Let's take the first following example

Click Here To Show Diagram Code

[go]$$W Black to kill.
$$ ------------------
$$ | . O . . . . X . .
$$ | a . b c O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Though I undestand the principle of the 3-move tsumego rule, I do not see clearly a real advantage at least for such position.

As far as I am concerned I first tried a black move at "a" but it doesn't work.
After that I envisaged the other moves having in mind the answer at "a" by white (in that sense it looks like the 3-move rule approach). But when I envisaged a black move at "b" or "c" my common sense tells me to consider they are miai and I didn't envisage at all an answer at "a".
BTW by handling the points "b" and "c" as miai I was unable to see how black can kill white stones.

Maybe I need some help here.

Click Here To Show Diagram Code

[go]$$W White is dead
$$ ------------------
$$ | . O . . . . X . .
$$ | W . B B O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

It is plain that if White plays at , the Black stones are necessary to kill. That means that if Black to play can kill, she must play on one of these points. Instead of 8 or 9 possible first moves we only have to look at 3. That's progress.

I guess this is what you mean by considering b and c as miai.

Click Here To Show Diagram Code

[go]$$B White lives
$$ ------------------
$$ | . O . . . . X . .
$$ | . . 2 1 O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$B White lives this way, too.
$$ ------------------
$$ | . O . . . . X . .
$$ | . . 1 2 O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

So if our reasoning is correct:

Click Here To Show Diagram Code

[go]$$B Only play to kill
$$ ------------------
$$ | . O . . . . X . .
$$ | 1 . . . O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

I guess you already got this far?

Let's turn the tables. If White can live from this position, can we apply the same reasoning?

Click Here To Show Diagram Code

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

If White can live, and Black plays at , the stones are necessary to make life. So one of these three points is the best play for to attempt to live.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Thank you for your help Bill. I am not quite sure but I believe I can now take the point.

Please tell me if am right in the way I would have solved this problem

Click Here To Show Diagram Code

[go]$$W Black to kill.
$$ ------------------
$$ | . O . . . . X . .
$$ | a . b c O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

1) First of all I try to kill white by playing at "a" which looks like the vital point.
2) Being not strong enough I fail to find how to kill white (but I note that some variants look quite difficult to analyse)
3) Because I was convinced the point "a" was the vital point I assume now that white will play at "a" and I look for two black moves in a row which will enable to kill white, and, good news, I find only(!) "b" and "c"
4) I try to kill white by beginning with "b" or "c" but, this time, it appears quite easy for white to live

Here is the point: I am not in a game but I try to solve a tsumego problem and I know for sure that I can kill white.
With this information and the 3-move rule I now know for sure that a black move at "a" will kill white, though I do not know yet how it is possible!!

5) I definitely choose black "a" and repeat the analysis from this new position

Well, I am no expert in tsumego, or even in this method. John Fairbairn reminded me of it, and the reason for so many posts on it is that I am trying to understand it better.



A common experience. I can relate.



Aside comment: It seems to me that we can tell that a is a vital point because we consciously or unconsciously see that White to play obviously makes life by playing there. Similarly, once White has a stone there, we see that b makes easy life and Black must prevent it. It is also obvious that Black b and c kill. Seeing that c is necessary is also something that we can do without much in the way of conscious processing.



Amazing, isn't it?



Having failed to kill with Black a before, it's good to know that you were on the right track. But how to proceed?

My first impulse for is to cut off Black a to enlarge the White eyespace and to guarantee an eye in the corner. That made me think, what if Black connected Black a? That wouldn't leave much room for White to live, so any two stones that did so would almost surely be necessary. Also, if the connection killed White, even allowing White two moves, would surely have to cut Black a off.

As Gödel said, logic is powerful.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Click Here To Show Diagram Code

[go]$$B Black to kill.
$$ ------------------
$$ | . O . . . . X . .
$$ | a . b c O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

3) Because I was convinced the point "a" was the vital point I assume now that white will play at "a" and I look for two black moves in a row which will enable to kill white, and, good news, I find only(!) "b" and "c"

When I wrote the point 3) above my intention was clearly to try and understand the proposed 3 move rule.

The point now is to see if it is really useful (at least for my level of play).

