Life In 19x19

Posted: **Wed Oct 12, 2016 7:17 am**

Bill,

in simple examples, such as mine, we must also be able to distinguish local gote from sente. In fact, in a different thread, you showed a similar example with White's local sente and four stones to be connected and we used the example to distinguish local gote from sente by tentative 'gote' counts and move values and conditions 2.

In such simple examples, G is the intermediate follower in the sequence from the initial position to the tentative 'sente' follower S. As a subproblem, we can look at G as a starting position, and G is a local gote endgame with the count G = (GB + GW) / 2.

We can apply conditions 1 or 2 to such a simple example to distinguish local gote from sente. The only trick here is to have a two-phase analysis: I) speak of tentative local gote or sente, II) determine the real local gote or sente status.

***

Now, IIUYC, you imagine different kinds of examples, in which a player, say Black, can start one gote sequence to the gote follower G or can start a different, sente sequence via some other intermediate position, say Q, to the sente follower S.

In such a different kind of example, I suppose we can use conditions 2. Again, we have G = (GB + GW) / 2. We do, however, not have S = GW. Therefore, for such a different kind of example, we may not apply conditions 1. Currently, I do not know yet if conditions 2 help us. I think we need more: MQ, which shall be the move value at the intermediate position Q, and is, IIUYC, the value of the threat, right?

Can you please show some such example and the necessary calculations to distinguish local gote from sente incl. the calculations involving the value of threat?

***

Then, for positions without kos, which method is generally applicable to all kinds of positions to distinguish local gote from sente?

***

My apology for apparently misrepresenting you. Has it been about you presuming the different kind of examples?

***

I am still at a loss trying to understand a) how certain conditions depend upon the threat carried by the sente and b) how your conditions "a local gote if S < (G + R)/2, Black's local sente if S > (G + R)/2" might be involved or derived.

***

EDIT: replace 'P' by 'Q' to fit your added tree.

Posted: **Wed Oct 12, 2016 10:15 am**

RobertJasiek wrote:Now, IIUYC, you imagine different kinds of examples, in which a player, say Black, can start one gote sequence to the gote follower G or can start a different, sente sequence via some other intermediate position, say Q, to the sente follower S.

Right. One player has two live options, one gote and one sente, while the opponent has only one, which is gote.

Can you please show some such example and the necessary calculations to distinguish local gote from sente incl. the calculations involving the value of threat?

I am pressed for time, these days, so I'll beg off on locating or constructing such a position. But they do show up in real games.

My apology for apparently misrepresenting you.

My apology for not being clear.

I will discuss this kind of position. Remind me in a few days if I have not.

Posted: **Wed Oct 12, 2016 12:01 pm**

Maybe tomorrow I will start discussion of your (so far) abstract example (a monkey with gote or sente options is not clean enough, I suppose) and see how far I come. Then, in a couple of days, you might fill the rest or correct my mistakes:)

Posted: **Thu Oct 13, 2016 12:16 am**

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Black has two options in P, one to G in gote and one to S in sente (through Q). White has one option, to R. BIG is big enough to make the play to Q sente. G > S > R (for Black).

Since G > S, if this is the last play on the board, Black to play should play to G. But is P characteristically sente or gote? We can answer that question by the method of multiples. (See http://senseis.xmp.net/?MethodOfMultiples )

Let there be 3 copies of P, with Black to play. Should Black play it as gote or sente? Except the last time, OC.

1) Gote. Black plays to G, then White plays to R, then Black plays to G. Result: 2G + R.

2) Sente. Black plays to Q, then White replies to S; Black plays to S with sente again; then Black plays to G. Result: 2S + G.

Black should play P as gote if 2G + R > 2S + G ; i.e., if (G + R)/2 > S . If S > (G + R)/2 then Black should play P as sente, except for the last one.

Let there be 5 copies of P, with Black to play. What are the results if Black plays it as gote versus the results if Black plays it as sente?

