This text begins a series of blog posts with a common theme: that of explaining what the proverb “sente gains nothing” means. My design is to start the explanation from easier, endgame-technical examples and little by little work towards more difficult concepts. If everything goes according to plan, at the end of the series I should have conveyed the idea of the proverb without (hopefully) having to explain it in abstract terms at all! (Having read essays on the art of human rationality on LessWrong, I wanted to try a similar concept on Go.)

At the same time, through the series I hope to improve on my writing skills. If the blog posts turn out well, I will probably refine them further into a part of an endgame theory book I have been planning for some time.

For readers unfamiliar with the terminology: please bear with me! I will also explain the technical meanings of *sente* and its opposite, *gote* in following posts.

### Part 1: Expected territory

Dia 1 shows a shape that could often come up in the late parts of a Go game. At some point late in the game, either White or Black plays at A, and then play switches elsewhere and soon the game is over.

For the purposes of this blog post, we will assume all black and white stones are alive at all times.

When asked, “How big is a move at A?” the most common answer seems to be: “One point! If Black plays A, Black has one point of territory, and if White plays A, Black has zero. The difference between these scenarios (Dia 2 and Dia 3) is one point, and thus A is a one-point move!”

If the same person is pressed for how much territory Black should expect back in Dia 1, the answer we are likely to get is, “Either Black has 1 or 0 points, but we cannot know which.”

Neither of these answers is wrong, but they are also not exact, and I think the way of thinking they represent is prone to confusion.

If Dia 1 was the last endgame point available on the board, then indeed Black would either get 1 or 0 point, depending on whose turn it is, and the game would end there. Throughout most of the game, however, there are a big number of valuable moves available, many of which are of equal sizes.

Enter my favourite trick for endgame analysis: shape duplication!

Consider Dia 4: A and B are clearly moves of equal size, both surrounding “either 1 or 0 points of black territory”. Inside the local situation of Dia 4, neither player can expect to get both A and B for themselves. Remember, a player gets only one move at a time! Thus, either Black gets A and White gets B, or Black gets B and White gets A. This does not change whether Black or White has the turn to move first. In Dia 4, then, Black always has exactly one point of territory.

Going back to Dia 1, which holds half the territory of the black shape in Dia 4, can you guess how much territory Black has?

That’s right! Half a point!

But wait, the counting gets trickier from here! If Black already has half a point in Dia 1, what does that make the value of a move at A?

Comparing the final points difference between White or Black having gotten A indeed gives an idea of the size of an endgame move, but that number does not tell us how the territory balance of the game changes. We should originally value the black territory in Dia 1 at 0.5 points. If White plays A, the black territory shrinks to zero, meaning a change of 0.5 points in the game score. Likewise, if Black plays A, the black territory increases to one point, similarly meaning a change of 0.5 points. In other words, the value of a single stone at A in Dia 1 is half a point!

This concludes Part 1 of the series, but I have homework for you!

How many points does Black expect in Dia 5? You will not need any tips other than the techniques above!

Is it 1.5 points? So the difference between black or white playing is also 0.5?

Finally Ten :3 You are coming back to writing, this was my favourite go site a few years ago <3.

Thanks for this

it’s 1.5 points, because black always has 1 point, but the question is wether he can get the other 1 point move.

50% * 2 + 25% * 1 + 25% * 0 = 1.25

3,5 / 3 = 1,1666… Three duplicates and black has 3 or 4 points depending on who starts first.

if white plays first then we revert to the example i.e 0.5 point

if black plays first we get 2 points

so the average is 1.25

Yep. Use Antti’s “shape duplication!” B get either 2 or 3 points. So it’s 2,5 with duplication.

Then just /2 for one diagram and = 1,25 just like Oota says.

Instead of messy calculations, I took a peek at earlier research. Page 5 of the sample pdf of “Mathematical Go” gives the answer 1+¼. The book itself is chock full of science and examples on how to win the one final point in endgame.

https://www.researchgate.net/publication/220691411_Mathematical_Go_-_chilling_gets_the_last_point

Doing the shape duplication Black always gets 2 points of territory and after playing the shape reduces to Dia 1, so we should originally value the black territory at 1.5 points. Then if White plays first, the black territory shrinks to 1, and if Black plays first it goes to 2, so the answer should be 0.5 again?

Hmm, the conclusion to dai1 makes sense as it seems there is a half chance that there is another move on the board worth the same value!

Dai 5

((0.5)(2points) + (0.75)(1point))/2 = 0.875 points?

Oh, it looks like the calculation is the amount black makes playing first (2p) plus whatever the value of the black move that makes one point (previously calculated to 1/2) divided by two.

(2)+(0.5)/2 = 1.25?

or:

(0.5)(2) + (0.5) ((0.5)(1)+(0.5)(0))= 1.25

interestingly, duplicates of anything other than a factor of four give other results to the one calculated above– in other even duplicates, you must average sente.

For an endgame with three spaces, a duplicate of factor 8 seems to be the ones which work.

Duplicate four times. In that case black can get himself 5 points with correct play and without regard to who initially had sente. 5/4=1,25

Many readers got the solution to the homework right! The correct answer was 1¼ points.

For my version of the explanation why, see part 2 of the series at http://gooften.net/2016/01/25/sente-gains-nothing-part-2-fair-exhanges/ .