# Chilling

In general, the difference between Territory and Area scoring is that stones on the board are counted in area scoring but not in territory scoring. Suppose that you play by area scoring but each board play costs one point. You would actually be playing by territory scoring, since each board play results in a stone on the board or captured. (Japanese and Chinese scoring also differ about whether to count points in seki, but that is another matter.)

It is a somewhat curious fact that, aside from exceptional cases involving seki or ko, winning play by territory scoring is winning play by area scoring. The difference between the two forms of scoring has very little effect on optimal play.

Similarly, if you impose a tax on territory scoring of 1 point per board play^{[1]}, you get a form of scoring such that, aside from exceptional cases involving seki or ko, winning play by it is winning play by territory scoring. In *Mathematical Go* Berlekamp and Wolfe refer to imposing this tax as *chilling* and solve endgame problems by "playing the *chilled game*". Chilling is like cooling.^{[2]}

**Why chill a game?**

Because for certain go positions how to play the chilled game is well known. Those not involving ko or seki with a miai value less than 1 chill to numbers. While it may be correct in some overall positions to make a play that is not the largest, numbers should not be played before hotter plays, and hotter numbers should be played before cooler ones. Also, plays with a miai value of 1 chill to infinitesimals, and correct play with many infinitesimals is known. The object is to get the last play (tedomari) at temperature 1. Getting that tedomari may not matter, but it doesn't hurt. :-)

**Numbers**

Some games (local go positions) that are not numbers chill to numbers. Conventionally, we just call them numbers. If there are no kos involved, it is never optimal to play in a number instead of a hotter play, and it is never wrong to play in the hotter of two numbers. With no kos involved, go numbers always have a power of 2 in the denominator. The ones with larger denominators are hotter; play in them has a miai value of 1 - 1/D, where D is the denominator.

**Infinitesimals**

Non-ko games with a miai value of 1 chill to infinitesimals. The point of play in infinitesimals in combinatorial game theory is to get the last move (tedomari). A good deal is known about best play with infinitesimals that carries over to go through chilling.

The simplest infinitesimal is STAR (*).

{ 0 | 0 } = *

STAR is a first player win. In territory scoring, a dame is a *.

- Let's see why. In ZeroInCGTTerms I have a zero position that I will use as standard. Then,

- The marked point is dame. It is the only legal play for either player, and results in zero. - Migeru

- Over to Bill.

Two *s are miai. In CGT terms,

* + * = 0

Zero is a second player win.

Considering territory scoring as chilled area scoring, in area scoring a dame is worth

{ 1 | -1 }

So if the number of dame is odd, getting the last one (tedomari) is worth 1 point.

An important infinitesimal is UP (^).

{ 0 | * } = ^

UP is positive, which means that Black (Left) wins, no matter who plays first.

UP is ambiguous, but is like a White sente. Black has the advantage because of the possibility of playing the reverse sente.

The negative of UP is DOWN (v).

Two games are said to be *confused* if their difference game is a first player win. UP and STAR are confused.

^ - * = ^ + * ; * + * = 0, so * is its own negative.

In ^ + *, also written ^*, Black moves in *, leaving ^, which wins, even though White has sente. White moves in ^ to * + *, which equals 0, so White wins, having gotten tedomari.

Double UP STAR (^ + ^ + *, written ^^*), however, is positive. Black moves to ^^, which is positive, and wins. White can do no better than to play to ^ + * + * = ^, which is also positive, and loses.

UP, STAR, and Double UP STAR are all chilled go positions, as are other infinitesimals.

For more on infinitesimals in chilled go and how to play them, see Infinitesimals.

-- BillSpight

[1] At least until the dame stage. At that point you stop taxing.

[2] The difference between chilling and cooling: Cooling by 1 point is the same as chilling, except that if cooling a game, G, by *less* than 1 point produces a value infinitesimally close to a number, x, the cooled game is x. For go positions the difference between chilling and cooling by 1 is academic. See /discussion

Wouldn't it make sense to move the definition of the various infinitesimals to infinitesimals and move the content of infinitesimals to __infinitesimals in chilled go__? or __infinitesimal example__?? - Migeru

I think it would and it might help Evpsych who commented that the discussion of Combinatorial Game Theory was difficult to follow. However, I think I will leave this to someone who has a more comprehensive understanding than I do. I have noticed that BillSpight seems to know a lot about this topic. - Randall

Charles Bill is an expert. We'd all like to see him connect up his contributions here into something more readable.