2007 AMC 10A 第 11 题

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

立方体每个顶点恰好在三个面上。

Each vertex of a cube lies on exactly three faces

将六个面的和相加，会把每个顶点数计算

3

次：

3(1 + 2 + \cdots + 8)

。

Adding all six face sums counts each vertex

3

times:

3(1 + 2 + \cdots + 8)

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}\text{。}

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.

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