Question

蚂蚁 Erin 从立方体的一个指定顶点出发，沿着恰好七条棱爬行，正好访问每个顶点一次，然后发现无法沿一条棱回到起点。有多少条路径满足这些条件？

Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?

$6$

$9$

$12$

$18$

$24$

答案：A

知识点：图论正方体分类讨论

难度评级：2030

解答：

前两条棱可用 $3\cdot2=6$ 种方式选择。这两条棱确定了立方体的一个起始面。

下一步必须访问该面上唯一尚未访问的顶点；否则以后到达它时，它的所有相邻顶点都已经访问过，路径将无法继续。剩余四个顶点都在对面，可以按两个环绕方向访问。

这两个方向中，恰有一个会终止在与起点不相邻的顶点。因此共有 $6$ 条合格路径。

所以正确答案是 A。

The first two edges can be chosen in $3\cdot2=6$ ways. These two edges determine an initial face of the cube. After those moves, there is one unvisited vertex on that initial face.

That remaining vertex must be visited next; otherwise Erin would later reach it after all of its neighbors had already been visited, and the path could not continue. The last four vertices are then on the opposite face and can be visited in two cyclic orders.

Of those two orders, exactly one ends at a vertex not adjacent to the starting point. Hence there are $6$ valid paths.

Thus, the correct answer is A.

2015 AMC 10B 第 20 题

答案：A

难度评级：2030

解答：

其他年份的第 20 题