Corridor Infinitesimals
Here are some examples of infinitesimals in chilled go that occur in corridors.
These corridors correspond to the following infinitesimals.
- The top one corresponds to * (STAR).
- The second one corresponds to ↑(UP).
- The third corresponds to ↑↑* (double up star).
- The fourth corresponds to ↑↑↑ (triple up).
Continuing, longer corridors of the same type correspond to ↑↑↑↑*, (quadruple up star), ↑↑↑↑↑ (quintuple up), etc.
Since the play in infinitesimals is about getting the last play (tedomari), you can see how Black would not be in a hurry to play in any of the longer corridors, as he still gets the chance to play last even if White plays there first.
There is a similarity between a fight for the last play and a race to capture. Each infinitesimal is roughly equivalent to some number of ↑s, called its atomic weight. Atomic weight is analogous to the liberty count in a semeai, but the contest is over tedomari. An advantage of at least 2 in atomic weight is enough to guarantee tedomari.
Playing corridor infinitesimals
White to play should play in one of the longer corridors, to reduce Black's atomic weight. Which one should he play in?
It turns out that it does not matter. A play by White in one of the long corridors gains the same thing. In terms of infinitesimals, it gains ↑*, reducing Black's atomic weight by 1.^{[1]}
See Go infinitesimals and Playing infinitesimals.
Why does the third corridor correspond to ↑↑*?
For convenience, I am just going to call these positions by the names of their corresponding infinitesimals. By definition,
- `` " ↑↑"ast = uarr + uarr + ast ``
i. e.,
Of course, the counts are different. What we want to know is if they are equivalent in terms of getting the last one point play.
To answer that question we can subtract the second position from the first. How do we do that? We reverse the colors of the second position and add it to the first. In other words, we take the difference game.
To say that this position equals 0, in terms of go infinitesimals, means that it is miai. With correct play, it does not matter who plays first, the second player will get the last play and the result will be the same (not necessarily 0).
We get to the same place by a different route. So the whole thing is miai, as claimed.
You may verify that this is optimal play by both sides.
Now, if this came up in a real game, you could read it out. But if you know your go infinitesimals, you can quickly see that it is miai.
Chilling corridors
Let's take the top one first.
If Black takes, the local score is +2 (which chills to +1 when we take away 1 point for each Black move). If White connects, the local score is 0 (which chills to +1 when we add one point for each White move). The chilled game looks like this:
- `` { 1 | 1 } ``
If a game is not a number, we may add a number to it by adding that number to both its Left and Right followers, so
- `` { 1 | 1 } = 1 + { 0 | 0 } = 1 ast ``
The top game chills to `` 1 + * `` (STAR). What the number is does not affect correct play at temperature 1, where we are simply trying to get tedomari, so it is customary to ignore the number and just call this position *.
(The values ``ast`` and ``uarr`` (UP) are introduced in the article Chilling.)
Now for the second one.
Black can move to a score of 3, which chills to 2. White can move to the first position, which chills to 2* (1* by itself, but White has made a move, which adds one to the chilled score). The chilled game looks like this:
- `` { 2 | 2ast } = 2 + { 0 | ast } = 2uarr ``
So the middle game chills to 2 plus ↑ (UP). Again, we ignore the number and just call this position ↑.
Now for the third one.
Black can move to a score of 4, which chills to 3. White can move to the middle position, which chills to 3↑. The chilled game looks like this:
- `` { 3 | 3uarr } = 3 + { 0 | uarr } ``
The infinitesimal, `` { 0 | uarr } ``, is the canonical form of ↑↑* (Double UP STAR)^{[3]}. So the bottom game chills to 3 plus ↑↑*, and we just call it ↑↑*.
The fourth one chills to 4↑↑↑, which may be derived in similar fashion.
The chilled value of the whole position is just the sum of its parts:
- `` 1ast + 2"↑" + 3"↑↑"ast + 4"↑↑↑" = 10 + "↑↑↑↑↑↑" + ast + ast = 10"↑↑↑↑↑↑" ``
Ignoring the number, as usual, we just call it ↑↑↑↑↑↑ (sextuple up).
[1]
The infinitesimals of the corridors form an arithmetic series. (By contrast, the fractions of the corresponding empty corridors form a geometric series.)
- `` ast + "↑"ast = "↑" + ast + ast = uarr ``
- `` uarr + "↑"ast = "↑""↑"ast ``
- `` "↑""↑"ast + "↑"ast = "↑""↑""↑" ``
- etc., etc.
This means that when White plays in one of the long corridors she gains ↑``ast`` in the chilled game, which is about as well as one can do by playing a go infinitesimal in the fight for tedomari. (Saving the stone gains ``ast``, which may be no gain at all!)
[2]
White scores are, by convention, negative.
[3]
Why `` "↑""↑"ast = { 0 | uarr } ``
- Consider the options for both sides in `` "↑""↑"ast ``:
- White (Right) can move to `` "↑""↑" `` or to `` uarr + ast + ast = uarr ``. Since `` uarr < "↑""↑" ``, `` uarr `` dominates.
- Thus White’s options can indeed be simplified to ``uarr``.
- Black (Left) can play to ``"↑"ast`` or to `` "↑""↑" ``; we shall find that both reverse:
- From ``"↑"ast``, White can play to ``0``.
- We know that ``"↑""↑"ast > 0 ``, so the play to `` "↑"ast `` reverses, i.e. we may replace the option `` "↑"ast `` with the Left options of ``0``.
- However, there are no Left options of ``0``, so we eliminate that option (`` "↑"ast ``) altogether.
- From ``"↑""↑"``, White plays to ``"↑"ast``, but ``"↑""↑"ast > "↑"ast `` because `` uarr > 0 ``, so that play reverses.
- From `` "↑"ast ``, Black can play to `` ast `` or `` uarr ``; both reverse:
- From `` ast ``, White plays to ``0``, so we eliminate `` ast ``, as we did with `` "↑"ast ``.
- From `` uarr ``, White plays to `` ast ``. `` "↑""↑"ast > ast ``, so that play reverses. From `` ast `` Black moves to ``0``.
- From `` "↑"ast ``, Black can play to `` ast `` or `` uarr ``; both reverse:
- Thus Black’s options can indeed be simplified to ``0``.
- From ``"↑"ast``, White can play to ``0``.
- White (Right) can move to `` "↑""↑" `` or to `` uarr + ast + ast = uarr ``. Since `` uarr < "↑""↑" ``, `` uarr `` dominates.
- Whew!