Why do we play on a 19x19 goban? I myself am content with the reasons given by Chang Nui around 1050 in "The Classic of Go"; see 19x19 Board.
But others aren't, apparently. I just read a small article entitled "Why the devil do we play on a 19x19 goban?" (in French). It started by remarking that, rather than giving historical reasons, a more "mathematical" approach could be imagined. Here's a short outline of this mathematics. (My own remarks come after this outline.)
Start with the well known proverb "The third line is the line of territory (Third Line), the fourth line is the line of influence (Fourth Line)". Then use this proverb as a hypothesis for a comparison of center versus side territory, namely by assuming Black plays only on the 3rd line, and White only on the 4th. (See Classical example of center versus side territory). An example on a 13×13 goban:
Then measure Black's and White's Efficiency: 92 points / 28 stones = 3.286 for Black, 25 points / 24 stones = 1.042 for White. Finally, the "importance of edge vs. center" measure (I) is Black's efficiency / White's efficiency, i.e. 3.15 in the case of the 13x13 goban.
Finally, a table is made, calculating this measure for different sized gobans. I'll give a formula for this instead:
Black's efficiency = 2 + 9 / (N - 6)
White's efficiency = (N - 7)^2 / 4 (N - 8)
For a 19×19 goban, this yields an I-value of 1.07, for larger gobans the value drops below 1. In other words: for a 19×19 board, Black and White play almost equally efficient, and this is how it should be. The conclusion: since I for 19×19 is closest to 1, we're mostly interested in playing on this size of board.
First a small remark: the exact layout you choose matters a lot (e.g. if you also count Black plays on 3-3, his efficiency becomes only 2 + 4 / (N - 5)). See Classical example of center versus side territory for some other layouts.
But the main remark is: this whole reasoning is flawed! I don't ever want to see somebody make a similar mistake again! Let's analyse what happens.
- We start with a well-known proverb about 3rd and 4th line. This proverb, mind you, implicitly assumes you're playing on a standard sized goban.
- We make a few more more assumptions and do some math based on the proverb.
- We find that 19x19 yields the nicest outcome value.
So, what to conclude? Some possibilities:
- There's a causal relation "use of 19x19 goban" => "proverb about 3rd and 4th line". In this case our math mainly confirms this causal relation, or rather confirms the validity of the proverb.
- There's a causal relation "proverb about 3rd and 4th line" => "use of 19x19 goban". This is what the article assumed. In this case our math confirms that we chose to play on 19x19 because it better matches with the proverb.
- There's no special relation between "use of 19x19 goban" and "proverb about 3rd and 4th line". Our math was much ado about nothing. In this case the proverb and the goban size just accidently match, while our math misleadingly tries to point at some causility.
Well, I'd rather believe 1 and 3 then 2.
I have to disagree with the above. The third line may have special significance in Go as the highest line which can easily generate secure territory along the side. On a larger goban it might be much more difficult to build territory along the side without ceding too much influence. It takes many more moves to prevent an opponent's invasion from easily living under a higher enclosure. If the third line truly has this property, then playing on a larger board would radically shift the relative importance of center and side territory.
AnonLinguist: I agree to disagree, here. From my own experience on 37x37, influence is all, and grabbing territory in the opening is nothing short of idiotic. It's like throwing away the game immediately. This makes for a completely new game, where existing joseki are based on a really bad idea. It's actually quite viable to build a wall on the 6th line and cede 5th line and below.
>ThorAvaTahr: Why 37x37? :P (doesnt 39x39 make a lot more sense... :P )
PlatinumDragon: A constant number when multiple to the side of a square board gives the "territory" line, and 5x5, 13x13, 19x19 are such special cases of them.
Phelan: Without commenting on the rest of your analysis, you're disregarding the fact that the small standard size is 9x9, and not 5x5. 5x5 is usually used as a simple beginner board, because games end faster, and that gives more game experience faster.
I (axd, not related to the other "I"s) ... think that the use of "I" in this page reduces its value to a personal point of view.