1) Gote. Result: 3G + 2R

2) Sente. Result: 4S + G

Black should play P as gote if 3G + 2R > 4S + G ; i.e., if (G + R)/2 > S . If S > (G + R)/2 then Black should play P as sente, except for the last time.

We get the same comparison with 7 copies, with 9 copies, etc.

So we classify P as gote when (G + R)/2 > S and as sente when S > (G + R)/2; when (G + R)/2 = S, P is ambiguous. (See http://senseis.xmp.net/?Ambiguous )

To put it another way, when the mean gote value of P is greater than the mean sente value of P, P is gote, and vice versa. When the two values are equal, P is ambiguous.

OC, even if we classify P as sente, there are times when it is correct to play it as gote. More on that later.

Posted: **Thu Oct 13, 2016 4:24 am**

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We view values from Black's perspective, know / presume that moving to G is a gote sequence and moving to S is a sente sequence and presume G > S > R.

We presume G, S > R so that Black has an interest in preventing White from starting a sequence to R at all. We presume G > S so that Black has an interest in considering choosing the option G at all and, if P is the last local endgame on the board, Black having the turn should start the gote sequence to G.

In the initial position P, Black can start

- the gote sequence with the move value (G - R)/2 or
- the sente sequence with the reverse-sente-preventing move value S - R.

Unless the move values are equal and P is ambiguous, Black chooses the option having the larger move value so he starts

- the gote sequence if (G - R)/2 > S - R <=> (G + R)/2 > S or
- the sente sequence if (G - R)/2 < S - R <=> (G + R)/2 < S.

This principle choice of Black is, IMO, good enough for a proof of conditions characterising P as a locale gote or local sente.

As it turns out, the conditions have inverted unequality signs when compared to my other type of positions more or less coincidentally because G and Q have different meanings. However, the formula structure is the same, which gives rise to hope that there might be a universal method applicable to several types of initial positions.

The environment affects whether a local gote or sente is a global gote or sente. If the enviroment is empty, Black plays P as a global gote. In an "ordinary" environmment (How exactly must it be?), Black plays P as a global gote / sente if it is a local gote / sente, respectively. If P occurs an odd number of N times and is a local sente, the first N-1 times P is played as Black's sente but the last time it might have to be played as a global gote, in such a case called tedomari. The proof above merely characterises P as a locale gote or sente but says nothing yet about correct global play considering the environment and a possible tedomari.

Bill conveys a sketch of an alternative proof using the method of multiples. It seems that a full proof would rely on correct play for the limit of N->oo copies of P. By dividing the count of the ensemble by N, we get the expected average count in P. Iterating N by 1, 2 or a different number on all integers, only even or only odd integers does not affect the construction of such a proof; for a given P, we would choose the most fitting iteration step and subset of all integers. Nevertheless, I remain to be convinced why any proof using the method of multiples is well-defined.

The proof using the method of multiples suggests that the last copy of P should be played as a gote. In reality, we have only one copy of P. We do not want to say that it must always be played as a gote because of being the last copy of P. In fact, we prefer to use the conditions for when, usually, it should be played as a sente. In other words, the proof using the method of multiples also presumes that the environment affects whether a local gote or sente is a global gote or sente.

Therefore, I do not find the proof using the method of multiples any more convincing that the simple proof for one copy of P comparing the move values for the gote sequence versus sente sequence. The method of multiples is an overkill for distinguishing local gote from local sente.

We do not need to calculate the value of the threat, which I think would be the move value of Black's move from Q to BIG, right? All we need initially is a calculation that BIG is indeed big (how big must BIG be?) and of the counts of the leaves shown.

Still missing are examples of actual positions for Bill's type of positions. Anyone? Examples without follow-ups at the leaves are preferred.

We know how to calculate counts if G, BIG, S or R are terminal or have follow-ups. Meanwhile we have methods for distinguishing local gote from local sente in these types of positions and can imply their symmetric cases with White having the options G and S:

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Which other types involving tentative sente options can exist? If a player has several gote options, we simply choose the dominating option. Same for several sente options of his. Hence, for Black's part of the trees, there are only the two types. White uses a gote sequence to reach R. For each of the two trees shown, there can be two additional types: a) White has White's sente option W only, b) White has White's sente option W and a reverse sente option R. Am I right about the types and the completeness of their classication? Which conditions arise for these other types and how do we derive them?

