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:
Black has (18 + 11 + 1) × 4 = 120 points.
White has 11 × 11 = 121 points.
Be happy to live with the third line!
(Used on RGG FAQ Part 1 Section 10 without tengen for jigo.)
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
Bill: This diagram is similar to the old, flawed diagram that gives Black more stones than White. Here each player has played 48 stones, to give a fairer comparison. Who stands better?
Bill: White has 121 points. If we estimate Black's territory by marking it off with xs, we get 110 points. Then we estimate White's lead as 11 points.
Velobici: Probably, this is worst possible case for Black.
Velobici: Rather than select the corners of the formation for the Black stones to remove, lets select the center of each side and compare the results.
White's territory is unchanged at 121 points. Black has 137 points. Black wins by 16 points minus komi.
This is probably the best possible result for Black.
Bill: As discussed, allowing gaps permits the Black stones to be more efficient than the White stones. In this case that is taken to the extreme, because Black encloses all four corners very efficiently. In addition, White's four stones are plainly wasted. We cannot simply compare best vs. worst and take the average.
Bill: Does this not make for an equitable comparison of the value of the third and fourth lines? The trouble is, it is hard to assess. Let's do this on all four sides.
Bill: Now the White stones work together in the center, while the Black stones exert influence in the corners. The size of the board matters.
Bill: The exchanges of and do not favor White and may favor Black to some extent. (After the miai are played, White will have two wasted stones.)
The aim is to get as equitable a comparison as possible between the third and fourth lines on a 19x19 board, not to favor Black or White. Allowing Black an efficient setup that gets all four corners makes a different comparison, center vs. corners plus sides. Yes, there is more territory there. But what about the effectiveness of the stones? So far, I have not seen such comparisons with an equal number of stones that have not made the stones on the third line more efficient than those on the fourth line via gaps.
Dieter: It occurred to me that the usage of a 19x19 board to show the relationship between third and fourth line, by calculating surfaces, is a completely false argument. If the relations between the surfaces had any meaning towards playing on the third or fourth line, then we would draw strange conclusions on different board sizes
On a 13x13, it is highly overconcentrated and doesn't come even close to White's third line territory.
This seems to suggest that on small boards, the fourth line isn't interesting at all, while the third lines are the lines of both influence and territory. Yet, on a 7x7, playing the centre (4-4) looks like a viable strategy (it's centre AND corner) and neither on 9x9 or 13x13 can I be convinced that opening at 4-4 puts you at an immediate disadvantage, or that "developing along the fourth line" would be bad.
Which leads me to the real meaning of third and fourth line, almost irrespective of the size of the board:
- the fourth line is the ideal line for development
- the third line is the ideal line for stability
Thorough proof, or more weakly stated, convincing reasoning behind this statement, and the definitions of stability and development, can be found in Minue's excellent writings Haengma tutorial for beginners and my own writings inspired by it Dieter Verhofstadt/Ideas on go theory. To summarize what's there:
- The third line is the highest line giving your group stability, because all undermining stones will be killed. In other words,. the area behind the third line is territory, hence the classical translation of the proverb.
- The fourth line is the highest line so that, when enemy stones undermine it, by attacking its eyespace on the side, enough pressure can be applied so that the group gains powerful influence towards the centre.
- AJP comment: in these arguments, it's not so much that the third line is the 'highest' line that gives stability, it's the only line that gives stability. on the fourth line, you get undermined, on the second line there's not enough room for eyes. likewise, the fourth line is the only line that is high enough to essentially force an invasion underneath it (otherwise it builds too much territory) but is still low enough that sufficient pressure can be applied downward on the invasion to gain good influence. contrast with the fifth line, which is also high enough to force an undermining invasion, but is too high to put sufficient pressure on the invasion to develop sufficiently good influence as compensation. (speaking generally in both cases of course)
- AJP comment: in these arguments, it's not so much that the third line is the 'highest' line that gives stability, it's the only line that gives stability. on the fourth line, you get undermined, on the second line there's not enough room for eyes. likewise, the fourth line is the only line that is high enough to essentially force an invasion underneath it (otherwise it builds too much territory) but is still low enough that sufficient pressure can be applied downward on the invasion to gain good influence. contrast with the fifth line, which is also high enough to force an undermining invasion, but is too high to put sufficient pressure on the invasion to develop sufficiently good influence as compensation. (speaking generally in both cases of course)
Dieter: White to kill.
I agree that on higher board sizes fourth line territory isn't that favourable anymore.
- The centre of the board is ideal for development, but it lacks a basis on large boards, hence it inverts the logical process of acquiring stability first, and developing next.
For small boards, such as a 7x7 where the 4-4 and the centre coincide, the above becomes a less interesting heuristic, because very soon tactical calculations take over.
However, maybe the surface argument can show why we eventually settled for 19x19, by reasoning the other way round. The borderline for territory/influence is 3/4, and by building territories along them such as the discussion on this page tries to do in a fair way, we can divide the board into roughly equal territories on a 19x19. But it's a big maybe, IMHO. Influence is potential territory indeed, but its overlap must be taken with care.
It wouldn't surprise me, for example, if an important argument behind 19x19 is the primality of 19, for cultural reasons, and/or reasons of symmetry and the possibility of breaking symmetry.
Bill: Dieter's diagrams show the importance of using an equal number of stones for each player to make the comparison.
Bill: For instance, this is the right comparison diagram for the 7x7 board.
Dieter: Quite so, but my point is that surface comparison doesn't teach us anything about the meaning of third and fourth line, whether you use an equal number of stones or not. It may teach us something about settling for 19x19 but I doubt it.
Bill: Oh, I think that the meaning of the third and fourth lines depends upon the size of the board. But it depends upon board size less and less as the board size increases. The 19x19 has a good balance between the third and fourth lines. I suspect that, as the board size increases further, the relative value of the fourth line does, too.
On 7x7, this is a likely opening. White stabilizes thanks to the eyespace provided by and on the 3rd line. Yet, Black applies severe pressure with the hanes of and . Black's influence meets the side very soon and perhaps all the area in between is already (4th line) territory.
This kind of analysis takes the true meaning of 3/4 into account. It speaks about the work stones are doing in a game. The constructions at the top of this page seem to be artificial results, not relevant to the game's principles. Let's compare with a Chess game.
, are the kings. Does this checkmate diagram constitute a proof of the principle that it is advantageouos to have more pawns () than your opponent in the endgame?
Phelan: Reread the diagram's explanation. Square marked stones () are pawns. The kings are the circled stones ( and ).
Phelan: Actually, Gess exists already, and is a bit different. ;) The above is an attempt to make a chess diagram in a site that only makes Go diagrams easily.
See also: