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Offline EdLee
 Post subject: Number of legal 18x18 Go positions
Post #1 Posted: Mon Mar 09, 2015 1:17 am 
Honinbo
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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. :mrgreen:
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Offline Sennahoj
 Post subject: Re: Number of legal 18x18 Go positions
Post #2 Posted: Mon Mar 09, 2015 1:42 am 
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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]
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Offline Sennahoj
 Post subject: Re: Number of legal 18x18 Go positions
Post #3 Posted: Mon Mar 09, 2015 1:44 am 
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oh seems EdLee edited away his question about how to get 57 for n = 2 :)
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Offline kivi
 Post subject: Re: Number of legal 18x18 Go positions
Post #4 Posted: Mon Mar 09, 2015 1:53 am 
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You forgot 1 white 1 black diagonal.
edit: Oh I see you count it. Thought you wanted to deal with symmetry as well.
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Offline EdLee
 Post subject: Re: Number of legal 18x18 Go positions
Post #5 Posted: Mon Mar 09, 2015 1:59 am 
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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 :black: and 1 :white: "

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. :) )
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Offline tentano
 Post subject: Re: Number of legal 18x18 Go positions
Post #6 Posted: Mon Mar 09, 2015 2:54 am 
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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.
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Offline Sennahoj
 Post subject: Re: Number of legal 18x18 Go positions
Post #7 Posted: Mon Mar 09, 2015 3:18 am 
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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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