Here is an sgf file for Three-Points-Without-Capturing, assuming no ko threats.
Here is the game tree: {-3|-3||-2}. This is a standoff, as neither player wishes to play first. In CGT this reduces to { |-2}. That is, White to play moves to a local score of -2. In go terms that is equivalent to filling your own territory. If filling your own territory costs one point, you had one more point to start with. { |-2} = -3. The fact that go players came up with the same value is very interesting, isn't it?
BTW, in CGT the dame when Black plays first is significant. {-3|-2} = -2½. You get a similar effect with rules that penalize Black one point for passing first, but not White. Then the first pass is worth {-1|0} = -½ in CGT.
Let's try to construct the thermograph for this position.
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The left scaffold is to the right of the right scaffold at or above temperature 0. This indicates a standoff. Neither player will wish to play first. All we can say at this point is that the score of the position lies between -2 and -3, inclusive. CGT, as we have seen, evaluates this game as -3. The Japanese 1989 rules evaluate it as -2, by forcing White to play at temperature 0 to avoid a seki. Before that, as the name indicates, Japanese rules evaluated this as -3 at the end of play. (The ko threat situation is significant, OC.) The fact that the left scaffold is vertical at temperature 0 cannot be seen, as it is inclined above temperature 0.
To handle such games Berlekamp extended thermography downward to temperature -1, which he called subterranean thermography. Below temperature 0 the vertical blue scaffold shows up.
- sanmoku888.png (2.86 KiB) Viewed 24568 times
Not only does the vertical blue line show up, the two scaffolds of the thermograph meet at a local score of -3. The mast rises vertically from that point.
- sanmoku000.png (2.36 KiB) Viewed 24568 times
At temperature -1 we have a tax of -1 for each play. In go that means that each stone played is worth 1 point. However, it is not exactly the same as stone counting. With stone counting this position would be worth 0, not -3.
Anyway, subterranean thermography gives us the correct CGT mast for Three-Points-Without-Capturing with no ko threats.
is a ko threat} we may consider the sequence,
-
, as a unit.