Let's try to compare with my "standard" method for solving such problem

1) First of all I try to kill white by playing at "a" which looks like the vital point.
2) Being not strong enough I fail to find how to kill white (but I note that some variants look quite difficult to analyse)

and now is new point 3:

3) I try to kill by playing another black move

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | j O h g c b X . .
$$ | a i e f O O X . .
$$ | d O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

I know the best procedure is to try the moves in the correct order, i.e. try first the moves which you feel are the best ones. It clearly depends on your level and your experience but as far as I am concerned I can easily choose the order b, c, d, e f, g, h, i , j

For most of the black moves, I choose as white answer a move at "a" because it looks like a good move for living but when I try a black move at "e" the answer white at "a" does not come to my mind and my common sense takes me to try first the answer white at "i" because I assure an eye in the corner while I restrict possibilities for black to use black stone at e.
BTW I am quite happy when seeing the following sequence with the corresponding ishi-no-shita

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | . O . . 4 . X . .
$$ | . 2 1 3 O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

As a conclusion black move at "e" cannot kill white and it is the same for the other black moves.

At this point I excluded all black moves except black at "a" for which I noted some difficulties with some variants. As a consequence here again I know for sure black move at "a" will kill white, though I do not yet know how it is possible.

My feeling is that this "standard" method imply much less analyses because I avoid analysing many positions resulting from a black stone at "e" or "f" and a white stone at "a".

Surely this 3 move rule method is very interesting for a theoritical point of view but in practice, at least for my level of play, I am not quite sure it could really help.

Because of the necessity requirement for the two Black stones, the method actually amounts to a search based, not on , but on . The two Black stones are necessary given at a only if at a refutes all other choices for .



You conclusion is right, but your premise is wrong. This is not to detract from seeing the ishi no shita play, but

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | . O 9 8 4 7 X . .
$$ | 6 2 1 3 O O X . .
$$ | 5 O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

is necessary to make the eye in the corner. But then is atari.



I don't know if there is a standard method, but one good search method is best first search. That is, you evaluate candidate moves using heuristics and pick the best candidate to explore.

What you did was find a good move for White and make a kind of best first search based upon instead of . When you got to e for you realized that a was not a good reply and tried something else. But OC, using the 3 move method would have led you to try e for .

The 3 move method includes what amounts to a 3 ply search based upon move 2. You need to have a good choice for move 2 for that to be effective. When you have that and make that move first, and then look for two moves in a row to refute it, and it is not hard to find them and to see that they are necessary, then I think this method pays off.

Add in your idea of trying each of the 2 moves by the first player to reply to the other one, and I think the payoff may well be even better.

As I said, I am learning about this method, myself.

When you have a good idea for your opponent's move, the proverb says that it's probably a good move for you, as well. That's probably the first thing to try. Then the 3 move method looks promising, as well.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Bill,

First of all, thank you for having corrected my mistake concerning the ishi-no-shita which does not exist. I already noted I have a lot of difficulties to see dame-zumari. I have to work hard to understand why I have such big weakness but it seems not so easy. For the time being I hope I am not the only man with such weakness!!!

Comming back to the theory:

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | . O . . . . X . .
$$ | 2 . b c O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Let's assume white and let's look at two black moves in a row and

When you say that b and c are obviously the only two moves allowing to kill white that means two things
1) Firstly you calculate (quite quickly?) than any other black move allows white to live
2) Secondly you calculate (here again quite quickly) the moves "b" and "c" allow black to kill white
and you conclude that, if black wants to kill white she must play a, b, or c

My point is the following : with only the first point above I am already able to say that only moves on a, b, c could kill white.

Assuming you put "a" aside because you fail to give an immediat sure result, you have in any case to look for a black move on "b" or "c" and you will discover they do not kill white (with white answering accordingly but not at )

When comparing the two move rule and my best first approach it appears that I avoid the point 2) above, without adding any other analysis.
For a more difficult problem it could be a great advantage couldn't it?

As for having a problem with damezumari, join the club. It seems to me that damezumari is involved in quite a lot of tsumego, and moreso as the problems get harder.

Another thing, though. Finding was quite good. But if you had looked at the position after as a problem of Black to play and kill, I think you would have seen how to do so immediately.



