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Chris: I would say "position". The game state also includes information regarding ko threats and captured stones, but "Stellung" only involes what is on the board, I think.
Position is already used for the distribution of black and white stones on the board.
Stellung is the position plus the respective prisoner count. State could be a possible translation. I'll ponder that a bit :)
What about "situation"?
(sign: Hicham)