AMC10 2022 B

AMC10 2022 B · Q19

AMC10 2022 B · Q19. It mainly tests Casework, Patterns & sequences (misc).

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: 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.)

一个 $5 \times 5$ 网格中的每个方格要么填充要么为空，每个方格最多有八个相邻邻居方格，其中相邻方格共享一条边或一个角。该网格按照以下规则变换：下图显示了一个变换示例。假设 $5 \times 5$ 网格有一个由空方格构成的边框，包围着一个 $3 \times 3$ 子网格。经过一次变换后，有多少种初始配置会得到变换网格只有一个中心填充方格？（同一配置的旋转和反射被视为不同。）

(A) 14 14

(B) 18 18

(D) 26 26

(E) 30 30

Answer

Correct choice: (C)

正确答案：（C）

Solution

There are two cases for the initial configuration: 1. The center square is filled. 2. The center square is empty. Together, the answer is $2+20=\boxed{\textbf{(C)}\ 22}.$

初始配置有两种情况： 1. 中心方格是填充的。 2. 中心方格是空的。总计，答案是 $2+20=\boxed{\textbf{(C)}\ 22}$。

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