AMC10 2020 A

AMC10 2020 A · Q19

AMC10 2020 A · Q19. It mainly tests Recursion & DP style counting (basic), Geometry misc.

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?

如图所示，一个正十二面体（由12个全等正五边形面组成的多面体）在空间中漂浮，有两个水平面。注意，顶部面相邻有5个倾斜面组成的环，底部面相邻也有5个倾斜面组成的环。从顶部面到底部面，通过相邻面的序列移动，有多少种方法，使得每个面至多访问一次，且不允许从底部环移动到顶部环？

(A) 125 125

(B) 250 250

(D) 640 640

(E) 810 810

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): Shown below is a planar graph representation of the dodecahedron with the top face in the center and the edges of the bottom face forming the border of the figure. Starting from the top face (the inner pentagon in this figure), there are 5 possible moves down to the next level (the top ring, which consists of the unshaded pentagons in the figure). For the next few moves there are $1+2\cdot 4=9$ possibilities because one can move directly down to the next lower level (the bottom ring), or one can remain on the same level and make 1 to 4 moves in either direction around the top ring before moving down. When the path moves to the bottom ring, there are 2 possible faces to choose from. Then again there are 9 possibilities for how the path moves before reaching the bottom face. This gives a total of $5\cdot 9\cdot 2\cdot 9=810$ possible paths from the top face to the bottom face.

答案（E）：如下所示，这是一个十二面体的平面图表示：顶面位于中心，而底面的边构成图形的边界。从顶面开始（本图中的内五边形），向下移动到下一层（顶环，由图中未阴影的五边形组成）有 5 种可能的走法。接下来的几步共有 $1+2\cdot 4=9$ 种可能性，因为可以直接向下移动到下一层（底环），也可以停留在同一层，并在顶环上沿任一方向绕行 1 到 4 步后再向下移动。当路径移动到底环时，有 2 个可能的面可选。然后在到达底面之前，路径的移动方式又有 9 种可能。因此，从顶面到底面的路径总数为 $5\cdot 9\cdot 2\cdot 9=810$。

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