Pio2001 wrote:
I have added the results for territory button go. They are the same as area button go.
Right.

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So, if I summarize button go :
It is an area + button rule set
It is a hybrid of area and territory rules. The score is typically the same as that by a form of territory scoring, but area rules allow questions of life and death and kos and superkos after the last dame has been filled to be settled by play.
The easiest way to implement button go is by adding a token worth ½ pt. to area rules. Implementing it via territory rules involves passing prisoners between the players.
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If I understand properly, we use nearly-pass stones (plus a button stone) and add them to the "apparent" territory in order to get the area, then we add the button to that area, which in turn tells us who's got the most "true" territory.
This sounds confusing to me.
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Unless a straightforward equivalence with territory or area can be clearly explained, I'm afraid that it doesn't meet my second requirement :
I am not sure what you are asking for. You have verified the equivalence of button go by area and territory scoring for typical situations. Do you want a more rigorous explanation? Or are you asking for something else?
Several years ago on the ancestor of this site I posted a fairly extensive treatment of button go, covering a variety of situations. Unfortunately, those notes got clobbered in the transition and are unrecoverable.

On the assumption that you want an explanation of why area button go and territory button go produce the same results, here is a start.
What is the basic difference between territory and area scoring? Black plays first and if Black gets the last board play at the dame stage, she will get one more point by area scoring than by territory scoring. Depending on other differences between the rules, such as counting territory in seki or not, and different ko or superko rules, there may be other differences, but this is the basic one.
Is there a way of modifying area scoring so that it does not matter who gets the last dame? Yes, there is. Each dame gains 1 point, so adding a play that is worth ½ of that, i.e., ½ pt., will do the trick. So let there be a token worth ½ pt. that a player can take. For instance, suppose that Black is to play when all that is left are dame. Then Black will gain ½ pt., no matter who gets the last dame. If White gets the last dame, so that each play has gotten the same number, Black will take the token (button) and gain ½ pt. If Black gets the last dame, and has gained 1 pt. against White, White takes the button, reducing Black's gain to ½ pt. So the area result will be the territory score + ½ pt. for Black.
What if we use territory scoring? AGA rules do that using pass stones. In that case the button is worth -½ pt., since a play by territory scoring gains 1 pt. less than by area scoring. Can we simplify that method? Yes, we can.

Herman Hiddema came up with the idea of adding a second button. At first, that sounds like an additional complication, but in the end it allows us to simplify.
At the end of the game by AGA territory scoring, the players pass and each pass costs 1 pt. The current AGA rules require White to make the last pass. Is there a way to modify the rules so that it does not matter who makes the last pass? That would allow us to do away with the last pass by White rule, which is still puzzling to some players, and even objectionable to some, and is a complication. Yes, there is. We add a last play at the end that loses only ½ pt. instead of 1 pt. Then it does not matter who makes the last pass. The reasoning is the same as having the ½ pt. button means that it does not matter who gets the last dame.
Here comes the simplification. If the player who took the button also gets the last pass, the result for that person is -½ - ½ = -1, i.e., the same as handing over 1 prisoner. So let that person hand over a prisoner when she takes the button, and if she also makes the last pass, let that pass be for free. If the person who took the button is different from the one who gets the last pass, the result for that person is -½ + ½ = 0, the same as if she hands over a prisoner when she takes the button, and her opponent hands over a pass stone on the last pass. Voila!

Edit: OC, the territory rules produce an integer result, so we have to us komi worth ½ pt. less to make the two results the same.
