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Infinitesimals
Path: Endgame · Prev: Numbers · Next: PlayingInfinitesimals Path: CGTPath · Prev: InfinitesimalsDiscussion · Next: MoreInfinitesimals
Keywords: EndGame
Here are some infinitesimals in chilled go.
Let's take the top one first. If Black takes, the local score is +2 (which chills to +1). If White connects, the local score is 0 (which chills to +1). 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* The top game chills to 1 plus * (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 *. (I have introduced * and ^ (UP) in Chilling.) Now for the middle 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 | 2* } = 2 + { 0 | * } = 2^ So the middle game chills to 2 plus ^ (UP). Again, we ignore the number and just call this position ^. Now for the bottom 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 | 3^ } = 3 + { 0 | ^ } The infinitesimal, { 0 | ^ }, it is the canonical form of ^^* (Double UP STAR). So the bottom game chills to 3 plus ^^*, and we just call it ^^*. Why { 0 | ^ } = ^^* White (Right) can play to ^^ or to ^ + * + * = ^. Since ^ < ^^, ^ dominates. Black (Left) can play to ^* or to ^^. Both reverse. From ^* White can play to 0. We know that ^^* > 0, so the play to ^* reverses. However, 0 has no Left option, so we eliminate that option. From ^^ White plays to ^*, ^^* > ^* because ^ > 0, so that play reverses. From ^* Black can play to ^ or *. Both reverse. From * White plays to 0, so we eliminate that option, as above. From ^ White plays to *. ^^* > *, so that play reverses. From * Black moves to 0. Whew! The chilled value of the whole position is just the sum of its parts: 1* + 2^ + 3^^* = 6 + ^^^ + * + * = 6^^^. Ignoring the number, as usual, we just call it ^^^ (Triple UP). Note that the infinitesimals of the corridors form an arithmetic series. (By contrast, the fractions of the corresponding empty corridors form a geometric series.) * + ^* = ^ + * + * = ^ ^ + ^* = ^^* Does the next corridor equal ^^^ ? ^^* + ^* = ^^^ Yes, it does. :-) Etc., etc. This means that when White plays in a corridor (except to connect to the stone) she gains ^* in the chilled game, which is about as good as you can do in the fight for tedomari. (Connecting gains *, which may be no gain at all!) Each infinitesimal is roughly equivalent to some number of ^s, called its atomic weight. Atomic weight is similar 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. More in Tinies and Minies, Playing Infinitesimals, Other Infinitesimals, and More Infinitesimals. Further discussion in Infinitesimals Discussion. [1] This is not a proof. There are other games that are confused with ^ that do not have an atomic weight of 1. Atomic weights are not necessarily numbers. Path: Endgame · Prev: Numbers · Next: PlayingInfinitesimals Path: CGTPath · Prev: InfinitesimalsDiscussion · Next: MoreInfinitesimals This is a copy of the living page "Infinitesimals" at Sensei's Library. ![]() |