ | . X . X O 1 O | $$ | X X X X O O O | $$ | O O X O . O . | $$ | O O O O O O O | $$ | 4 . O B X 2 . | $$ ---------------[/go]

After

loses 0.25 points, making Black ahead by 0.5. The next three moves each lose 0.5 points. :w6:

and

show why the top left eye is worth 3 points of territory. They are miai and the obvious result is 3 points. The same is true if Black plays at 7 and then White plays at 6. The score is obviously even, and White to play loses.

Author: Bill Spight [ Fri Jan 08, 2016 10:13 pm ]

Post subject: Re: This 'n' that

Study 3.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . O . . X . . | $$ | X X X X X X X | $$ | . X . X O X O | $$ | X X X X O O O | $$ | . O X O . O . | $$ | O O O O O O O | $$ | . . O X . . . | $$ ---------------[/go]

Black has a count of 2.75 in the top left corner plus 1.5 points in the top right, for a total of 4.25.

White has a count of 2.875 in the bottom right corner plus 1.5 points in the bottom left, for a total of 4.325.

There is a dame.

The net count is -0.125*. But Black to play wins, because the top left and bottom right are not yet territory.

Click Here To Show Diagram Code: [go]$$B Black to play $$ --------------- $$ | . O 1 . X . . | $$ | X X X X X X X | $$ | . X . X O X O | $$ | X X X X O O O | $$ | 3 O X O . O . | $$ | O O O O O O O | $$ | . . O X . 2 . | $$ ---------------[/go]

gains 0.25 points, making 3 points of territory in the top left, for an overall count of 0.125*.

:w2:

gains 0.125 points, making 3 points of territory in the bottom right, for an overall count of *.

:b3:

fills the dame, for a net score of 0. White to play loses.

----

In the original position, White to play wins, OC.

Click Here To Show Diagram Code: [go]$$W White to play $$ --------------- $$ | . O 1 . X . . | $$ | X X X X X X X | $$ | . X . X O X O | $$ | X X X X O O O | $$ | 3 O X O . O . | $$ | O O O O O O O | $$ | . . O X 2 . . | $$ ---------------[/go]

gains 0.25, for an overall count of -0.375*.

:b2:

gains 0.125, for an overall count of -0.25*.

This position is basically Study 1. White wins.

----

OC, this is all strange to us. But if this were the usual form of go instead of our game, it would be familiar to experienced players.

Author: Bill Spight [ Sun Jan 10, 2016 10:25 pm ]

Post subject: Re: This 'n' that

Some observations about no pass go

There a number of different forms of no pass go, not just the straight form I have been discussing lately. They include the Capture Game, along with its variants, Capture-2, Capture-4, etc. Also, Cats and Dogs (played on the go board with go stones), in which no stone is allowed to touch a stone of the opposite color, and No Capture Go, in which the first player to capture a stone loses. Also, No Pass Go with Prisoner Return, which is equivalent to territory scoring with a group tax, which appears to be the most ancient form of go for which we have scored game records.

All known forms of no pass go are games of territory, although the definition of territory may be different from game to game. Also, even in the context of a given form of no pass go, territory may have more than one meaning. The basic meaning of territory in all no pass go games is this: an empty point on the board where one player may play without eventually getting captured and can prevent the other player from ever playing there without playing there herself. The territory belongs to the player who can play there. In all known forms of no pass go a one point eye may be territory for the player whose stones surround the single empty point. OC, the opponent might be able to play there by capturing one or more of the stones that form the eye. Each point of basic territory represents one move for the player who owns it. We know from CGT that we can equate one move with one point of a player's score; likewise, we can equate one point of basic territory with one point of the score of the player who owns the territory.

We have already seen, in straight no pass go, regions of the board that may be scored for one player, even though the opponent can play there. We are also familiar with such regions in regular go. We call them territory, and we can also call them territory in no pass go, even though they are not basic territory. Doing so allows us to define territory as regions of the board that are scored for one player or the other.

What do we mean by a score? A score is an number. By convention, scores for Black are positive, scores for White are negative. Zero is also a score. Playing on a point of basic territory loses that point; the player can never play on it again. It reduces one's score by one point. That is a characteristic of the broader meaning of territory. Playing there reduces one's score or increases the opponent's score. We have seen examples in straight no pass go. Also, in regular go, playing inside either one's own territory or the opponent's territory (with gote) loses one point net. (There are also broader meanings of territory such that playing inside it can gain something, but they are not our concern here.)

Once play reaches the point where only scorable points (of territory) are left, the players may halt play and agree on the score. That is in general true of all no pass games, not just no pass go. In fact, with experienced players that will be the general practice. Neither player will wish to reduce her net score.

