Kyu Exercise 96 / Solution

Solution  

The optimal solution is to play the placement at , followed by the endgame profit at , before starting a throw-in ko at . (Of course, if Black cannot win this ko immediately, can play at a for sente seki, or even tenuki instead.)

The are other ways to start a ko, but none of them can get the endgame benefit of in sente.

Bill: The trouble is, is normally a mistake.

Solution  

Normally, White should play to make a throw-in ko. This allows Black to take the ko first, but, win or lose, White has saved a move and a point by not making a play at a.

Bill: Later note. The savings of a move and a point are by comparison with a Black tenuki after Wa, and then a White throw-in at . As Kirby and xela show below, White does not get those savings by comparison with a Black throw-in at after Wa. If that Black throw-in is correct, then Wa is indeed superior. :)

xela: Are you sure about this? This position appears in many books, and the authors are unanimous (if my memory is correct) that the first diagram above is the preferred solution. I'm sure it's possible to construct a whole board position where the second method is better--it depends on what ko threats exist--but generally I think taking the ko first is significant, worth more than one point.

Bill: I suspect that this is given as a life and death problem in the books. As such, taking the ko first gives a better chance to live. But as an endgame problem life and death is not the only consideration. If White connects at a she loses a move (and a point, in addition).

Throw-in  

Connection  

Bill: The throw-in position is worth, on average, at least as much as the connection position. I hope that that is fairly obvious, on inspection. :) (In two net moves Black can take the corner in either diagram. In the throw-in diagram White can take the corner in one move, while in the connection diagram it takes two net moves.)

Bill: In general, the connection is better only when it matters who takes the ko first *and* we are in the middle endgame, not the large endgame. In the large endgame White can get adequate compensation if she loses the ko.

Kirby: Here is my take on the situation. Please feel free to correct me if I am mistaken.

Assuming that either W or B will immediately take the throw-in for the ko, there are really two cases that we are dealing with - Case A and Case B.

Case A  

B has 1 white capture. White must find first ko threat.

Case B  

W has 1 black capture. Black must find first ko threat.

For each case, there are two scenarios: 1.) W loses the ko threat 2.) W wins the ko threat

W loses ko

Case A  

No more white captures, but 8 dead stones.
Result: B had 1 white capture at start, plus 8 dead stones after ending ko = 9 white captures.

xela: black has captured 8 stones now on the board plus the original white throw-in stone, and has 11 points of territory in the corner (including the spaces occupied by the dead stones), for a total of 20 points.

Kirby: Here, and for all of the other diagrams, I've only counted the stones. The territory is the same in both cases (I've also taken into account the extra stone in the case of connecting).

Case B  

(one additional white capture from b taking the ko) 1 white stone capture + 9 other dead w stones = 10 captures.
Result: W had one inital black capture, but B has 10 captures from winning the ko. The net result is 9 white captures.

xela: Black has 9 net captures plus 11 territory in the corner, total 20, same as case A.

W wins ko

Case A  

W gets 1 additional black capture, and fills in.

Result: White has 1 black capture. Nothing dies. Black has 1 capture from the definition of Case A. Net result = 1-1 = 0.

xela: Case A is the throw-in scenario: black captured a white stone to start with, now white has recaptured, so the net ko captures are zero. White has net two black prisoners plus 5 territory, total 7 points in the corner.

Kirby: I had forgotten to include the initial captures specified in the description of Cases A and B. For Case A, B has 1 capture. So the 1 white capture and the 1 b capture cancel out for a net result of zero. I've updated this in what I wrote above.

Case B  

W fills in the ko, having 0 additional black captures. White has one less point than in the Diagram A case, due to the marked stone.

Result: White has 0 black captures here, 1 black capture from the definition of Case B, and -1 points for the marked stone. Net result = 0+1-1 = 0.

xela: In case B, the ko started with black throwing in, so white has net three black prisoners (two on the board plus the ko capture) and 4 territory for 7 points in the corner, same as case A.

Kirby: As in the case above, I forgot to add the capture noted in the definition of Case B. I've updated it now.

Conclusion: It seems that both Cases A & B are equal in points. In Case A, white must find the first ko threat, which could result in him losing the ko. Therefore, I feel it is better to play Case B in all situations.

Have I made mistakes in this analysis?

xela: Now I look at it again, is seems to me that the score is exactly the same either way.

Kirby: I've updated the conclusion to reflect the captures that I forgot from the definitions of Cases A & B for the case of white winning the ko.

Bill: OK, let's do a bit more in depth analysis. :)

First, suppose that White is komaster.

Throw-in  

Then, of course, White will make the ko with . What is this position worth?

White wins the ko  

If White wins the ko, is sente. The net score in the corner is -7. (White has taken one more stone than Black in the ko.)

Black wins the ko  

If Black wins the ko the net score in the corner is +22. (Since White is komaster, we do not expect Black to win the ko, but this is what White is preventing, and so it is used in determining the value of the ko.)

Throw-in  

The value of this position is then (22 - 7 * 2)/3 = 2 2/3. Since this is a simple ko, the value is the same regardless of who can win it.

fractic: I'm confused about how you got that value. If nobody was komaster it would have a value of (22+7)/3 right? So how do you evaluate kos when someone is komaster (or even komonster)? I couldn't find on SL with a quick search.

Bill: You are referring to the value of a move, which is indeed 9 2/3, regardless of who is komaster. With one move White can play to a position worth -7, so the value of this position is -7 + 9 2/3 = 2 2/3. :) It is not easy to explain komaster evaluation without thermography. There are a number of examples on rec.games.go in the 1990s. There are also a couple of examples on SL of komaster computations. One is for the approach ko on the komaster page, another is on the BQMRJ000 page.

fractic: Thanks for the answer and the links. I just got confused about the notation you used.

Now let us suppose that Black is komaster. White could still play the throw-in to get the same value. However, suppose that White plays the connection.

Connection  

What is this position worth?

Throw-in  

Black can, of course, play a throw-in and win the ko fight.

Black wins ko  

If Black wins the ko the net score in the corner is +22, as above.

But we cannot assume that Black will make and win the ko.

Why not? Because Black may be satisfied to have induced White's connection and play somewhere else. After all, what does White threaten?

White's threat  

lives.

How much is this position worth?

White's corner  

White can take the corner, after which is sente. The net score in the corner is -6.

Seki  

Or Black can make seki, after which is still sente. The net score in the corner is 0.

Small corner  

So the value of this position is (-6 + 0)/2 = -3.

Connection  

And the value of this position is (22 - 3 * 2)/3 = 5 1/3. Which is better for Black than the immediate throw-in.

What this means is that White should throw in, even if Black wins the ko. Now, there are exceptions, of course. For instance, the connection is right when neither player has a ko threat. But in general the throw-in is right.

Kirby: I had felt thorough about my analysis before, but after reading this, I realize that I did not consider at least two aspects of the problem - namely, the potential for black to make seki and the fact that black may not immediately start out the ko. Even without doing additional math, this shows that the throw-in is worth more points if white can win the ko. I won't go so far to say that the throw-in is correct "in general", because an additional ko threat could make a big difference in a lot of games. But I think it is fair to say that the connection is worth fewer points if you assume that the additional ko threat does not make a (significant) difference.

xela: Yes, it turned out to be much deeper than I initially expected--I'm glad I asked! Thanks Bill for answering our questions so patiently.