Question

一个整数被分配给立方体的每个顶点。一条边的值定义为它所连接的两个顶点的值之和，一个面的值定义为围成该面的四条边的值之和。立方体的值定义为六个面的值之和。若分配给各顶点的整数之和为 $21$ ，这个立方体的值是多少？

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21.$ What is the value of the cube?

$42$

$63$

$84$

$126$

$252$

答案：D

知识点：双重计数正方体

难度评级：1270

解答：

按关联次数计数。每条边在 $2$ 个面上，所以六个面的值之和是所有边值总和的 $2$ 倍。每个顶点在 $3$ 条边上，所以所有边值总和是顶点值总和的 $3$ 倍。连起来，立方体的值为 $2 \cdot 3 \cdot 21 = 126$ 。因此，答案是 D。

Count by incidences. Each edge lies on $2$ faces, so the six face values together are $2$ times the total of all edge values. Each vertex lies on $3$ edges, so the total edge value is $3$ times the vertex sum. Chaining these, the cube's value is $2 \cdot 3 \cdot 21 = 126.$ Therefore, the answer is D.

2023 AMC 10A 第 6 题

答案：D

难度评级：1270

解答：

其他年份的第 6 题