In mathematics, we know there's a difference between:

• Textbook methods of calculation, for pen-and-paper work or for theory building;
• Methods that work well for calculating in your head.

I suspect the same is true for endgame counting, especially after seeing this recent conversation.

I'd like to share a couple of things that I'm finding useful at my current level. And then the stronger players can tell me why my methods are wrong, and we'll all learn something!

Tip 1: avoid negative numbers

I've seen in a number of places that you should count black's score or move values as positive numbers and white's as negative. And this is exactly right for writing down equations and making sure the theory all works. But then Endgame 2 - Values contains a section on common mistakes, and it strikes me that a lot of those mistakes are either sign errors or confusing differences with averages, things that happen if you try to do "pure" symbolic manipulation inside your head without applying what I think of as "number sense".

Let me give some examples.

Quick, what's the average of 6 and 10? At a glance, I can see that 8 sits exactly in the middle of the two. If I try to unpack what happened in my mind, it's something like: 6 and 10 are 4 units apartQuick, what's the average of 6 and 10? At a glance, I can see that 8 sits exactly in the middle of the two. If I try to unpack what happened in my mind, it's something like: 6 and 10 are 4 units apart. Half way in between means I want the number that's 2 units past 6, or 2 units before 10, whichever is easier to find. So I do 6+2 = 8.

What about the average of 27 and 41? A bit harder, but I can see that they're 14 apart, so I want whichever of 27+7 or 41-7 is easier. Either way I get 34, and I didn't have to do any "big" sums like 27+41 = , um, it's more than 60, and then I'll have to divide the result by 2, more work than I want to do in my head.

Average of +3 and -5? Wait, do I add 3 and 5, or do I take one away from the other? Easy to get confused where negative numbers are involved.

In a go game, average of B+3 and W+5? Well you can see right away that white is doing better, so the answer will be W+something. In fact, white's gain (compared to an equal position) is 2 points more than black's gain, half the time, so the average will be W+1. (And if I've calculated a count of W+1, now it's easy to compare W+1 with W+5 and get a move value of 4, without having to go up to "big" numbers and measure 8-point swings.)

Average of B+12 and W+5? It's going to be B+something. Black is 7 points better off, half the time, so I get B+3.5 on average.

Average of 96394 and 30482? This is where you get out your calculator and press the buttons for 96394 + 30482 and so on -- do this one by the textbook method.

Tip 2: count relative to a "reference position".

I think this is implicit in a lot of older writing, but I've never seen it spelled out, except in Endgame Values - 2, page 124, where Robert Jasiek calls it an "average position" and suggests that it's not very useful. But I disagree! So let's have an example.

Click Here To Show Diagram Code

[go]$$W Value of white 'a' or black 'b'?
$$---------------------+
$$ . . . . . . . . . . |
$$ . . . . a b . . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

First, here's the traditional method of counting.

Click Here To Show Diagram Code

[go]$$B Variation 1: black follower
$$---------------------+
$$ . . . . . 7 5 6 . . |
$$ . . . . 3 1 2 8 . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

After in the above diagram, it's black's privilege to play . Hold this position in your head for a moment while we do something else.

Click Here To Show Diagram Code

[go]$$Wc Variation 2: white follower
$$---------------------+
$$ . . . a . . . . . . |
$$ . . . 2 1 3 . . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Here (above), white a is gote. So we have to consider two follow-ups.

Click Here To Show Diagram Code

[go]$$Wc Variation 2.1: white continues
$$---------------------+
$$ . . 2 1 3 . . . . . |
$$ . . . X O O . . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Click Here To Show Diagram Code

[go]$$Bc Variation 2.2: black continues
$$---------------------+
$$ . . . 3 1 2 . . . . |
$$ . . . X O O . . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Now comparing variation 2.1 with variation 1 (I hope you haven't forgotten it!), how many points has white gained?

Click Here To Show Diagram Code

[go]$$Wc Result for white, variation 2.1.
$$---------------------+
$$ . . B W W x x x . . |
$$ . . . B O O x x . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Circled stones are places where black had territory in variation 1; the xs are places where white has gained territory compared with variation 1. So white has gained 9 points.

Remember that number. Now do the same calculation for varation 2.2. You should get a gain of 7 points in that case.

So on average, white gains 8 points. It's an 8-point move by deiri counting. How are you doing with that cognitive load?

