Endgame Problem 40 / Attempts

Black to play  

KarlKnechtel: This is the original board position, we are asked to give the best option for black - a, b, or c. I will attempt to calculate the miai values of the moves.

Black at 'a', White next  

is White's sente. and eventually (or the other way around) are forced and the local tally is 4 points.

Black at 'a', black next  

Black threatens x with a net result of 6 points. This is thus sente, so white will play 4 at x to limit the result to 4 points.

Since an exchange by both players in either order leads to 4 points after , the tally is 4 points after .

White at 'a', black next  

Black has this option if white plays first starting at a. ( at 5 is of course inferior). and are sente so this is the limit of the analysis. The result is -2.

White at 'a', black next  

Alternately, black starts a ko. Black has had 2 stones captured, but has a solid point of territory and is about to begin the capturing. Black reaches this position in sente I believe.

What is the value of ko here?  

Call the local score at this point x. With white first, the score moves to -3. With black first...

What is the value of ko here?  

Call this score y. If white plays next, the score is -1/3, or x, whichever is less (favourable to white). If black plays again...

What is the value of ko here?  

Call this score z. If white plays next here, the score is y or 2 2/3, whichever favours white.

What is the value of ko here?  

If black plays yet again, and also connects, the score is 4 (3 white stones captured, plus 3 points territory, less 2 black stones captured). If black and white trade here, the score is z again.

Putting all that together: When white chooses to simplify the ko in all cases, I get a value of 28/9 for z, and hence -29/36 for x. When white keeps the kos complicated (does not play passively), I get -8/5 by solving the simultaneous equations in x,y and z (and assuming that sente doesn't apply to ko since you don't necessarily get to respond anyway. I suppose this ko could depend on who is komaster, too).

Putting all that together, black will want to choose the ko in response to at a (in the original diagram), and white will play the ko aggressively, yielding a score of -8/5 after -.

Bill: See Three Stage Ko.

White at a, white next  

Finally, if white is allowed two moves in the starting diagram, we get this situation. Black at a is sente and leads to a score of -3. White at a ends up with a score of -4 points.

Putting everything together, the tree becomes consistent with a value at the top of (-7/2 + -8/5)/4 + 4/2 = .725, and thus a miai value of 3.275 points for a play at a. Playing here is not sente for either player.

Now then, let's try the miai value of ''b''?. :)

White avoids ko  

If , White does better to avoid the ko with . could also be at a.

Karl Knechtel: I could have sworn I thought about a line of play like that, but didn't think to analyse it. :) So the tally is -2 here right? That would make the miai value 3 3/8 instead, with the original score being 5/8.

Bill: The tally here is -2. If the Black sagari is correct, then the tally after it is +4, as you figured. :-)