# Endgame Question 1

Consider the following diagram in which all outside stones are considered alive.

According to my understanding of endgame theory, a play at *a* is a three point sente play for Black (miai value three). On the other hand, a play at *c* is a five and a half point double gote play (miai value two and three quarters).

I make errors all the time with evaluating these things. Does it get easier with practice?

Have I got this one right?

I calculated these values using thermographs, but this is quite difficult to do in one's head. I feel sure that there must be a traditional technique for doing it, but I can't quite get it from the Ogawa book or from the pages here.

With the knowledge that *a* is Sente for black, the value of *a* is easy to deduce, you just assume that if Black plays *a*, White has to respond at *b* giving White seven points compared to the ten she would have got from playing *a*. The difference of the ten and the seven gives the three point value for *a*.

Similarly, with the knowledge that *c* is gote, its value is easy to deduce. If Black gets it, then you count the average of the two scores:

- for if Black gets
*d*too - for if White gets
*d*.

This is the average of zero and five points for White. That is, two and a half for White. Contrarywise, if White plays *c* she gets eight points. The difference of the eight and the two and a half gives the five and a half point value for *c*. However, since this move is Gote, you have to divide by two to give a miai value of two and three quarters if you want to compare it to the value calculated for *a*.

However, if you apply the Sente calculation to the Gote move, or the other way around, you get the wrong answer. I realise that this answer will not be very inaccurate for moves that are close to the Gote / Sente divide.

Goodness, what a lot of waffle. My question is: I can work out the value of moves (sometimes) if I know whether they are Sente or Gote. However, in order to work out which they are, I have to draw thermographs in my head. Is there an easier way?

Thanks.

Jono I'm also trying to work this out, however using area scoring. Which brings us to EndgameQuestion2.

evand: I believe the easier way is to calculate it both as sente and as gote, and whichever gives a smaller value is correct. Take the play at a, for example: if we assume sente, it's a 3 point miai value. If we assume gote, the a B play at a gives a score of W + 3.5 (average of 7 and 0), or a W play at a gives a score of W + 10, for a gote value of 3.25. Since the value is smaller if W answers than if not, it is correct play for W to answer; that is, the play is in fact sente. If the values are the same, then the play is ambiguous. Perhaps someone more versed in CGT theory can tell me whether this is always correct or not.

Bill: Tom, you got these values right. Evand, your way works. (With caveats. See below.) :-)

Another approach is to assume one way or the other and check it. If you are wrong, you will find out. For instance, let's assume that both plays are 3 point sente plays. To check that, figure the size of the followup after Black takes. The play at *b* after Black plays at *a* is worth 3.5 points, which checks, since it is larger than 3. The play at *d* after Black plays at *c*, however, is worth only 2.5 points, which is less than 3, and shows that Black's play is gote.

One thing you have to watch out for is a reverse. For instance,

This is the well known 1 point hane-tsugi. If you work things out, you will find that is a larger play than , and assume that is sente. It is, in a way, but it is a losing sente. Black has to play at , and the whole thing is gote.

Another caveat is that you cannot use the lower value method to compare different options. For instance, a losing sente option can give you a lower value than the correct gote option.

Tom:Thanks people. How then should I evaluate (a|b)|(c|d)?

I could go through all four possible combinations of Sente/Gote for Black or White's initial move and take the lowest score, but this seems a bit long winded (though it may be necessary).

Alternatively, I could hypothesise that White's move is sente and Black's gote, check whether this makes sense with respect to the deduced temperature of the initial position, and if it doesn't then try another of the four possibilties etc. It isn't obvious to me that this would give the right answer, and even if it did, it might take a long time.

- Bill: { a | b || c | d }, when a > b > c > d, has a count of b, c, or (a + b + c + d)/4 = m. If m > b, then it is a Black sente and the count is b; if c > m, then it is a White sente and the count is c; if b > m > c, then it is gote and the count is m. If m = b or m = c the play is ambiguous. If b = c it is miai with a miai value of 0.

- If it is Black's sente the miai value is b - (c + d)/2; if it is White's sente the miai value is (a + b)/2 - c; if it is gote the miai value is (a + b - c - d)/4.

Tom:Terminology: I use the word *privilege* below to mean describe the attribute of a position described by saying it is Black's sente, White's sente or double gote. If there is already a word for this, or you can think of a better one, please change it. (As I understand it, the *double sente* description does not apply to an isolated position, but to a position in the context of a game.)

- Bill: Privilege is a term introduced, I believe, by Richard Bozulich to refer to the fact that if a play is sente for one player, that player will normally be able to make the play with sente. (That's not all there is to it. See ambiguous play and two way ko threat.)

A more general question (I'm getting my money's worth). If I have a complicated tree to evaluate, then is there a general algorithm for getting the temperature, privilege, and count of it, other than the thermograph one. Reading the Ogawa and Davies book, I get the impression that they essentially have one, but I can't work out what it is.

- Bill: There is. For non-kos, work bottom up with the initial assumption that each play is gote, correcting for sente and reverses, i. e. for sequences of play where the temperature rises temporarily.

My trouble with these things is that people who are good at them seem to be able to say that such a move is gote or sente which then lets them chop off huge bits of the tree. I, on the other hand, have to work carefully down all the branches of the tree (which can be a suprisingly long way down in Go), and only on my way back up can I deduce the privilege etc. Sometimes, of course a follow up is very large or small, and I can prune, but I want to know how to do it more often.

- Bill: A key word in what you say is
*seem*. In practice, it appears that pros typically guess whether a play is sente or gote and calculate accordingly. They often guess wrong, but the errors are usually within 1/6 of a point. (This level of error is apparent from reading yose books and articles. Even if the purported author did not do the calculations, he did not catch the errors, either.) This level of error is good enough for practical purposes. Remember that the largest play is not always the best play, anyway. :-)