Question

如下图所示，一个正十二面体，也就是由 $12$ 个全等正五边形面组成的多面体，漂浮在空间中，并有两个水平面。注意，顶面相邻有一圈五个倾斜面，底面相邻也有一圈五个倾斜面。从顶面出发，经由一系列相邻面移动到底面，且每个面至多访问一次，并且不允许从底部环移动到顶部环。这样的走法有多少种？

As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?

$125$

$250$

$405$

$640$

$810$

答案：E

知识点：图论乘法原理分类讨论

难度评级：2460

视频讲解：

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文字解答：

离开顶面后，先选择 $5$ 个顶部环面之一。由于不允许从底部环回到顶部环，每条有效路径都由顶部环阶段、一次向下移动、底部环阶段组成。

固定第一个顶部环面。在顶部五环上，可以不重复地沿五环走若干步后停止；选择不动有一种，选择一个方向后可走一到四步，所以共有 $1+2\cdot4=9$ 种顶部环路径。停止处有 $2$ 种向下移动选择，因此顶部部分有 $18$ 种。

到达底部环后，同理有 $1+2\cdot4=9$ 种方式在底部五环上不重复移动并进入底面。总数为 $5\cdot18\cdot9=810$ 。正确答案是 E。

After leaving the top face, choose one of the $5$ top-ring faces. Because moves from the bottom ring to the top ring are forbidden, every valid path has a top-ring phase, then one move down to the bottom ring, then a bottom-ring phase.

Fix the first top-ring face. On the top ring, the path can move around the 5-cycle without revisiting a face and then stop at any point: there are $1+2\cdot4=9$ possible top-ring paths. From the stopping face, there are $2$ possible downward moves to the bottom ring, so the top part has $18$ choices.

Once in the bottom ring, the path can move around the bottom 5-cycle without revisiting a face and then enter the bottom face; this gives $1+2\cdot4=9$ choices. The total is $5\cdot18\cdot9=810$ . Thus, E is the correct answer.

2020 AMC 10A 第 19 题

答案：E

难度评级：2460

视频讲解：

文字解答：

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