When we will have identified and calculated all possible types including tentative sente options, let me ask again: For arbitrary such positions P without ko, how to spell out a general method of distinguishing local gote, local sente and the ambiguous case?

How much in this thread is current research and what did already exist, possibly in more general contexts, among CGT researchers or by you, Bill?

Posted: **Thu Oct 13, 2016 8:35 am**

RobertJasiek wrote: Bill conveys a sketch of an alternative proof using the method of multiples. It seems that a full proof would rely on correct play for the limit of N->oo copies of P. By dividing the count of the ensemble by N, we get the expected average count in P. Iterating N by 1, 2 or a different number on all integers, only even or only odd integers does not affect the construction of such a proof; for a given P, we would choose the most fitting iteration step and subset of all integers. Nevertheless, I remain to be convinced why any proof using the method of multiples is well-defined.

The method of multiples has been used for many years, maybe more than a century, to find the mean value of games. IIUC it was used to define the mean value of games. (Independent go positions are games.) Here I used it to indicate that Black will choose the sente option under certain conditions, the gote option under others. That is enough to tell us how to characterize the original position. But without a formal definition of sente and gote, it is not a formal proof. (I have developed formal definitions, which is why I came up with the classification of ambiguous.) Also, the full derivation of the mean value by the method of multiples would mean showing that the mean value converges as the number of multiples goes to infinity and showing that it is the same when White plays first.

The method of multiples is an overkill for distinguishing local gote from local sente.

Thermography is a simpler technique, but I think that the method of multiples is easier to comprehend.

When we will have identified and calculated all possible types including tentative sente options, let me ask again: For arbitrary such positions P without ko, how to spell out a general method of distinguishing local gote, local sente and the ambiguous case?

The method of multiples works and is perfectly general. As I indicated, you can also use thermography, but you have to use colored masts to distinguish ambiguous positions. (I came up with the idea of colored masts circa 2000.)

How much in this thread is current research and what did already exist, possibly in more general contexts, among CGT researchers or by you, Bill?

Except for not having the concept of ambiguous positions, top go players 200 years ago had ways of distinguishing between sente and gote in this type of play.

Thermography was invented some 40 years ago. Around that time I had my own methods to reach the same conclusions (except for not yet having the ambiguous classification).

Posted: **Thu Oct 13, 2016 9:05 am**

Bill Spight wrote:Here I used it to indicate that Black will choose the sente option under certain conditions, the gote option under others. That is enough to tell us how to characterize the original position.

Is it? We characterise a local (non-ambiguous) position as either local gote or local sente so that then we can calculate the correct count and move value. I am sure that you would also do the latter but apparently you use a different procedural step when and how to calculate them, do you?

Thermography is a simpler technique, but I think that the method of multiples is easier to comprehend.

I think that using neither but directly distinguishing local gote from local sente is the simplest.

top go players 200 years ago had ways of distinguishing between sente and gote in this type of play. :)

Any explicit ways?

Thermography was invented some 40 years ago.

Thermography is a methodology with a much broader scope of application (not that I understand much of it). I have wondered about specific methods for distinguishing local gote from local sente, before we even consider environments.

Posted: **Thu Oct 13, 2016 1:20 pm**

RobertJasiek wrote:
Bill Spight wrote:Here I used it to indicate that Black will choose the sente option under certain conditions, the gote option under others. That is enough to tell us how to characterize the original position.
Is it? We characterise a local (non-ambiguous) position as either local gote or local sente so that then we can calculate the correct count and move value. I am sure that you would also do the latter but apparently you use a different procedural step when and how to calculate them, do you?

