Number of legal 18x18 Go positions

[EdLee]

Number of legal 18x18 Go positions

So the number included all symmetrical positions.

$$
$$ . . ? 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 ? ? ? ? ? ? ? ? ?

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[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.

[Sennahoj]

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

$$ 1 (empty board)
$$ --
$$ | . . |
$$ | . . |
$$ --

Click Here To Show Diagram Code

[go]$$ 1 (empty board)
$$ --
$$ | . . |
$$ | . . |
$$ --[/go]

$$ 4 (a single black stone)
$$ --
$$ | X . |
$$ | . . |
$$ --

Click Here To Show Diagram Code

[go]$$ 4 (a single black stone)
$$ --
$$ | X . |
$$ | . . |
$$ --[/go]

$$ 4 (two adjacent black stones)
$$ --
$$ | X X |
$$ | . . |
$$ --

Click Here To Show Diagram Code

[go]$$ 4 (two adjacent black stones)
$$ --
$$ | X X |
$$ | . . |
$$ --[/go]

$$ 2 (black diagonal)
$$ --
$$ | X . |
$$ | . X |
$$ --

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[go]$$ 2 (black diagonal)
$$ --
$$ | X . |
$$ | . X |
$$ --[/go]

$$ 4 (three black stones)
$$ --
$$ | X X |
$$ | X . |
$$ --

Click Here To Show Diagram Code

[go]$$ 4 (three black stones)
$$ --
$$ | X X |
$$ | X . |
$$ --[/go]

$$ 4 (single white stone)
$$ --
$$ | O . |
$$ | . . |
$$ --

Click Here To Show Diagram Code

[go]$$ 4 (single white stone)
$$ --
$$ | O . |
$$ | . . |
$$ --[/go]

$$ 4 (two adjacent white stones)
$$ --
$$ | O O |
$$ | . . |
$$ --

Click Here To Show Diagram Code

[go]$$ 4 (two adjacent white stones)
$$ --
$$ | O O |
$$ | . . |
$$ --[/go]

$$ 2 (white diagonal)
$$ --
$$ | O . |
$$ | . O |
$$ --

Click Here To Show Diagram Code

[go]$$ 2 (white diagonal)
$$ --
$$ | O . |
$$ | . O |
$$ --[/go]

$$ 4 (three white stones)
$$ --
$$ | O O |
$$ | O . |
$$ --

Click Here To Show Diagram Code

[go]$$ 4 (three white stones)
$$ --
$$ | O O |
$$ | O . |
$$ --[/go]

$$ 12 (one black, one white)
$$ --
$$ | X O |
$$ | . . |
$$ --

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[go]$$ 12 (one black, one white)
$$ --
$$ | X O |
$$ | . . |
$$ --[/go]

$$ 8 (two black, one white)
$$ --
$$ | X X |
$$ | O . |
$$ --

Click Here To Show Diagram Code

[go]$$ 8 (two black, one white)
$$ --
$$ | X X |
$$ | O . |
$$ --[/go]

$$ 8 (two white, one black)
$$ --
$$ | O O |
$$ | X . |
$$ --

Click Here To Show Diagram Code

[go]$$ 8 (two white, one black)
$$ --
$$ | O O |
$$ | X . |
$$ --[/go]

[Sennahoj]

oh seems EdLee edited away his question about how to get 57 for n = 2

[kivi]

You forgot 1 white 1 black diagonal.
edit: Oh I see you count it. Thought you wanted to deal with symmetry as well.

[EdLee]

oh seems EdLee edited away his question about how to get 57 for n = 2

Yea, instead of asking, I just brute forced it.

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. )

[tentano]

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.

[Sennahoj]

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

$$
$$ --
$$ | O . - X . |
$$ | . . - . . |
$$ --

Click Here To Show Diagram Code

[go]$$
$$ --
$$ | O . - X . |
$$ | . . - . . |
$$ --[/go]

So there are only 8 "essentially unique" positions

$$
$$ --
$$ | . . - X . - X X - X . - X X - X O - X . - X X|
$$ | . . - . . - . . - . X - X . - . . - . O - O .|
$$ --

Click Here To Show Diagram Code

[go]$$
$$ --
$$ | . . - X . - X X - X . - X X - X O - X . - X X|
$$ | . . - . . - . . - . X - X . - . . - . O - O .|
$$ --[/go]