Now let's try modern endgame theory using Robert Jasiek's method in Endgame Values - 2. His idea of a "locale" is a very useful aid to calculation!

Back to the initial position:

Click Here To Show Diagram Code

[go]$$W The locale
$$---------------------+
$$ . . x x x x x x . . |
$$ . . y x a b x x . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Here the xs are the intersections that change status depending on who plays first: this is the locale. Note the point y: this is black's territory in all variations, so it doesn't need to be part of the locale. Do you want to include it in the locale anyway, so that you have a nice rectangular shape that's easy to remember? Or do you want to exclude is so that you're using a minimal locale and keeping the numbers small? You'll get the same final answer either way, so make your choice and stick to it. You can exclude a and b] from the locale on the principle that they'll always have stones on them, or you can include them to keep the shape simple, you'll get the same answer either way. And make sure to remember that the right edge of the locale stops two lines before the edge of the board, no more, no less. And remember where the left edge is.

Click Here To Show Diagram Code

[go]$$B Variation 1: black follower
$$---------------------+
$$ . . . . . 7 5 6 . . |
$$ . . . . 3 1 2 8 . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Take another look at variation 1, and mentally superimpose your minimal locale. Black has kept the four xs on the left, but the xs on the right are now covered up, so the count within this locale is B+4. (Or if you used the rectangular locale, it would be B+5.)

Variation 2.1, same thing, you should get W+5. (Or W+4 if you chose the other locale). And variation 2.2, W+3. Variation 2 is W+4 on average, so the count is the average of W+4 and B+4, i.e. it's a count of zero for this locale. Black moving first changes zero to B+4, so the move value is 4 (using miai counting this time, so we should get half of the traditional value).

Compared with the traditional method, you no longer have to hold two different board positions in your head and superimpose them. You just have to visualise one position at a time and remember the shape of the locale.

So now we get to the idea of a reference position. The idea is to have the simplest settled position you can imagine -- something that pops into your head straight away when you look at the current board position; it doesn't need to be a realistic move sequence. In most cases you can just take both colours straight down to the edge to get a reference position.

Click Here To Show Diagram Code

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

Now look at variation 1 and imagine the reference position under it:

Click Here To Show Diagram Code

[go]$$B Variation 1: black follower with reference position marked.
$$---------------------+
$$ . . . . C B X O . . |
$$ . . . . B B O O . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Circles in the above diagram are the edges of the reference position. You should be able to look at the board, imagine the follower position in your head and superimpose. So white is five points down compared to the reference position (white loses 4 points of territory and black gains 1). The only things you have to hold in your memory are one follower position at a time and the numbers.

Same thing for variations 2.1 and 2.2, white comes out 4 or 2 points ahead compared to the reference position.

It's pretty similar to using a locale, but you no longer have the cognitive burden of remembering exactly what shaped locale you used, so less room for mistakes with this method.

On reflection, perhaps my "reference position" and Robert's "average position" are actually two different things. For me, the reference position doesn't need to be an exact mathematical average of the black and white options, it just needs to be something that's very easy to visualise. For example:

Click Here To Show Diagram Code

[go]$$B move value of 'a'?
$$--------------
$$ . . . . a . .
$$ . . . X O . .
$$ X X X X O . .
$$ . . . . O O O[/go]

For the above diagram, a reference position might look like this:

Click Here To Show Diagram Code

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

This isn't an average in any strict sense, but it makes it easy to see that a is a three-point move (comparing B+2 with W+1).

On the other hand, going back to my previous post, variations 2.1 and 2.2: you can average them out to get this diagram:

Click Here To Show Diagram Code

[go]$$Wc Variation 2 averaged
$$---------------------+
$$ . . . B W . . . . . |
$$ . . . X O O . . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

This is a true average of the follower positions, and you can compare it to a locale or to a reference position and get accurate counts. So I would call this one an average but not a reference position.

Minor comments on your messages:

"I've seen in a number of places that you should count black's score or move values as positive numbers and white's as negative." I think that you do not mean "move values" themselves but "other values from which move values are calculated".

Average position is not so useful but maybe your concept of reference position is more useful. (You have noticed that these are two different concepts.)

"Locale" had been invented by others but I gave this concept its name and written descriptions. Before, quite a few might have overlooked the concept because it had not been mentioned explicitly.


Major comments:

Aids for mental calculation can be used if
- the user is comfortable with them,
- the user saves time by using them,
- the user applies them correctly,
- they are correct and
- the user verifies that their application is correct.