Calculation of variations is not necessarily involved, or may be so in a minor way. For instance,

Click Here To Show Diagram Code

[go]$$B White is dead
$$ ------------------
$$ | C O . . . . X . .
$$ | O . B B W W X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

White has only one potential eye at . The stones are goners, with only 2 dame to 3 dame for the stones.

Two pieces of advice from E. A. Znosko-Borovsky in How Not to Play Chess:



White wants to live. So, without doing any tree search, consider this position.

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | C O . . . . X . .
$$ | W C W . O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

With the placement of the stones White has two one point eyes at the points. Is it possible to place three Black stones in this corner so that White is dead? Hardly any reading, if at all, is required to say no.

In that case, if White has a stone on one of the points, and Black kills, Black must have a stone on the other one. Like so.

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | C O . . . . X . .
$$ | W . B a O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Black wants to kill, and we can see, if we have not done so already, that a Black stone on a will do so. Since White wants to live, let's put a White stone there.

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | C O . . . . X . .
$$ | W . B W O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Here it is obvious that White is alive because White has a second eye by capturing the stone. However, White has played one more stone than Black, and it is Black's move. Is it possible to place a Black stone in the corner so that Black to play can save the stone? Obviously not, with little or no reading.

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | C O . . . . X . .
$$ | 2 . 3 4 O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

The stones are numbered for convenience. If White plays , a Black stone on is necessary. Then if White plays , there is no place for that kills. Therefore, in order to kill Black must play at 2, 3, or 4. We can reach this conclusion by considering what Black and White want to do, with little or no calculation.

As for the utility of the 3 move rule for solving problems or finding plays over the board, the proof of the pudding is in the eating.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

This time I like very much your reasoning for proving the killing black move must be play at 2, 3, or 4.
But I note your approach is a little different from the 3 move rule approach for two reasons:
1) you identified two vital point
2) you used two null moves in a row

Basically I like very much the idea of one or several null move but I do not like very the three move rule as it stands because it looks too restrictive.

Let's take this time a far easier position

Click Here To Show Diagram Code

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

with a pure three move rule approach I imagine the following reasoning:

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | O . O . a . X X .
$$ | O O O O b O . X .
$$ | O O X . c . . X .
$$ | X X X X X X X X .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

if black let white play at "a", then black must play at "b" and "c" => the killing move msut be a, b or c

Taking you idea to use two vital points which assure easily two eyes the result is far more interesting.
In fact when you found previously the white vital point "a" you had also in mind to follow by a white play at "b" building two sure eyes.
Why not using such information (the second vital) point as you did but as three move rule didn't ?

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | O . O . a . X X .
$$ | O O O O b O . X .
$$ | O O X . . . . X .
$$ | X X X X X X X X .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

You see clearly that if white can play "a" and "b" then white gets two sure eyes.
Now, noting black cannot avoid a white play "a" or "b" you can take separetly the two possibilities

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | O . O . 2 . X X .
$$ | O O O O b O . X .
$$ | O O X . c . . X .
$$ | X X X X X X X X .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Here black has to play her two first move and at "b" and "c"

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | O . O . a d X X .
$$ | O O O O 2 O . X .
$$ | O O X . . . . X .
$$ | X X X X X X X X .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Here on the other hand black has to play her two first moves and at "a" and "d"


As a consequence, in order to kill white black must play her move at "a" or "b".

Very good.

Here is an example where the double application of the 3 move rule finds the unique first move. https://lifein19x19.com/viewtopic.php?p=236420#p236420

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Well what about trying go a little farther

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | O e Q . a . X X .
$$ | O f Q Q b O . X .
$$ | O O X . . . . X .
$$ | X X X X X X X X .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

When you try at "a" or "b", because of the possibility for black to play and at "c" and "d" you cannot apply the three move rule.

Let me try another reasonning to try and prove that the only moves that can kill white are at a, b or e.

1) As black I consider point "e" and "f" equivalent and I can always suppose black will choose a play at "e" rather than a play at "f"
2) As white I consider points e" and "f" as miai and I can decide to always answering a black play at "e" by a white play at "f". You can see that, by imposing for white to answer a black move at "e" by a white move at "f", I am really restricting the white possilities. The point is the following : if under such restriction I prove that black must play at a, b, or e for killing white, this is still true without restricting white plays!
3) Of course I cannot exclude a black move at "c" because after at "c", at "b" the triangle white stones lose one liberty and I know for sure that such damezumari may have great importance!
4) Because of the three first points the position above is equivalent to the following

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | O . Q . a . X X .
$$ | O O Q Q b O . X .
$$ | O O X . . . . X .
$$ | X X X X X X X X .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

just keeping in mind that black has a local hidden ko threat ( at e, at f)


You recognize the position I studied earlier. Because no ko threat were necessary in this analyse the conclusion is still valid (only black moves at a or b can kill white).

