Question

将 $1$ 到 $8$ 的数字放在一个立方体的顶点上，使每个面上四个数字之和相同。这个共同和是多少？

The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?

$14$

$16$

$18$

$20$

$24$

答案：C

知识点：双重计数正方体

难度评级：1280

解答：

每个顶点恰好属于三个面，所以把六个面的数字和加起来得到 $\begin{aligned} 3(1 + 2 + \cdots + 8) &= 3 \cdot 36 \\ &= 108. \end{aligned}$

共有六个面，所以共同和为 $108 \div 6 = 18$ 。

所以正确答案是 C。

Each vertex belongs to exactly three faces, so summing the numbers over all six faces gives $\begin{aligned} 3(1 + 2 + \cdots + 8) &= 3 \cdot 36 \\ &= 108. \end{aligned}$

There are six faces, so the common sum is $108 \div 6 = 18.$

Thus, the correct answer is C.

2007 AMC 10A 第 11 题

答案：C

难度评级：1280

解答：

其他年份的第 11 题