When I was first learning thermography I figured out the correct plays and then drew the thermograph. Berlekamp said that that was backwards. Thermographs tell us what typically correct plays are. Go players often start by characterizing plays as sente and gote and then figuring out the count and move value. If you start by figuring out the count and move value, which is what the method of multiples does, you can avoid some mistakes and confusion.

Thermography is a simpler technique, but I think that the method of multiples is easier to comprehend.
I think that using neither but directly distinguishing local gote from local sente is the simplest.

But somewhat error prone.

top go players 200 years ago had ways of distinguishing between sente and gote in this type of play.
Any explicit ways?

Seat of the pants, as far as I can tell.

But usually right.

Thermography was invented some 40 years ago.
Thermography is a methodology with a much broader scope of application (not that I understand much of it). I have wondered about specific methods for distinguishing local gote from local sente, before we even consider environments.

The method of multiples does that, unless you think of the copies as an environment.

Posted: **Thu Oct 13, 2016 1:21 pm**

P versus other plays

First let us simplify things a bit. Let Black have a move to a position worth G, a positive number, with gote, but let White have a move to a position worth -R, a negative number, and let Black also have a sente move to a position worth 0. Now we know that if G > R, the position is gote, if R > G, it is sente, and if G = R, it is ambiguous.

Next, let us add a play, a simple gote where Black can move to a position worth T and White can move to a position worth -T. (All variables are greater than 0.) T is smaller than BIG, so Black still has a sente play.

Black to play.

1) Black plays sente, then plays to T. Result: T.

2) Black plays to G, then White plays to -T. Result: G - T.

3) Black plays to T, then White plays to -R. Result: T - R.

T > T - R, so option 3) is out. Comparing 1) to 2) we find that if G > 2T then Black should play to G (gote) and if 2T > G Black should play sente first.

(This was my approach in the 1970s.

)

Now let us add another simple gote, where Black can move to T1 and White can move to -T1, with T >= T1 > 0.

1) Black plays sente, then plays to T, then White plays to -T1. Result: T - T1.

2) Black plays to G, then White plays to -T and Black plays to T1. Result: G - T + T1.

3) Black plays to T, then White plays to -R, then Black plays to T1. Result: -R + T + T1.

4) Black plays to T, then White plays to -T1, then Black plays sente. Result: T - T1, the same result as 1).

5) Black plays to T, then White plays to -T1, then Black plays to G. Result: G + T - T1, which is at least as good as the result for 2).

4) and 5) show that if a White reply to -T1 is correct, a Black play to T dominates the other plays. However, if 3) is inferior to 1) or 2), then Black should not play to T. So we can ignore White’s response to -T1.

Comparing 2) to 3), we find that 2) is better if G + R > 2T. I. e., if playing P as a gote is better that playing T. T1 does not enter the picture.

Comparing 1) to 2), we find that playing to G is better than the sente if G > 2(T - T1).

Comparing 1) to 3), we find that playing sente is better than playing to T if R > 2T1. T does not enter the picture.

Now let us add other simple gote similar to T and T1 to make an environment, such that T >= T1>= T2 >= . . . > 0

We still have three comparisons.

1) Black plays sente. Result: T - T1 + T2 - . . . .

2) Black plays to G. Result: G - T + T1 - T2 + . . . .

3) Black plays to T, then White plays to -R. Result: -R + T + T1 - T2 - . . . .

Comparing 2) to 3) we get that Black should not play to T if G + R > 2T. Same as before.

If 2T > G + R we compare 1) to 3) and get that Black should play sente if R > 2(T1 - T2 + . . .).

If G + R > 2T we compare 1) to 2) and get that Black should play gote if G > 2T - 2(T1 - T2 + . . .). I have separated T from T1, T2, etc., because T does not appear in the comparison between 1) and 3). It really should not be considered part of the environment.

We may estimate T1 - T2 + . . . as T1/2. The estimate does not affect the comparison between 2) and 3), but it does affect the others. Using the estimate we get

1) vs 3): Compare R to T1.

1) vs 2): Compare G to 2T - T1.

