schilds wrote:As for your examples, I can play this game too
. Make the board bigger again, at some point we might find that a white invasion leaves black with a 3rd wall again ideally distanced from one of the two original walls.
I don't think it works that way. The bigger the board, the less severe the attack on the invasion is going to be, so it's unlikely black will be able to build another wall on a bigger board. The bigger the board, the better the invasion move will be. On a 19x19 board, invading immediately and deeply is usually a mistake because you're going to be attacked severely. On the 27x27 board, an immediate deep invasion is certainly playable. Black will get less profit from the attack, because defending is easier. Above a certain board size, there will be no attack at all. When the side is that big, why play near the wall at all? There will never be a second, third and so on wall. White will just build a stable position at a safe distance, and the thickness will be wasted.
Of course, all my examples are assuming that the players will just play normal 19x19 sequences. Generally one can say that what's joseki on 19x19 is not necessarily joseki on other board sizes. In fact I think on a big board, black should block on the other side... Why? Because the side hoshi stone is too far away for the wall to be effective. That's similar to it being a mistake blocking on this side without having a stone on the side hoshi on 19x19, because the corner gets larger without a clear benefit. And that again leads us to the original point: The outside influence is not able to compensate the additional territory in the corner on a larger board (in that particular situation).
schilds wrote:Also, one would have to question the positioning of that center stone on the larger board with respect to both walls (which I gather is the kind of thing Bill is thinking about). If we assume both walls need an extension, and that the walls project the same strength regardless of board size, then the ideal extension is the same regardless of board size, then on a large board simply putting a stone smack in the middle of two walls is being wishy washy and you shouldn't be basing your strategy around the happy coincidence of the star point being ideally distanced from possible walls created near two corners (because there is no longer such a happy coincidence) :p.
But the hoshi stone was there before the corner invasions.
This is just another example of influence being less useful on larger boards. Let's assume the hoshi stone was not there. On 19x19 you have one move that's an ideal extension from both walls. On a larger board, you don't have that play. That's another reason why the walls are less useful on the larger board
Of course, when I say "less useful", I mean in comparison with territory.
Let's look at a simple example: Let's say white invades at the 3-3 point under a 4-4 point stone, on an otherwise empty board. White gets about 10 points in territory. We don't know what the wall is worth exactly, but that's not so important, lets just call that A points. Later, black may want to make an ideal extension from the wall. Let's say that's about B points gain. A longer extension will be worth less.
Let's now assume black has a ni-ren-sei formation, and white invades both corners for 20 points, that's the same regardless of board size. Let's again assume later black wants to extend.
On a 19x19 board, one move is the ideal extension from both walls, so we can say it gains about twice as much, that is 2*B points, for a total profit of 2*A + 2*B points for black resulting from both wall plus the extensions.
On a larger board, let's just assume the walls themselves are rally worth A points. Now black has two possible choices: Play the middle, that's not optimal for both sides, for a total of 2*A + 2*(B-something). Or play the ideal extension from one wall, for 2*A + B + a little bit (because the ideal extension from the one wall may still affect the other wall a little bit
). That's less than on the 19x19 board, but white still got exactly the same about of territory.
So even if you assume the value of the walls is the same regardless of board size, the total usefulness compared to other plays on the board is still lower. Not because the walls are worth less, but because other plays (such as invasions) are worth more.
