Classical example of center versus side territory
The following diagram is often used to "show" that sides are more efficient in making territory than the center.
This diagram is criticized (see Bill's comment below) because Black makes 8 moves more than White.
- Bill: I think that Fujisawa Hideyuki was the first to point that out.
We can also look at the efficiency of stones as the territory/stones ratio. This is how I've seen it presented.
Black has invested 52 stones for 140 points of territory.
Territory per stone: 2.69
White has invested 44 stones for 121 points of territory.
Territory per stone: 2.75
This really is quite close. (The numbers are different if one uses area scoring and fills in the dame for perfect squares. Then black's efficiency along the third line is 3.43 and white's efficiency in the center is 3.52.)
The first flaw that came to my mind here is simply that white, creating a square (which maximizes area while minimizing the perimeter) has a more efficient formation anyway; black is making rectangles which are less efficient with respect to area when we are concerned with what we are investing in the perimeter. The number of stones seems irrelevant since this seems to be a begged question of efficiency: we have given white the most efficient enclosing shape.
Nevertheless, I set about deciding whether this really was a begged question or not.
I went about creating a formula for relative efficiency of the 3rd and 4th lines. White creates a fourth line square, black gets the third line, and area counting is used. The number I am I about to give is (efficiency of white stones) / (efficiency of black stones) for a boardsize when black takes third line territory. Thus, a number smaller than one indicates the importance of edge area while a number larger than one indicates the importance of center area.
9: 0.25
13: 0.544
17: 0.86
19: 1.02
21: 1.19
...
101: 7.8
Technically white will not truly get a live shape in the center until 11x11, but since this is just academic anyway the point doesn't really matter.
The center area grows faster than the edge and black's efficiency never catches up after 19x19. When viewed as a continuous function, a board size of 18.67 yields parity.
When coupled with the idea that underneath the 4th line there is often enough room to live--that is, when we suggest that even ignoring area counts there is a genuine distinction between the third and fourth lines in terms of the survivability of stones (so that we would never consider a "4th line is the line of territory, 5th the line of influence" situation), then 19x19 really does seem to be optimal. Other board sizes place more emphasis on either the side or the center.
As a final note on this efficiency matter, third line area efficiency never improves as the board size increases without bound. As boardsize approaches infinity, the efficiency of the third line (calculated in the manner described above) decreases, approaching 3.
Suggestions for improvement include:
Each side has invested 52 stones. White has 121 points. Black has 361-121-104 = 136 points.
--Dieter
Bill: Like the truly classical example, in which each side had a solid square, this is unfair to White. In the classical example, Black had 8 more stones than White. Here Black has 8 gaps while White has none. Clearly Black's stones are more efficient than White's for that reason, not because of center vs. edge territory.
As White voluntarily played 4 dame points in the last position, Merlijn Kuin (Spirit) proposed this diagram. He also commented that because of the arbitrary postitions of the white incursions, one may assume that Black will have to add two more moves inside his territory to cover any defects. Making the final balance 130 to 121.
Bill: In this version, Black still has 8 more gaps than White, which makes his stones more efficient.
Spirit: Why do gaps increase ones efficiency?
Nightvid? How about this, which secures the center?
While in the Gap diagram, B 8 would be better at 9, Black is at least 1 point better off with the gap (the 2 circled points minus the squared point), and actually more, since White does not threaten a hane, as with no gap, while Black threatens at least a magari.
In your example, Black has 8 more gaps than White, and is ahead by 11 points. If we conservatively estimate that each gap was worth only 1 point for Black, Black is only 3 points ahead after adjustment. In the original diagram, with no gaps, let us conservatively estimate that each extra stone is worth only 2 points. After adjustment, that puts White 1 point ahead. That is so close in either case that these diagrams plainly do not show what they propose to show.
Better examples may be at First corners then sides then center.
Authors
- Dieter copied the examples from a discussion on the Dutch Go mailing list[1], posted by Paul Van Galen, Pieter Cuijpers and Merlijn Kuin, with my apologies to the confusion caused due to my failure to notify the Dutchmen that their discussion was copied here.
[1]: Jan: There is a Dutch Go mailing list? Why do I not know of this? What's the address?
Dieter: The address is mailto://go-nl@listserver.tue.nl I'm not surprised you don't know it: the path from the frontpage of the Dutch Association to the subscription page for the mailinglist is long and strange.
This is a game I have actually played on DGS:
White has 11 * 11 = 121 points.
Black has (18+11+1) * 4 = 120 points.
Be happy to live with the third line!
JamesA: Just a couple of points... In the diagram above Black has played four moves more than White, which may explain the higher score! Also, I don't think it is a good general rule to use the third line when invading - very often this will lead to a heavy group. Thirdly, I don't agree that when extending between positions you should play on the third line as a general rule - think about the sanrensei fuseki (opening). Last point, if you are worried that your group may come under attack, it's often worth playing on the fourth line to get out into the centre and avoid being sealed in.
Gorobei: Black has actually played the same number of stones as White. Giving the corners to Black is a little unfair, though. My diagram was only an illustration of the surprising strength of the third line (thanks to BigNose for pointing out the break-even line as a concept.)
Goran: I still think Black played more stones :) 13*4 > 12*4
JamesA: Urm, sorry Gorobei but Black definitely has four more stones on the board than White - just count them! The diagram is therefore invalid... that's why I'm pressing the point.
Gorobei: OK, so I was never good at counting :)
HolIgor: Moreover, you can take away four more white stones without changing the score while all black stones work hard. But that is not the point. A third line move really helps to get a basis for a group. Too low but solid.
Storm Crow: I've updated the diagram so both sides have made an equal number of moves. I've seen this diagram done this way in books. Done this way, black has 120 points (one less than white's 121, but it demonstrates the balance of the third line).
Bill Spight: What it illustrates is the inefficiency of White's layout. White has 16 inefficient stones!
The original diagram, with white stones on the fourth line and black stones on the third line, appeared in textbooks to show how central territory is not as big as it looks. Fujisawa Hideyuki pointed out, many years ago, that Black had 8 more stones than White.
In my opinion, that diagram, and its derivatives, should have disappeared from textbooks by now.
See Fourth line vs. Third line.
Tjalveboy: Hey, black doesn't win... white has one more point than black, has noone realized?
JamesA: In the diagram below, would anyone be happy taking Black? I wouldn't!
Bill: Suppose that you use 12 stones to surround territory with a single wall.
You can surround 30 points in the corner, 12 points on the side, and 4 points in the center. Building a wall up to the fourth line on the side makes the same amount of territory as going up to the third line.
What if we use 16 stones to surround territory on the side?
Now the wall on the fourth line surrounds 24 points, while the wall on the third line surrounds only 20 points. Going up to the fourth line is more efficient. :-)
Years ago I read something by Ishida referring to an article in Kido magazine by a mathematician about territory on the side. The most efficient shape for territory on the side is a rectangle with a 2:1 ratio of length to height.
If you make a wall on the fourth line, your opponent may still have room to play underneath on the second line. But with a wall on the third line, you may reasonably estimate the territory by dropping perpendicular lines at the edges.
That leads to this diagram as the basic efficient shape for a third line wall to make territory. A wall of 6 stones forms a 4 by 2 territory (a 6 by 3 area).
The rule of thumb then, is not to form a longer wall on the third line, or to force your opponent to do so. It also means that it is OK, as a rule, to stretch (nobi) along the third line up to 5 points from the opposite end of your third line territory.
I don't completely buy that argument, but it's good enough for Ishida. ;-)
-- Bill Spight
See also: