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Published online by Cambridge University Press: 27 June 2025
Most of the application of combinatorial game theory to Go has been focussed on late endgame situations and scoring. However, it is also possible to apply it to any other aspect of the game that involves counting. In particular, life-and-death situations often involve counting eyes. Assuming all surrounding groups are alive, a group that has two or more eyes is alive, and a group that has one eye or less is dead.
This naturally raises the question of which game-theoretical values can occur for an eyemaking game. We define games that provide a theoretical framework in which this question can be asked precisely, and then give the known results to date. For the single-group case, eyespace values include 0, , and several ko-related loopy games, as well as some seki-related values. The eye is well-understood in traditional Go theory, and only a little less so, but, and
may be new discoveries, even though they occur frequently in actual games.
For a battle between two or more opposed groups, the theory gets more complicated.
1. Go
1.1. Rules of Go. Go is played on a square grid with Black and White stones. The players alternate turns placing a stone on an unoccupied intersection. Once placed, a stone does not move, although it may sometimes be captured and removed from the board.
Stones of the same color that are adjacent along lines of the grid are considered to be connected into a single indivisible unit For example, if one takes the subgraph of the board grid whose vertices are the intersections with Black stones and whose edges are the grid lines that connect two such vertices, the Black units are the connected components of that subgraph.
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