2022 AMC 12B 第 18 题

Question

5 \times 5

网格中的每个小方格或填充或为空，并且至多有八个相邻方格，相邻指共边或共顶点。网格按如下规则变换：

任一填充方格若有两个或三个填充邻居，则保持填充。任一空方格若恰有三个填充邻居，则变为填充。其他所有方格保持为空或变为空。

下图给出一个变换示例。

假设

5 \times 5

网格有一圈空方格包围一个

3 \times 3

子网格。有多少种初始构型会在一次变换后得到只有中心一个填充方格的网格？（旋转或反射得到的构型视为不同。）

Each square in a

5 \times 5

grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:

Any filled square with two or three filled neighbors remains filled. Any empty square with exactly three filled neighbors becomes a filled square. All other squares remain empty or become empty.

A sample transformation is shown in the figure below.

Suppose the

5 \times 5

grid has a border of empty squares surrounding a

3 \times 3

subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)

14

18

22

26

30

答案：C

知识点：过程模拟分类讨论

难度评级：2000

小提示：

只有内部

3 \times 3

子网格可以被填充；中心若原为空，则需恰有

3

个填充邻居，若原已填充，则需有

2

或

3

个填充邻居。

Only the inner

3 \times 3

can be filled; the center becoming filled needs exactly

3

filled neighbors (if it was empty) or

2

-

3

(if it was filled)

大提示：

关键限制是没有边界方格获得恰好三个填充邻居；检查可知每个有效构型恰有三个填充格。

The binding constraint is that no border square gains exactly three filled neighbors; check that every valid configuration has exactly three filled cells

解答：

只有内部

3 \times 3

子网格可以初始填充。要使中心在变换后填充，若它原为空，则需恰有

3

个填充邻居；若它原已填充，则需有

2

或

3

个填充邻居。

其他所有方格在变换后都必须为空。关键限制是任何边界方格都不能恰好获得三个填充邻居，这就排除了把

3 \times 3

子网格某条外边上的三格全部填充的情况。

若中心初始为空，则它的

8

个邻居中必须恰有

3

个被填充。在正方形的对称下检查这些三元组，只剩四种类型。以中心为

(0,0)

建立环坐标，代表元为

\begin{gathered} \{(-1,-1),(-1,0),(1,1)\},\\ \{(-1,-1),(-1,1),(1,-1)\},\\ \{(-1,-1),(-1,1),(1,0)\},\\ \{(-1,-1),(0,1),(1,0)\} \end{gathered}\text{。}

它们的对称轨道大小依次为

8,4,4,4

，共给出

20

种构型。

若中心初始已填充，同样的邻居检查只留下两种构型，其中另外两个填充格是一对相对的角邻居。因此总数为

20+2=22

。

所以正确答案是 C。

Only the inner

3 \times 3

squares can start filled. For the center to be filled afterward, if it began empty it needs exactly

3

filled neighbors, and if it began filled it needs

2

or

3.

Every other square must end empty. The key restriction is that no border square may acquire exactly three filled neighbors, which rules out filling all three squares along an outer edge of the

3 \times 3.

If the center starts empty, exactly

3

of its

8

neighbors must be filled. Checking these triples up to square symmetry leaves four types. With ring coordinates centered at

(0,0),

representatives are

\begin{gathered} \{(-1,-1),(-1,0),(1,1)\},\\ \{(-1,-1),(-1,1),(1,-1)\},\\ \{(-1,-1),(-1,1),(1,0)\},\\ \{(-1,-1),(0,1),(1,0)\}. \end{gathered}

Their symmetry-orbit sizes are

8,4,4,4,

giving

20

configurations.

If the center starts filled, the same neighbor check leaves only the two configurations in which the other filled cells are opposite corner neighbors. Hence the total is

20+2=22.

Thus, the correct answer is C.

2022 AMC 12B 第 18 题

答案：C

难度评级：2000

小提示：

大提示：

解答：

其他年份的第 18 题