Go Infinitesimals

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  Difficulty: Intermediate   Keywords: EndGame

In combinatorial game theory (CGT) infinitesimals are games of temperature zero where the whole point is to get the last move. (The winner always moves to a score of 0, and so does not win on points.)

The simplest infinitesimal is {0|0}, called STAR (*). Each player can play to a position worth 0. A * in go is a dame.

I (Bill Spight) have heard that other infinitesimals have been constructed, that involve seki which alters the parity of the dame on the board, but I have not seen one. There seem to be no other infinitesimals in regular go.

However, plays with a miai value of 1 may be treated as infinitesimals. Then the CGT theory of infinitesimals may be applied to them. Berlekamp and Wolfe did this in Mathematical Go: Chilling Gets the Last Point.

Such plays are related to infinitesimals by a process called Chilling. Chilling in effect reduces their temperature to 0, which allows us to apply what has been learned about CGT infinitesimals to them. Understanding them is useful in a fight for the last play, which may decide the game.

Plays with a smaller miai value than 1 chill to numbers.

Some of Berlekamp and Wolfe's problems stumped the pros, even 9 dans! However, the theory makes them fairly easy to solve. A few heuristics should enable the knowledgeable go player to get the last 1-point play if possible, in nearly all cases. (See Ongoing Game 1 #1 for an example.)

Learning about infinitesimals can help go players because it may be important to get the last move in certain situations where the plays are larger or even the last move in a close game.

For examples of chilled go infinitesimals, see

Further discussion in Infinitesimals Discussion.

--Bill Spight


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Go Infinitesimals last edited by Bill on May 1, 2008 - 17:05
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