In many forms of no pass go, not every one point eye is a point of territory. That is so for straight no pass go, the capture game, and no pass go with prisoner return. Consider the following position.

Click Here To Show Diagram Code: [go]$$ Not territory $$ -------------- $$ | . X X . X O . $$ | X X X X X O . $$ | O O O O O O . $$ | . . . . . . .[/go]

Given that the White stones are alive, the two one point Black eyes are not territory. True, White cannot play in either one, but Black cannot play there without getting captured. If Black fills one of the eyes, White can capture and get some points. That is worse for Black than not playing at all. If Black has no alternative, then Black may as well resign.

Click Here To Show Diagram Code: [go]$$ Group tax $$ -------------- $$ | . . . . X O . $$ | X X X X X O . $$ | O O O O O O . $$ | . . . . . . .[/go]

The Black eye surrounds 4 board points. If the surrounding stones were immortal, Black would have 4 points of territory there in the capture game, 3 points of territory in straight no pass go. However, if the corner were played out, since the Black stones are not immortal, Black would have to form 2 one point eyes in order to live, eyes which are not territory. So to find the actual points of territory we subtract 2 points from the points Black would have if her stones were immortal. These 2 points are called a group tax.

Click Here To Show Diagram Code: [go]$$ Dead stones $$ ------------------ $$ | . O X O . O X O . $$ | X X X X X X X O . $$ | O O O O O O O O . $$ | . . . . . . . . .[/go]

.

You may verify that in straight no pass go the corner is worth 3 points for Black, 5 points minus the "group tax". Without the dead White stones it would be worth 1.5 points plus an infinitesimal. The dead stones add to the score, even though prisoners do not. So dead stones are part of the territory.

That may come as a surprise to someone who is used to regular go, as in regular go dead stones are on a par with prisoners. However, it helps to explain why at the end of the game dead stones are removed for counting without capturing them. We may think of dead stones in situ in regular go as worth 2 points each. But it is also possible to think of each dead stone as worth 1 point and the board point that it rests upon as worth 1 point. The same holds for no pass go with prisoner return, where returning a prisoner to the opponent is a possible move. The corner is worth 6 points in that game.

To recap:

No pass go is a game of territory, with the definition depending upon the game.

It is normal to stop play in no pass go by agreement and count the score.

Forms of no pass go in which groups may be captured may have a "group tax". One point eyes that are necessary for life do not count.

In such forms of no pass go dead stones may count.

Author: Bill Spight [ Mon Jan 11, 2016 3:47 pm ]

Post subject: Re: This 'n' that

The Segoe-Takahashi 10,000 year ko rules crisis

I have written about this before, the first time for the AGA Journal in 1977. Also on Sensei's Library ( http://senseis.xmp.net/?canceledit=TenT ... ulesCrisis ). The AGA article was about the recently adopted Taiwan rules, which were, AFAICT, the first Ing rules. It was the article in which I suggested the use of pass stones, which I called bookkeeping stones, to score by area scoring using territory counting.

Basing my article on the Taiwan rules booklet in English, I got the point of the Japanese rules crisis wrong. It wasn't about possibly continuing play in order to resolve the ko, which the booklet implied could be done under area scoring, but about ending play at all. Even the Japanese pros had that problem; it's not just beginners. :lol:

The curious ruling, White won but Black did not lose, may well have had more to do with politics than anything else, as the game was part of an East-West Japan match.

Here is the diagram when Segoe suggested that the game was over, after :b2:

. The ko is on the right (marked).

Click Here To Show Diagram Code: [go]$$W Segoe (W) - Takahashi (2 stones) $$ --------------------------------------- $$ | . . . O O O . . . O . O O O O X . . . | $$ | . O O . . O X . . . O . O X X X . . . | $$ | . X X O O X . O O O . O X . . . . . . | $$ | . X . X O O . . . O O X X X . X . X . | $$ | . X X . O O X X X O O X . . X . . . . | $$ | . O . X . O X O O O X X . X O . O X . | $$ | O O O X X X O O . X X O X X O . . X . | $$ | . O X O X O . . O X . O X X . O O X . | $$ | . . . O O O O . O . O O O X X O X X X | $$ | . O O O X X X O O O O X O X O X X O O | $$ | O O X X O X O O X X O X X O O O O O C | $$ | X O O X . X X X X 3 O X X X X O O X O | $$ | X X X X X . X . O O X X X . X O X X X | $$ | . . . . O X X X X O O X X X O O . X . | $$ | . . O O O O X O O . O X O X X O O O O | $$ | 4 O O X O X X X O , O O O X O X X X X | $$ | X X X X O O O O O O . . O X . X . . . | $$ | . . X X X X O . . . O O X 2 X . . . . | $$ | . X . O . X O . . . . O 1 X . . . . . | $$ ---------------------------------------[/go]

Usually Takahashi, the weaker player, would agree, but not this time. He stared at the ceiling. After a while Segoe filled a dame ( :w3:

) and Takahashi did likewise ( :b4:

), the players soon filling all the dame in alternation. Then play reached an impasse with White to play.