Usually, aids for mental calculation simplify or rely on assumptions of simplified presuppositions. Aids might be faster if one presumes that they work correctly without verifying that their application is correct.

However, the assumptions are not always fulfilled and such is not infrequent. For endgame calculations, I have often made the experience that verifying correctness of application is necessary and more time-consuming than the time saved by using aids for mental calculation at all.

With some exceptions, such as the following. Shorthand equal-distance calculation of averages of small numbers. Locale works well if only it is large enough. Speaking of "White's points" instead of considering negative points works in the simplest cases, especially if White's start results in White's excess territory, there is one simple boundary and no / little iteration. Even then, it might still be faster to consider negative points.

Usually doing strict calculations and using negative numbers was one of the things improving my endgame calculations. As an early learner, I still shared your desire for aids for mental calculation but soon I became aware of the frequent necessity of verifying correctness of application, which is slow, cumbersome and error-prone. It already takes much time to reflect what verification is needed before actually doing the verification.

Eventually, I have found that calculating (close to) accurate values is faster than "rounding values combined with verifying validity of the rounding and correctness of comparing rounded values to other (possibly also rounded) values".

I am very weak at endgame, but FWIW here are my thoughts:

• I like to use locales whose shape are easy to remember, like rectangles.
• Reference positions are useful if it is easy to keep track mentally of their boundary, for instance when the boundary is a straight line.

Click Here To Show Diagram Code

[go]$$W Value of white 'a' or black 'b'?
$$---------------------+
$$ . . . . . . . . . . |
$$ . . . . a b . . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

There is a distinct possibility that Black a or White b can and should be played with sente. Leaving that question aside, here is a semi-traditional approach to evaluating this position.

Click Here To Show Diagram Code

[go]$$W White first
$$---------------------+
$$ . . . 6 7 3 . . . . |
$$ . . . 2 1 a . . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

After is the normal follow-up. OC, traditional theory does not have the concept of ambiguous plays, such as , and assumes at a to be the default. That's why I say, semi-traditional.

Click Here To Show Diagram Code

[go]$$B Black first as gote
$$---------------------+
$$ . . . . 3 9 5 7 8 . |
$$ . . . . . 1 2 6 0 . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

Traditional theory assumes as a default that White cannot afford to make the ko after . In that case - is sente, raising the local temperature to 5.

Edit: BTW, is an error, as the local temperature has dropped to less than 3. But the quick and dirty assessment is still correct that - is sente.

Click Here To Show Diagram Code

[go]$$B Black first as sente
$$---------------------+
$$ . . . . 3 6 4 . . . |
$$ . . . . 7 1 2 . . . |
$$ . X X X X O O . . . |
$$ . . . . X O . O . . |[/go]

is the expected sente follow-up to .

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

Visualize whirled peas.

Everything with love. Stay safe.

Most traditional endgame writing includes a trick that, I suppose, pros use in practice in real game calculations. That trick is to guess whether a play is sente or gote. If it is sente, then you just have to mentally play the sente sequence to get the count, and then play the reverse sente sequence to get the move value or temperature. Where this approach breaks down is when you assume that a position is a double sente. Otherwise, the assumption that a play is sente or gote can be corrected. As in the example above, where the assumption that - is gote produces a follow-up that is too big.

Another non-traditional trick is to play the local position out to the the end with alternating play with each player playing first. Except for kos this gives boundaries on the count, which may be helpful. It also may help to identify sente and gote, and to detect reversals. Reversal is a modern concept about which traditional players had some intuition, but no theory.

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

Visualize whirled peas.

Everything with love. Stay safe.

The book "the Endgame" explains the entire reasoning behind it carefully:
(And a couple of years ago, I posted these 2 pictures on twitter,
because I personally believe that this text is the underestimated shortcut to amateur dan-level play)
Link: https://twitter.com/zbaduk/status/1002439000160907264
And most of all, I love the book, because it teaches skills.
Knowledge gets outdated or may change over time, but skills are timeless.

TL;DR;
For hane on the 2nd line I know the following by hard.
(having calculated it many times in games)
- if the move is gote for both sides, and also the follow-ups are gote: then the value is 6 points.

