Counting Crawls

  Difficulty: Intermediate   Keywords: Strategy, Theory

This page was inspired by the Ready Reckoner.

It is a curious fact that we can determine the asymptotic value of some plays without doing any reading at all. Here we do that for crawls.

1st line crawl  

We suppose that the White territory extends infinitely far to the right. How much does B1 gain?
Plainly, 1 point, as it reduces White's potential by that much.[1]
If White's territory is finite, B1 gains less than that. If the White stones extended only one point beyond the Black stones, it would be worth nothing, merely filling a dame. If they extended 2 points beyond, it would gain ``1/2`` point. If they extended 3 points beyond, it would gain ``3/4`` point, etc. The asymptotic value is 1 point, the exact value is ``1-1/2^n`` with n+1 being the length of the corridor.

2nd line crawl  

Now B1 gains 3 points: it adds 1 point of Black territory and takes away 2 points of White territory.
OC, on an infinite board the crawl is not the best play. The Monkey Jump is. However, on a finite board the crawl is sometimes better. Then it is usually worth 2+ points.

3rd line crawl  

This is easy. It gains 5 points.
In fact this kind of crawl on the ``N``th line gains ``2N - 1`` points in the limit, up to some N where the opponent can play underneath your wall to some effect.

This is not of any great practical value, since this kind of crawl is typically wrong. But it does illustrate the idea behind miai counting, which asks how much a play gains in comparison to the original position.

[1] For the mathematically inclined: Yes, I know that infinity minus infinity is not determinate, and therefore that the argument would be better in terms of taking the limit. However, in terms of games and Conway numbers, we may consider White to have made an infinity (``omega``) of plays to make the territory. Then after B1, White's territory is ``omega - 1``. See ONAG and Winning Ways.

-- Bill Spight

Counting Crawls last edited by PJTraill on January 31, 2020 - 16:15
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