Question

坐标平面中的正方形 $ABCD$ 的顶点为 $A(1,1), B(-1,1), C(-1,-1)$ 以及 $D(1,-1)$ 。考虑下面四个变换：

• $L$ ：绕原点逆时针旋转 $90^{\circ}$ ；

• $R$ ：绕原点顺时针旋转 $90^{\circ}$ ；

• $H$ ：关于 $x$ 轴反射；

• $V$ ：关于 $y$ 轴反射。

每个变换都把正方形映到自身，但标记顶点的位置会改变。例如，先应用 $R$ 再应用 $V$ ，会把顶点 $A$ 从 $(1,1)$ 送到 $(-1,-1)$ ，并把 $(-1,1)$ 处的顶点 $B$ 送回自身。从 $\{L, R, H, V\}$ 中选择 $20$ 个变换组成序列，有多少个序列会把所有标记顶点送回原来的位置？例如， $R, R, V, H$ 是一个长度为 $4$ 的序列，会把顶点送回原位。

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:

• $L,$ a rotation of $90^{\circ}$ counterclockwise around the origin;

• $R,$ a rotation of $90^{\circ}$ clockwise around the origin;

• $H,$ a reflection across the $x$ -axis; and

• $V,$ a reflection across the $y$ -axis.

Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)

$2^{37}$

$3\cdot 2^{36}$

$2^{38}$

$3\cdot2^{37}$

$2^{39}$

答案：C

知识点：变换奇偶性

难度评级：2060

视频讲解：

Click to load, then click again to play

文字解答：

$L,R,H,V$ 中的每个变换都会把每个顶点移动到相邻角点。

因此经过奇数次变换后，标记处于四个奇状态之一。任取前 $19$ 个变换后，正方形处于奇状态；从每个奇状态出发， $L,R,H,V$ 中恰有一个变换会把标记顶点送回原位。因此前 $19$ 个变换的每个序列都恰有一个有效的最后变换。

前 $19$ 个变换有 $4^{19}=2^{38}$ 种选择，所以有效序列共有 $2^{38}$ 个。

所以正确答案是 C。

Each of $L,R,H,V$ moves every vertex to an adjacent corner of the square. Therefore after an odd number of transformations the labeling is in one of the four odd-parity states, and after an even number it is in one of the four even-parity states.

After any first $19$ transformations, the square is in an odd-parity state. From each odd-parity state, exactly one of $L,R,H,V$ sends the labeled vertices back to their original positions. Thus every sequence of the first $19$ transformations has exactly one valid final transformation.

There are $4^{19}=2^{38}$ choices for the first $19$ transformations, so there are $2^{38}$ valid sequences.

Thus, C is the correct answer.

2020 AMC 10B 第 23 题

答案：C

难度评级：2060

视频讲解：

文字解答：

其他年份的第 23 题