But it can be bigger if the followups are sente !
So, I just imagine the outcomes for both sides, and evaluate if there are sente follow-ups.
In those cases I add the value of the follow.

e.g. a follow-up of 2 points in sente, means the total value will be 8 points.
e.g. if both sides have a followup of 2 points in sente, then the total value will be 10 points.
e.g. if neither side has a follow-up in sente: it's just 6 points
e.g. if one side has a follow-up of 4 points in sente (happens occasionally) then the value is 10 points.
...

_________________
Enjoy LeeLaZero and KataGo from your webbrowser, without installing anything !
https://www.zbaduk.com

Depending on what you refer to, either "yes" or "explains only a tiny fraction of the entire reasoning".



See
https://lifein19x19.com/viewtopic.php?p=252612#p252612

As a minimum, I expect a theory book on the endgame to explain the difference between the following two concepts:

(first, firm definitions of count and value, which I think have been adequately developed in modern endgame theory and which are available on SL and in books like Rational Endgame and Robert's)

- a move which, when unanswered, leaves a follow-up move with a higher value than the original move
- a move which requires an immediate answer (i.e. playing elsewhere, even another local sente, does not offset the damage done by the follow-up)

Let's call the first "local sente" and the second "global sente". There may be multiple local sente but at most one global sente. When a local sente is answered by another local sente elsewhere (and subsequently both answer the local sente, or continue following up), we can speak of the process of mutual damage.

If what I'm stating here is not true, let me know.

If it is correct, then let me know which books have this notion.

"The endgame" by Ogawa doesn't. It discusses mutual damage but otherwise treats "sente" as if all local sente were global sente.

For practical endgame, I would benefit more from clear concepts than from correct calculations. The latter follow from the former anyhow and are merely a matter of precision than clarity.

I haven't studied endgame in great depth, but it seems highly dubious to suggest there can be no more than one global sente.

Look at this position:

Click Here To Show Diagram Code

[go]$$c Black has global sente at a, b, c, and d.
$$ ---------------------------------------
$$ | O . . c X . . . . . . . . . X a . . O |
$$ | . O O O X . . . . . . . . . X O O O . |
$$ | . O O O X . . . . . . . . . X O O O . |
$$ | d O O O X . . . . , . . . . X O O O b |
$$ | X X X X X . . . . . . . . . X X X X X |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

If black plays any of a, b, c, or d, white must respond to that particular one. They're all global sente (assuming the group being captured is bigger than other endgame by this stage of the game). The point is, even if there can't be two moves that are effectively worth infinity (instantly game winning for whoever plays it), there can dozens of moves whose followups are effectively worth infinity.

IMHO, it's the other way around. Clear concepts follow from correct calculations, and unclear concepts lead to incorrect calculations.

There are endgame techniques and concepts which produce perfect play and do not rely upon any calculation, except counting final scores. I am referring to difference games, reversals, and to the concepts of miai, tedomari, and go infinitesimals.

Calculations, however, only lead to heuristics, to playing the averages. The concepts of tedomari and temperature help to show when it may be wrong to play the averages.



First, I would say gain instead of value, as value is an ambiguous term.

Second, the first is not necessarily sente. Whether it is depends upon whether the sequence of best local play before the local temperature drops is odd or even. Only if it is even is the sequence sente.

Third, the second asks too much, in general. To ascertain whether a play requires an immediate response (whatever that means) is asking for what endgames calculations do not as a rule provide. They only tell us how to play the averages. Even in the case of go infinitesimals, where correct play can be determined, you have to analyze the whole board. The fuzzy concept of global sente is much more broad than the idea of go infinitesimals.

To borrow from TelegraphGo, here is a possible global sente.

Click Here To Show Diagram Code

[go]$$Bc
$$ -------------------
$$ | O . 2 1 X . X . .
$$ | . O O O X X X . .
$$ | . O O O X . X . .
$$ | . O O O X X . . .
$$ | X X X X X . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

But suppose that the rest of the board includes this position, as well.

Click Here To Show Diagram Code

[go]$$Bc
$$ ----------------------------------------
$$ | O . . 1 X . X . . . . . O . O 2 . . X |
$$ | . O O O X X X . . . . . O O O X X X . |
$$ | . O O O X . X . . . . . O . O X X X . |
$$ | . O O O X X . . . , . . . O O X X X . |
$$ | X X X X X . . . . . . . . . O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]

In that case dominates the local reply. Is still global sente? That depends on your definition.

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

Visualize whirled peas.

Everything with love. Stay safe.

See https://lifein19x19.com/viewtopic.php?p=252641#p252641