Life In 19x19 :: Number of legal 18x18 Go positions

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Author:	EdLee [ Mon Mar 09, 2015 1:17 am ]
Post subject:	Number of legal 18x18 Go positions
Number of legal 18x18 Go positions So the number included all symmetrical positions. Click Here To Show Diagram Code [go]$$ $$ . . ? X . ? . X ? . . ? . . ? O . ? . O ? . . ? . . ? X X $$ . . ? . . ? . . ? . X ? X . ? . . ? . . ? . O ? O . ? . . $$ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? $$ . X ? . . ? X . ? X . ? . X ? O O ? . O ? . . ? O . ? O . $$ . X ? X X ? X . ? . X ? X . ? . . ? . O ? O O ? O . ? . O $$ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? $$ . O ? X O ? . X ? . . ? O . ? O X ? . O ? . . ? X . ? X . $$ O . ? . . ? . O ? O X ? X . ? . . ? . X ? X O ? O . ? . O $$ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? $$ . X ? O . ? . O ? X X ? . X ? X . ? X X ? O O ? . O ? O . $$ O . ? . X ? X . ? . X ? X X ? X X ? . X ? . O ? O O ? O O $$ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? $$ O O ? X X ? . X ? O . ? X O ? X X ? O X ? . O ? X . ? O O $$ O . ? . O ? O X ? X X ? X . ? O . ? . X ? X X ? X O ? . X $$ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? $$ . O ? X . ? O X ? O O ? X O ? . X ? O . ? ? ? ? ? ? ? ? ? $$ X O ? O O ? O . ? X . ? . O ? O O ? O X ? ? ? ? ? ? ? ? ?[/go] I'll leave the 12,675 cases of the 3x3 for others.

Author: Sennahoj [ Mon Mar 09, 2015 1:42 am ]

Post subject: Re: Number of legal 18x18 Go positions

Here are the possible states, it seems they add up to 57:

Click Here To Show Diagram Code: [go]$$ 1 (empty board) $$ -- $$ | . . | $$ | . . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 4 (a single black stone) $$ -- $$ | X . | $$ | . . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 4 (two adjacent black stones) $$ -- $$ | X X | $$ | . . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 2 (black diagonal) $$ -- $$ | X . | $$ | . X | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 4 (three black stones) $$ -- $$ | X X | $$ | X . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 4 (single white stone) $$ -- $$ | O . | $$ | . . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 4 (two adjacent white stones) $$ -- $$ | O O | $$ | . . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 2 (white diagonal) $$ -- $$ | O . | $$ | . O | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 4 (three white stones) $$ -- $$ | O O | $$ | O . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 12 (one black, one white) $$ -- $$ | X O | $$ | . . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 8 (two black, one white) $$ -- $$ | X X | $$ | O . | $$ --[/go]

Click Here To Show Diagram Code: [go]$$ 8 (two white, one black) $$ -- $$ | O O | $$ | X . | $$ --[/go]

Author:	Sennahoj [ Mon Mar 09, 2015 1:44 am ]
Post subject:	Re: Number of legal 18x18 Go positions
oh seems EdLee edited away his question about how to get 57 for n = 2

Author:	kivi [ Mon Mar 09, 2015 1:53 am ]
Post subject:	Re: Number of legal 18x18 Go positions
You forgot 1 white 1 black diagonal. edit: Oh I see you count it. Thought you wanted to deal with symmetry as well.

Author:	EdLee [ Mon Mar 09, 2015 1:59 am ]
Post subject:	Re: Number of legal 18x18 Go positions
Sennahoj wrote: oh seems EdLee edited away his question about how to get 57 for n = 2 Yea, instead of asking, I just brute forced it. kivi wrote: You forgot 1 white 1 black diagonal. Do you mean Sennahoj ? He counted them, under "12 for 1 and 1 " An interesting factoid is the number of unique 2x2 positions would have been the number of Sennahoj's diagrams, 12, except for the above case, so it's actually 1 more, at 13. (Because he merged the adjacent and diagonal cases into 1 diagram; but he kept them separate for 2 stones of the same color. )

Author:	tentano [ Mon Mar 09, 2015 2:54 am ]
Post subject:	Re: Number of legal 18x18 Go positions
If they could just make that diagram on page 5 of their paper for 19x19 and figure out a way for me to memorize it perfectly, that would be nice.

Author: Sennahoj [ Mon Mar 09, 2015 3:18 am ]

Post subject: Re: Number of legal 18x18 Go positions

EdLee but if we are talking about symmetries, then these should also be considered the same:

Click Here To Show Diagram Code: [go]$$ $$ -- $$ | O . - X . | $$ | . . - . . | $$ --[/go]

So there are only 8 "essentially unique" positions

Click Here To Show Diagram Code: [go]$$ $$ -- $$ | . . - X . - X X - X . - X X - X O - X . - X X| $$ | . . - . . - . . - . X - X . - . . - . O - O .| $$ --[/go]

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