GO sequence

[y.kohmoto]

    Hugo wrote :
    >As I understand it, modern Go refines the Ko rules to:
    >you may not make a move that would repeat a previous board state
    >..under which I don't think it makes sense to talk about the 'period'
    >of a game. However, I may have misunderstood your sequence

    Sorry I edited my mail, I wrote as follows in the first mail.
    >>If we play GO, four cases are possible.
    >>1. Black stone win
    >>2. White stone win
    >>3. It stopped at "Seki"
    >>4. Periodic
    >>Comment :
    >>If it becomes Seki, each players are not able to put a stone, because
if they do so then they will not win.

    >>But we should treat all cases as "periodic", because the periods of 1
and 2 are one.
    >>And why don't players continue the game at "Seki"? Even if they will
not win.     .

    >>I want calculate the longest periods on  n x n.

    In a real GO game, you are right.
    The cases 1.2.3. the game ends  and in the case 4 which contains "Ko",
the rule make it stop.
    So, there is no periodic game.
    I meant that I considered about a little abstract GO game whose rule are
only two as follows.
    o If a set of a player's stones has no "open edge" then the other player
gets the set of stones.
    o If the sets of both player's stones has no "open edge" in a
configuration, then a player who made this configuration gets the set of the
other player's stone.

    I ashume these rules and I enumerated the "longest" period of each games
on 1 x n board.
    I wrote in the last mail :
    >>     I  enumerated the period of GO game on 1 x n.

    I had lacked the word "longest period" in the last mail. It was a typo
mistake.

    You understand that the game never ends, because the state of "Win" is
not defined.
    So, all cases become periodic.

   Hugo wrote :
    >For example, it isn't clear in the above why column '1' shows only the
    >case of black making her first move in the first position: moving to
    >the centre position would also be a legitimate move.

    You mean :

    t  1  2  2
    +  +  o  +
    +  x  x  x
    +  +  +  +
    If black player put a stone on the center position, the white player
must put a his stone on first or third.
    The results are the same ,so we consider a case of the first position.
See the table.

    The white stone has no "open edge", so immediately black player gets it.
    In the same time "2",  a state becomes the same as time "1" 's state.
    So, the period is 2-1=1.
    It is not the longest period on 1 x 3 board.
    I calculate "the longest period", it is the reason why black put a stone
on the first position.

    Time table of n=4 :
    0 1 0 0 2 0 2 0 2 0 0 1 0 1 0 1
    0 0 2 2 2 0 0 1 0 1 1 1 0 0 2 0
    0 0 0 0 0 1 1 1 0 0 0 0 2 2 2 0
    0 0 0 1 1 1 1 1 0 0 2 2 2 2 2 0
    where "0,1,2" represent "+,x,o".
    the process where a player gets stones are ignored

    fifteenth column is the same as the first column
    So, the period is 15-1=14.

    S_GO :    1, 2, 6, 14, 18
    I think these number are the longest.

    If you know longer period for each n , tell me about them.

    I think the abstract GO game on 1 x n board is a cell automaton with
three states.

    Yasutoshi