Finally I know for sure that, in order to kill white, black has only the possibilties a, b and e.

Do you agree with such reasonning?

Indeed you are. In terms of eye-space values (See http://www.msri.org/publications/books/ ... andman.pdf ) defined by Howard Landman, the eye in the corner has an average value of 1 eye for White, with a sente by Black. Normally we simply assume that it is an eye for White and ignore it otherwise. A Black move at e or f creates a half eye for White, thus gaining ½ eye, on average. A White reply would also gain ½ eye. However, if White has a alternative that gains more than ½ eye, White will normally take it instead of answering Black e, In such cases Black will typically also have a larger alternative and Black e will be a mistake. While it is true that sometimes, as here, Black e will kill, ignoring that fact will simplify our task. In terms of eyes, all that Black e gains with correct play is that it prevents White from playing at e or f as a reverse sente to leave 1 eye.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I write ΔT

and call it the alternating sum.



I write -ΔT



Please disambiguate the order of priorities of the conjunctions, thanks!

To all: Happy New Year!

It's quite likely to be better than last year, isn't it?

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Cher Robert,

Not sure what you mean. These are logical expressions. {shrug}

Also, a link to the quoted note would help. Thanks.

I'm rather busy now. Don't know when I can get back to you on this.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

The citations are from this thread, May 05 or 06 in 2019.

"3) 5 gote
Play in r if (s > a) or ((s > a - d + e) and (t > a + e)) or ((s > a - b + c - d + e) and (t > a + c - d + e))."

Which is it?

[(s > a) or ((s > a - d + e) and (t > a + e))] or ((s > a - b + c - d + e)] and (t > a + c - d + e))

(s > a) or [((s > a - d + e) and (t > a + e)) or ((s > a - b + c - d + e) and (t > a + c - d + e))]

"Now we can predict the conditions for 7 gote:
Play in r if (s > a) or ((s > a - f + g) and (t > a + g)) or ((s > a - d + e - f + g) and (t > a + e - f + g)) or ((s > a - b + c - d + e - f + g) and (t > a + c - d + e - f + g))."

Which is it?

[(s > a) or ((s > a - f + g) and (t > a + g))] or ((s > a - d + e - f + g) and (t > a + e - f + g)) or ((s > a - b + c - d + e - f + g) and (t > a + c - d + e - f + g))

[(s > a) or ((s > a - f + g) and (t > a + g)) or ((s > a - d + e - f + g) and (t > a + e - f + g))] or ((s > a - b + c - d + e - f + g) and (t > a + c - d + e - f + g))

(s > a) or [((s > a - f + g) and (t > a + g)) or ((s > a - d + e - f + g) and (t > a + e - f + g))] or ((s > a - b + c - d + e - f + g) and (t > a + c - d + e - f + g))

(s > a) or [((s > a - f + g) and (t > a + g)) or [((s > a - d + e - f + g) and (t > a + e - f + g)) or ((s > a - b + c - d + e - f + g) and (t > a + c - d + e - f + g))]]

"We can predict the conditions for 6 gote.
Play in r if ((s > a - f) and (t > a)) or ((s > a - d + e - f) and (t > a + e - f)) or ((s > a - b + c - d + e - f) and (t > a + c - d + e - f))."

Which is it?

[((s > a - f) and (t > a)) or ((s > a - d + e - f) and (t > a + e - f))] or ((s > a - b + c - d + e - f) and (t > a + c - d + e - f))

((s > a - f) and (t > a)) or [((s > a - d + e - f) and (t > a + e - f)) or ((s > a - b + c - d + e - f) and (t > a + c - d + e - f))]

Logical 'or' is associative, so those are all equivalent (where there is no mistake in brackets).

_________________
A good system naturally covers all corner cases without further effort.

Thanks. Let me format it differently.



