Gérard TAILLE wrote:
Oops I am now a little lost.
The results you calculated are wrong (instead of 6 + 8 + 6 + 2 = 22 you must read -6 + 8 + 6 + 2 = 10) but I corrected them easily.
In your example with u,v,y,z = 4,3,2,1 you showed that playing in the environment is best.
Could you clarify when you have to provoque a ko in the corner?
Note that in your example you have both u+v < 13 and u < 6½ and black prefers playing in the environment.
Thanks for the correction.
Edit: To be clear, I did not show that playing in the environment first is best, I showed that playing in U dominates playing the sente, given the ko. The environment is Y + Z.
The main assumption is that Black is the ko winner, killing the corner in 3 net local plays, if she plays first. To keep things simple, we are ignoring the possibility of seki. Obviously, to be the ko winner Black needs ko threats which White answers. These threats are not shown. Any other threats in the ko fight are not shown, either.
Let's back up to where there is only the corner and the environment with temperature t, where the last player to play in it gains t/2 in the end, an ideal environment.
Black to play has three choices, to start the ko, to play in the environment, or to play the sente, allowing White to live in the corner, and then play in the environment.
2) Black plays in the environment, White saves the corner, and then Black plays in the environment again.
Result: -6 + 1½t
3) Black plays the sente first, White replies, and then Black plays in the environment.
Result: -5 + t/2
Black is indifferent between these plays when
-6 + 1½t = -5 + t/2
t = 1
(We knew that already.
)
If t > 1 then Black prefers to play in the environment first.
What if Black starts the ko first? Black kills the corner in 3 net plays, while White plays three times in the environment in exchange.
Result: 20 - 2½t
If Black is indifferent between that and playing in the environment first, then
-6 + 1½t = 20 - 2½t
4t = 26
t = 6½
Black is komaster because she has not had to ignore any ko threats to win the ko. From this we derive the average value of a play in the ko when Black is komaster to be 6½ points.
So far, so good.
But it is unusual, except at low temperatures, to have several plays worth exactly the same. So we look at the corner plus U = {u|-u}, where u > t. And then we look at the corner plus U and V = {v|-v}, where u ≥ v > t.
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Bill Spight
is needed to avoid a potential ko
allows white to play tenuki BEFORE the beginning of the ko. That is the point; white is able to destroy black ko threats and saves his two groups.
takes t/2