L10 is represented in ternary in the 10x10 position above, with empty representing 0, and black and white stones representing 1 and 2 respectively. The trits are in row-major order, with the most significant trit in the top-left. Of the 17 values below, only L2, L3, L7 and L10 can be represented as legal positions themselves. Here's hoping that L19 can be as well...

Click on the left links to find tables for L(m,n).

n | number of legal n*n positions |

1 | 1 |

2 | 57 |

3 | 12675 |

4 | 24318165 |

5 | 414295148741 |

6 | 62567386502084877 |

7 | 83677847847984287628595 |

8 | 990966953618170260281935463385 |

9 | 103919148791293834318983090438798793469 |

10 | 96498428501909654589630887978835098088148177857 |

11 | 793474866816582266820936671790189132321673383112185151899 |

12 | 57774258489513238998237970307483999327287210756991189655942651331169 |

13 | 37249792307686396442294904767024517674249157948208717533254799550970595875237705 |

14 | 212667732900366224249789357650440598098805861083269127196623872213228196352455447575029701325 |

15 | 10751464308361383118768413754866123809733788820327844402764601662870883601711298309339239868998337801509491 |

16 | 4813066963822755416429056022484299646486874100967249263944719599975607459850502222039591149331431805524655467453067042377 |

17 | 19079388919628199204605726181850465220151058338147922243967269231944059187214767997105992341735209230667288462179090073659712583262087437 |

18 | ~0.017322427762 * 3^324 ~ 6.697231143 * 10^152 |

19 | ~0.011957528698 * 3^361 ~ 2.081681994 * 10^170 |

The results for n=14,15,16 and 17 were obtained in a joint effort between Michal Koucký and John Tromp.

A preliminary version of our paper ``Combinatorics of Go'' that contains these and many other results.

This gzipped tar contains various programs used to compute the exact numbers. Many thanks to Gunnar Farnebäck and Michal Koucký for their contributions. Gunnar wrote a legal counting program in pike, while Michal suggested the use of Chinese Remaindering and implemented a file based program.

This small program approximates the probability of a random n*n position being legal.

Additions and corrections are welcome. tromp@cwi.nl