Fourth line vs. Third line

   

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Table of diagrams
3d line vs. 4th line
Miai
Estimate
3d line vs. 4th line: Best Case for Black
3d line vs. 4th line, top side
3d line vs. 4th line (3)
3d line vs. 4th line, revisited
7x7
9x9
13x13
corner
7x7
A 7x7 opening
Checkmate
37x37 diagram

[Diagram]

3d line vs. 4th line

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?

[Diagram]

Miai

Bill: To help assess the position, note that B1 - W4 are miai.

[Diagram]

Estimate

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.


[Diagram]

3d line vs. 4th line: Best 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 on classical example of center versus side territory, 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 white+circle stones are plainly wasted. We cannot simply compare best vs. worst and take the average.

[Diagram]

3d line vs. 4th line, top side

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.

[Diagram]

3d line vs. 4th line (3)

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.

[Diagram]

3d line vs. 4th line, revisited

Bill: The exchanges of black+circle and white+circle 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

[Diagram]

7x7

On a 7x7 board, the "fourth line" is captured.

[Diagram]

9x9

On a 9x9, it is dead.

[Diagram]

13x13

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:

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)

Dieter: White to kill.

[Diagram]

corner

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.

[Diagram]

7x7

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.

[Diagram]

A 7x7 opening

On 7x7, this is a likely opening. White stabilizes thanks to the eyespace provided by W2 and W4 on the 3rd line. Yet, Black applies severe pressure with the hanes of B5 and B9. 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.

[Diagram]

Checkmate

black+circle, white+circle are the kings. Does this checkmate diagram constitute a proof of the principle that it is advantageouos to have more pawns (white+square) than your opponent in the endgame?

Phelan: Reread the diagram's explanation. Square marked stones (white+square) are pawns. The kings are the circled stones (black+circle and white+circle).

Phelan: Actually, [ext] 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.



[Diagram]

37x37 diagram

kevinwm: And on a 37x37 board, the 4-4 stones enclose far more territory, though b might live in the center...

Actually, the 4-4 stones don't have any territory, because black can play inside the "territory", like at a, and make a live group.





This is a copy of the living page "Fourth line vs. Third line" at Sensei's Library.
(OC) 2012 the Authors, published under the OpenContent License V1.0.
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