Here is the position with all the dame filled.

Click Here To Show Diagram Code: [go]$$W Dame filled $$ --------------------------------------- $$ | . . . O O O . . . O . O O O O X . . . | $$ | . O O . . O X . . . O . O X X X . . . | $$ | . X X O O X . O O O . O X . . . . . . | $$ | . X . X O O . . . O O X X X . X . X . | $$ | . X X . O O X X X O O X . . X . . . . | $$ | . O . X . O X O O O X X . X O . O X . | $$ | O O O X X X O O O X X O X X O . . X . | $$ | . O X O X O . . O X X O X X . O O X . | $$ | . . . O O O O . O O O O O X X O X X X | $$ | . O O O X X X O O O O X O X O X X O O | $$ | O O X X O X O O X X O X X O O O O O . | $$ | X O O X . X X X X O O X X X X O O X O | $$ | X X X X X . X O O O X X X . X O X X X | $$ | X O X X O X X X X O O X X X O O . X . | $$ | X O O O O O X O O . O X O X X O O O O | $$ | X O O X O X X X O , O O O X O X X X X | $$ | X X X X O O O O O O . . O X . X . . . | $$ | . . X X X X O . . . O O X X X . . . . | $$ | . X . O . X O . . . . O O X . . . . . | $$ ---------------------------------------[/go]

Segoe connected his groups with the first dame in the center, but did not prevent Takahashi from connecting his groups in the bottom left. Apparently Segoe was not preparing for No Pass Go, with its group tax.

At this point the referee, Iwasa Kei, told Takahashi to take and fill the ko, but Takahashi did not, with support from his team captain, Kubomatsu Katsukiyo. OC, for Takahashi to take the ko and fill it while Segoe did nothing would mean that Takahashi would have made three moves in a row (starting with the last dame). By some accounts Segoe said that he (Segoe) could relinquish the right to make a move (pass, as we would now say), but Takahashi said that making a move was an obligation, not a right.

I kind of doubt those accounts, although that was a question that came up at some point. I don't think that Takahashi would have argued with Segoe, but would have let Kubomatsu do the talking. Also, at first blush why shouldn't Segoe fill a point or two of territory while Takahashi captured and filled the ko? He would still win by almost 20 points. But of course if Segoe filled a point of territory Takahashi would not take and fill the ko, either. To do so would mean defeat.

Now if they continued to play inside (someone's) territory, on the theory that the players could not pass, could White eventually force Black to take and win the ko, or resign? Basically, could White win straight no pass go? I am pretty sure that White could, but that's a guess. After all, White is about 20 points ahead. However, 6 of those points are net captured stones, which do not count in straight no pass go, and 9 of those points are net dead stones, which only count partially. Still, Black is 4 points behind on group tax, and the top right and bottom right corners have some wide open spaces, so they are not worth near as much as they would be in regular go.

But suppose that White had won that way. That would have caused a real rules crisis, reverberating far beyond this single game. It would basically change the nature of the game to straight no pass go, because there would be a number of games in which the loser when you count prisoners could force the opponent to resign. The Taiwan rules booklet suggested that modern area scoring was the answer, but that is because the referee could have given Takahashi the choice of having the game scored as is, or having it scored after he had taken the ko, whether he filled it or not. Obviously Takahashi would lose by less if he took the ko. But Iwasa could have given Takahashi that choice, anyway. We now know that Honinbo Shusai believed that a point in an open ko at the end of the game after the dame have been filled was a point of territory. Why did he not make that ruling? My guess is that Baron Okura wanted a political decision that would not offend players in the West of Japan.

Edit: I have since found out that there had been an earlier argument between Shusai and Kubomatsu about filling a ko at the end of play. It would have been impolitic for Shusai to rule against Kubomatsu in this case, based upon the idea that if Black did not take the ko White would get the point of territory in it.

