Torazu sanmoku = Three points without capturing.
The traditional Japanese ruling (by Honinbo Shuwa, according to the Esperanto link below) was that this corner position was worth 3 points for Black at the end, without his having to capture the marked white stones. The Japanese '89 rules make this seki if it is left on the board at the end of the game.
See http://jega.hp.infoseek.co.jp/terminaro/esp_tekn_S-.htm
for an Esperanto explanation.
Mathematical Go defends the 3-point ruling in terms of combinatorial game theory.
Bill:
The question of whether this corner position is worth 3 points for Black (assuming no ko threats) may be decided, in theory, by a difference game. Having 3 prisoners is equivalent to having 3 points, so for the difference game we can give White 3 black prisoners.
For this kind of difference game, instead of making a play on the board a player may return a prisoner to the opponent. There are no ties. If the score would be a tie, the second player wins. Two games are equal if their difference game is zero, a second player win.
The difference game is played only in the 6 points in question in the corner. By convention, the outer stones are alive.
First, the play when Black plays first:
White has 3 prisoners.
White has 3 prisoners, Black has 4.
White has 6 prisoners, Black has 4.
at
.
Now White has 6 prisoners and Black has 5.
Continued, White gets the last play, and wins.
Second, the play when White plays first:
White has 3 prisoners.
at
.
White has 4 prisoners, Black has 5.
at
.
Now each player has 5 prisoners, with White to play.
Continued, Black gets the last play, and wins.
Since the second player wins the difference game, it is worth zero, and the corner is equivalent to having 3 white prisoners, i. e., 3 points, as advertised.
Shouldn't that be worth TWO points for black??
Take this example: Komi is 0, no prisoners, if the left corner is counted as seki, the score is balanced 9-9. Black should capture, and the only white move making sense after that is to cut-off.
This is how black should continue. Black playing first at 3 leads to a win by just one point.
After 3, no side makes extra territory but the prisoner balance is 5-3 for black.
There is nothing black can do to win 3 points. Or is it? (Please correct this if I'm wrong.)
Bill: But if White plays first the result in the corner is 3 points for Black. Each player would prefer for the other to go first. That is why this is an impasse.
It is analogous to a snapback, except that it is more complicated. But as with a snapback, the result differs by one point if either player continues, playing first.
This issue is being discussed on the /Discussion page. :-)
(Minor point: There is a problem with White's carpenter's square shape in the corner that complicates the example.)
Thank you. Now I understand. I believed the corner could not be invaded (so does GnuGo). But now I have tried invading at (2,2). Bad example.