Continuous thinking on the rational change of Go (1)

In this article, I have discussed some issues about the rational number of GO games. At the end of this article, I have raised the limit questions about V (N)/3 ^ (N * n. I started to think it would be more than 0.5, but on the Forum, someone suggested it should be 0. I tried to find some proof methods to prove that it was not 0, but it was not successful. Now I am more and more inclined to this value is 0, but there is still no perfect proof. Below are some of my existing ideas.

One way is to check the number of reasonable changes. For example, it is reasonable to place a Sunday on the Board as long as it is not full. The N * n board has P sunspots and contains C (P, n. Then, you can place the white on the board and consider how to place the white on it to ensure that the Board is reasonable. The simple idea is that the white child is not adjacent to the black child. This situation must be reasonable. A maximum of 4 * P adjacent points can be found in P sunspots. So there are N * n-4p points, white can be randomly placed or not put, a total of 2 ^ (N * n-4p) method. So the overall rational situation is at least
C (P, n) * 2 ^ (N * n-4p), where p changes from 0 to N * n, each accumulate (expressed in Σ is clearer, but cannot be entered here ). However, this valuation is too small. I tried other methods of valuation, but I can't get rid of the problem that the limit is 0. Of course, you can also consider the upper limit, but now I have no time to think about it.

Another idea is to examine the number of unreasonable changes. For example, if there is a Sunday in the upper left corner of the board, and then there are two white ones to enclose it and it becomes "dead", it is unreasonable to place any other part of the board. So on the N * n board, unreasonable changes (recorded as D (N), D = dead) have at least D (n)> = 3 ^ (N * n-3 ), in this case, D (N)/3 ^ (N * n)> = 1/27. Seeing this result, I can't help but sigh that if V (n) has such a simple estimate. But now this result does not help us much, because our goal is to prove that D (N)/3 ^ (N * n) is equal to or less than a certain constant.

Another idea is to establish a one-to-N or N-to-one relationship between reasonable and unreasonable changes. If this relationship exists, you can immediately know that V (N)/3 ^ (N * n) is greater than or equal to 0. I think this idea is very promising. Unfortunately, this relationship is still not easy to establish.

Record it first and wait for the future ......