# How often do random games of go end in illegal moves?

Suppose that moves are generated from two players in accordance with three rules: each move is chosen uniformly at random among places on the board ($$19 \times 19$$, $$9 \times 9$$, or $$k \times k$$ with small $$k$$ say) which are not occupied, one player can pass at most twice in a row, and a sequence of two passes from both players is considered an illegal move (so that consideration of the game does not coninue any further). What proportion of these random games when played to at most $$n$$ moves end in an illegal move (that is one player suicides while not simultaneously capturing a group or plays in a ko after the last player played it)? Is there a certain value of $$n$$ where almost all games of size larger than $$n$$ end with illegal moves?

If this question is better suited elsewhere, feel free to suggest closing/migrating it.

Edit: The conditions about passing are in place mainly to avoid very long games where little stones actually appear on the board. The same questions could be asked without passing occurring at all, or with some other set of rules on passing if convenient. Also, superko rules are being ignored here, but the question with or without these implemented might be considered depending on which is more convenient.

• Is there a reason to not simply try to answer this question with a monte carlo simulation? – Will Sawin Jan 1 at 15:26
• @WillSawin It would be satisfactory to check it for all cases on a small board, but yes it should be possible to statistically estimate the proportion. It is known that the number of board positions which are legal is about $1.2$% of all possible board positions (of which there are on the order of $10^{170}$ such) by the computations of Tromp. It is interesting if there are any comparable computations for game sequences (the number of which is extremely large in comparison). – Josiah Park Jan 1 at 15:33