Why did Iwasa tell Takahashi to take and fill the 10,000 year ko? My guess is that that was the custom. After all, 10,000 year kos are not that infrequent, and surely similar positions had occurred without incident in the three centuries of modern Japanese go to that point. Despite Shusai's opinion about the open ko, there was obviously a widespread feeling, apparently shared by Iwasa, that single kos should not be left open at the end of the game, and that became part of the Japanese rules some two decades later. I have some ideas about why that makes sense, but this post is long enough, and contains enough speculation as it is.

Author: Bill Spight [ Tue Jan 12, 2016 3:33 pm ]

Post subject: Re: This 'n' that

Speculations on the prehistory of go

We know that both store scoring and territory scoring with a group tax were ancient forms of weiqi. Apparently territory scoring was the form that came to Japan. Some experts believe that territory scoring evolved from stone scoring, but the two were so similar that it could have happened the other way. So far I have shown how territory scoring can arise from no pass go. As well as a group tax and counting dead stones as territory. Also stopping play and scoring instead of filling in territory. No Pass Go with Prisoner Return is, as far as we can tell, equivalent to ancient territory scoring with a group tax, except that the latter allows jigo. Ancient territory go could easily have evolved from that form of no pass go, instead of from stone scoring. OTOH, why should no pass go have prisoner return?

I think that ancient go may have evolved from a no pass form of the capture game, where the object of play was not originally to surround territory, but to capture stones. As we have seen, the concept of territory emerges naturally from no pass go. Here is an example of territory with group tax in the Capture Game, which I have shown earlier.

Click Here To Show Diagram Code: [go]$$Bc Equal territory $$ ----------- $$ | . O . X . | $$ | . O . X . | $$ | . O . X . | $$ | . O . X . | $$ | . O . X . | $$ -----------[/go]

Each player has 3 points of territory, and there are 5 dame. Black to play will get the last dame, leaving a net score of 0, which means that White to play at that point will lose.

Click Here To Show Diagram Code: [go]$$Bc Filling the dame $$ ----------- $$ | . O 1 X . | $$ | . O 2 X . | $$ | . O 3 X . | $$ | . O 4 X . | $$ | . O 5 X . | $$ -----------[/go]

The players could score the game and agree to the outcome, but play could also continue.

Click Here To Show Diagram Code: [go]$$Wcm6 Filling territory $$ ----------- $$ | 3 O X X . | $$ | . O O X 2 | $$ | 1 O X X 6 | $$ | . O O X 4 | $$ | 5 O X X . | $$ -----------[/go]

Now White resigns instead of playing self atari.

In No Pass Go with Prisoner Return, returning a prisoner as a move means that each prisoner is worth one point of territory, which also means that each dead stone is worth one point of territory, as well as the point it is on, just as in regular go. But there is no prisoner return in the Capture Game, so how doe the equivalence between prisoners and territory arise in it?

Consider the following 5x5 Capture game, in which Black, as the weaker player, gets reverse komi of one point. That is, Black only has to capture one of White's stones to win, while White must capture two of Black's stones.

Click Here To Show Diagram Code: [go]$$Bc $$ ----------- $$ | . . . . . | $$ | 8 5 3 . 7 | $$ | . 4 2 1 . | $$ | . . . 6 9 | $$ | . . . 0 . | $$ -----------[/go]

Click Here To Show Diagram Code: [go]$$Bcm11 $$ ----------- $$ | 0 1 . 9 . | $$ | O X X . X | $$ | 2 O O X 5 | $$ | . 4 . O X | $$ | 8 3 6 O 7 | $$ -----------[/go]

Black takes a cue from straight no pass go and plays a stone inside White's territory at move 13. If Black does not make eyes with :b19:

White can make seki with a move there.

Click Here To Show Diagram Code: [go]$$Bcm21 $$ ----------- $$ | O X . X . | $$ | O X X . X | $$ | O O O X X | $$ | . O . O X | $$ | O . O O X | $$ -----------[/go]

Still, after :w20:

each player has 1 point of territory, for a zero game with Black to play. Black still loses.

Here is the variation at :b13:

without the reverse komi.

Click Here To Show Diagram Code: [go]$$Bcm11 $$ ----------- $$ | 8 1 . 3 . | $$ | O X X . X | $$ | 2 O O X 5 | $$ | . 4 . O X | $$ | . 6 . O 7 | $$ -----------[/go]

After

White has 2 points to 1, and wins. This shows how the sacrifice of :b13:

saved one move for Black.

This game illustrates how in Capture-N, each player should sacrifice in order to come down to what is in effect Capture-1 at the end of the game. It also shows how each sacrifice stone is worth one point of territory, establishing the one to one equivalence of stones and territory. This is another way of doing that instead of via prisoner return.

It is easy to see how something like Capture-10 or Capture-20 could lead to territory scoring with a group tax.

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