——

We may also consider a generalized environment with a temperature of T. In that case thermography will tell us that the comparison between 1) and 2) is between G and T. It applies when P is classified as sente or ambiguous.

My method is more powerful than thermography, as it suggests that the environment should start with T1, and it applies to the case with a losing sente, i. e., when G > R and P is classified as gote.

The sente option does not show up in the thermograph in that case.

Posted: **Thu Oct 13, 2016 4:51 pm**

Bill Spight wrote:
directly distinguishing local gote from local sente is the simplest.
But somewhat error prone.

What kinds of errors are you speaking of?

***

(Currently I am studying the distinction of local gote / sente. E.g., I want to search some examples. I will study value calculations also involving environments a bit later and need to catch up some very old posts and articles, too.)

Posted: **Thu Oct 13, 2016 5:05 pm**

RobertJasiek wrote:
Bill Spight wrote:
directly distinguishing local gote from local sente is the simplest.
But somewhat error prone.
What kinds of errors are you speaking of?

IMX, one of the most common errors is continuing a line of play past the point where the local temperature has dropped. There are also errors of calculation, but they are typically small, so little harm is done. There are errors of misclassification, especially calling a play a (local) double sente. (Locality is implied, because the whole board is not shown.) See, for instance, http://senseis.xmp.net/?YoseErrorsInMagicOfGo .

Posted: **Fri Oct 14, 2016 1:59 am**

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Example for this type of positions:

Click Here To Show Diagram Code: [go]$$B $$---------------------------- $$|X X X X X X . . . . X . X . $$|X X X X X O X . O . O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . O X X X X O X . . $$|. . . . . O O O O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$B Black's gote option, G = -33 $$---------------------------- $$|X X X X X X . . 2 3 X . X . $$|X X X X X O X . O 1 O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . O X X X X O X . . $$|. . . . . O O O O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$B Black's sente option, S = -44 $$---------------------------- $$|X X X X X X . 4 1 3 X 5 X . $$|X X X X X O X 6 O 2 O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . O X X X X O X . . $$|. . . . . O O O O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$W White starts, R = -48 1/3 $$---------------------------- $$|X X X X X X . . . 1 X . X . $$|X X X X X O X . O . O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . O X X X X O X . . $$|. . . . . O O O O O O X . .[/go]

Conditions:

The local endgame is a local gote if (G + R)/2 > S.
The local endgame is a local sente if (G + R)/2 < S.

Application:

(G + R)/2 > S <=> (-33 + (-48 1/3)) / 2 > -44 <=> -81 1/3 / 2 > -44 <=> -40 2/3 > -44.
The local endgame is local gote. Usually, Black chooses the gote sequence.

Posted: **Fri Oct 14, 2016 7:19 am**

Thanks. A very nice example.

Here is a close call.

Click Here To Show Diagram Code: [go]$$B $$---------------------------- $$|X X X X X X . . . . X . X . $$|X X X X X O X . O . O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Posted: **Sun Nov 13, 2016 8:01 am**

Click Here To Show Diagram Code: [go]$$B $$---------------------------- $$|X X X X X X . . . . X . X . $$|X X X X X O X . O . O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Since nobody has done the exercise yet, here is my call. I use the obvious locale for the counts.

Click Here To Show Diagram Code: [go]$$B Black's gote option, G = (GB + GW) / 2 = -33.5 $$---------------------------- $$|X X X X X X . . 2 3 X . X . $$|X X X X X O X . O 1 O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$B G's black follower, GB = -33 $$---------------------------- $$|X X X X X X . . O X X . X . $$|X X X X X O X . O X O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O 1 O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$W G's white follower, GW = -34 $$---------------------------- $$|X X X X X X . . O X X . X . $$|X X X X X O X . O X O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O 1 O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$B Black's sente option, S = -38 $$---------------------------- $$|X X X X X X . 4 1 3 X 5 X . $$|X X X X X O X 6 O 2 O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$W White starts, R = -42 1/3 $$---------------------------- $$|X X X X X X . . . 1 X . X . $$|X X X X X O X . O . O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$B intermediate position Q = (QB + QW) / 2 = -45 / 2 = -22.5 $$---------------------------- $$|X X X X X X . . 1 3 X . X . $$|X X X X X O X . O 2 O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$B Q's black follower, QB = -7, move 1 is move 4 as continuation $$---------------------------- $$|X X X X X X . 1 X X X . X . $$|X X X X X O X . O O O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

