In this article, I discussed some questions about the number of playful changes, and the limit of the limit of V (n) / 3 ^ (n * n) was proposed. I started feeling more than 0.5, but later in the forum, some people should be 0. I try to find some proved methods prove that not 0, but it is not successful. Now I am more and more inclined to this value is 0, but there is still no full proof. Let's write a few ideas in my existing.
One method is to investigate the number of reasonable changes. For example, the chessboard is put on the black son, then as long as it is not full, it is reasonable. N * N's chessboard, pendulum P-neck, there are c (p, n) swing. Can then be placed on the chessboard, consider how to put a white child still guarantee the chessboard reasonable. A relatively simple idea is that all the whitebs are not adjacent to the blacks, so that the situation is definitely reasonable. P-neighborhood has 4 * P adjacent points. Therefore, there are n * n-4p points, and the whitebs can be placed or not placed, a total of 2 ^ (n * n-4p) methods. So the total reasonable situation is at least
C (p, n) * 2 ^ (n * n-4p), where P changes from 0 to N * n, and the accumulation is accumulated (with σ indication will be relatively clear, but it cannot be entered here). But this estimate is too small. I tried other valuation methods, but I couldn't get rid of the limit of 0. Of course, I can also consider how much it is, but I have no work now.
Another idea is to examine the quantity of unreasonable changes. For example, a black child is placed in the upper left corner of the chessboard, and then uses two whitebs to become "dead shape", so that the panel is arbitrarily changed. So the N * N board, unreasonable changes (remember D (N), D = DEAD) at least D (N)> = 3 ^ (N * N-3), which d (n) / 3 ^ ( n * n)> = 1/27. Seeing this result, I can't help but sigh, if V (n) is also so simple enough. But now this result is helpless to us, because our goal is to prove that D (N) / 3 ^ (n * n) is equal to one or less than a certain constant.
There is also a way to establish a relationship between a pair of n or N or N to a reasonable change and unreasonable changes. If there is such a relationship, you can know that V (n) / 3 ^ (n * n) is greater than or equal to 0. I think this kind of idea is very hopeful, but this relationship is still not very easy to build.
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