Claro?

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Thank you both!

I still do not understand difference games comparing two positions P and -Q.

Click Here To Show Diagram Code

[go]$$B Example 1: P + (-Q)
$$ ---------------------------------
$$ | . . . . . . O . X . . X O . . |
$$ | X X X X O O O . X X X O O O O |
$$ | . . . . . . . . . . . . . . . |[/go]

Click Here To Show Diagram Code

[go]$$B Black starts I, score 0
$$ ---------------------------------
$$ | . . . . 1 . O . X . 2 X O . . |
$$ | X X X X O O O . X X X O O O O |
$$ | . . . . . . . . . . . . . . . |[/go]

Click Here To Show Diagram Code

[go]$$B Black starts II, score 0
$$ ---------------------------------
$$ | . . 3 2 4 . O . X . 1 X O . . |
$$ | X X X X O O O . X X X O O O O |
$$ | . . . . . . . . . . . . . . . |[/go]

Click Here To Show Diagram Code

[go]$$W White starts I, score 0
$$ ---------------------------------
$$ | . . 2 1 . . O . X . . X O . . |
$$ | X X X X O O O . X X X O O O O |
$$ | . . . . . . . . . . . . . . . |[/go]

Click Here To Show Diagram Code

[go]$$W White starts II, mistake 2, score -1
$$ ---------------------------------
$$ | . 4 3 O 5 . O . X . 2 X O . . |
$$ | X X X X O O O . X X X O O O O |
$$ | . . . . . . . . . . . . . . . |[/go]

Click Here To Show Diagram Code

[go]$$W White starts III, score 0
$$ ---------------------------------
$$ | . . . . 2 . O . X . 1 X O . . |
$$ | X X X X O O O . X X X O O O O |
$$ | . . . . . . . . . . . . . . . |[/go]

Black starts and achieves the score 0.
White starts and achieves the score 0.

What does this mean for the difference game P - Q >= 0?

Is P - Q >= 0 characterised by "Black starts and wins or ties by achieving at least the score 0 AND White starts and does not win by achieving at least the score 0" or by what else?

What does this mean for the difference game P - Q > 0?

Is P - Q > 0 characterised by "Black starts and wins by achieving a score larger than 0 AND White starts and does not win by achieving at least the score 0" or by what else?

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[go]$$B Example 2: P + (-Q)
$$ ---------------------------------
$$ | . . . . . . X . O . . O X . . |
$$ | O O O O X X X . O O O X X X X |
$$ | . . . . . . . . . . . . . . . |[/go]

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[go]$$B Black starts I, score 0
$$ ---------------------------------
$$ | . . 2 1 . . X . O . . O X . . |
$$ | O O O O X X X . O O O X X X X |
$$ | . . . . . . . . . . . . . . . |[/go]

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[go]$$B Black starts II, mistake 2, score 1
$$ ---------------------------------
$$ | . 4 3 X 5 . X . O . 2 O X . . |
$$ | O O O O X X X . O O O X X X X |
$$ | . . . . . . . . . . . . . . . |[/go]

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[go]$$B Black starts III, score 0
$$ ---------------------------------
$$ | . . . . 2 . X . O . 1 O X . . |
$$ | O O O O X X X . O O O X X X X |
$$ | . . . . . . . . . . . . . . . |[/go]

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[go]$$W White starts I, score 0
$$ ---------------------------------
$$ | . . . . 1 . X . O . 2 O X . . |
$$ | O O O O X X X . O O O X X X X |
$$ | . . . . . . . . . . . . . . . |[/go]

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[go]$$W White starts II, score 0
$$ ---------------------------------
$$ | . . 3 2 4 . X . O . 1 O X . . |
$$ | O O O O X X X . O O O X X X X |
$$ | . . . . . . . . . . . . . . . |[/go]

Black starts and achieves the score 0.
White starts and achieves the score 0.

What does this mean for the difference game P - Q <= 0?

Is P - Q <= 0 characterised by "Black starts and does not win by achieving at most the score 0 AND White starts and wins or ties by achieving at most the score 0" or by what else?

What does this mean for the difference game P - Q < 0?

Is P - Q < 0 characterised by "Black starts and does not win by achieving at most the score 0 AND White starts and wins by achieving a score smaller than 0" or by what else?