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 numbersI'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.