Click Here To Show Diagram Code: [go]$$W Q's white follower, QW = S = -38, move 1 is move 4 as continuation $$---------------------------- $$|X X X X X X . 1 X X X 2 X . $$|X X X X X O X 3 O O O X X . $$|. . . . . O . O O X O X . . $$|O O O O O O O O . X O X . . $$|. . . . . . . O O . O X . . $$|. . . . . . . . O . O X . . $$|. . . . . . . . O O O X . .[/go]

We need to do three things:
1) verify that the supposed gote sequence to G is a gote sequence indeed,
2) verify that the supposed sente sequence to S is a sente sequence indeed,
3) identify whether the local endgame is a local gote, ambiguous or local sente.

(1) The supposed gote sequence to G is a gote sequence because the move value of the follow-up is 0.5 and therefore much smaller than the gote move value of the initial position, which is (G - R) / 2 = (-33.5 - (-42 1/3)) / 2 = (-33.5 + 42 1/3)) / 2 = (8 5/6) / 2 = 4 5/12.

(2) The supposed sente sequence to S is a sente sequence due to the sente condition S < (Q + R) / 2
<=> -38 < (-22.5 + (-42 1/3)) / 2 <=> -38 < (-64 5/6) / 2 <=> -38 < -32 5/12.

(3) The local endgame is a local gote due to the condition S < (G + R) / 2
<=> -38 < (-33.5 + (-42 1/3)) / 2 <=> -38 < (-75 5/6) / 2 <=> -38 < -37 11/12.

***

Bill Spight wrote: Suppose that the ambient temperature is T, i. e., that the gain from making the largest play elsewhere on the board is T.
[...] Then we can estimate the gain from playing out the rest of the board as T/2.

Why, and by which proof, T/2?

I am reminded of arguments related to komi compensating the right of moving first by being half the miai value of the first move. That was ca. 15 years ago, so I do not recall the proof by heart. But my real concern here is that you presume move values T >= T1 >= T2.. >= Tn > 0 and I wonder how, and due to which assumption of move value decrements, to sum up to get the excess value T/2.

Exception 2: In a gote position to play the sente option.

This exception requires a large gote (ending in a local count of A if you play, B if your opponent plays, A > B).

Here, I do not understand what gote you mean. Is this
- the gote option G of the local endgame P, with A being the count of G's black follower and B being the count of G's white follower,
- a big gote in the environment, where there are also other, significantly smaller gotes with values T and smaller,
- something else?

Case 1. The sente threat is at least as large as the other gote: H - S >= A - B.

Conditions:

1) S - R > T

2) A - B > G - S + T

Case 2. The sente threat is smaller than the other gote: A - B > H - S

Conditions:

1) H - G > T

2) H - R > A - B + T

Why does the term A - B occur? The counts A and B are of a (large) gote, so what is A - B? The gote's count is (A + B) / 2 and move value is (A - B) / 2. Have you already cancelled division by 2? If so, from which unequations do you start?

Why do we need A and B at all? Is it not sufficient to use one parameter for either the (large) gote's count or move value, without referring to its followers' counts A and B?

let White have a move to a position worth -R, a negative number

Is -R negative or is R negative and you'd better write R < 0? The annotation -R confuses me here.

***

EDIT 4: overlooked the tiny follow-up. Now corrected. Oops. Fractions. Once more corrected. Then forgot to alter the gote sequence verication.

Posted: **Sun Nov 13, 2016 9:57 am**

Now, after 4 corrections, I might have solved Bill's exercise, provided the theory is right. :)

Life In 19x19

Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays