AMC10 2007 A

AMC10 2007 A · Q11

AMC10 2007 A · Q11. It mainly tests Basic counting (rules of product/sum), Area & perimeter.

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?

将数字 $1$ 到 $8$ 放置在立方体的顶点上，使得每个面上的四个数字之和相同。这个公共和是多少？

(A) 14 14

(B) 16 16

(D) 20 20

(E) 24 24

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Each vertex appears on exactly three faces, so the sum of the numbers on all the faces is $3(1+2+\cdots+8)=3\cdot\frac{8\cdot 9}{2}=108.$ There are six faces for the cube, so the common sum must be $108/6=18$. A possible numbering is shown in the figure.

答案（C）：每个顶点恰好出现在三个面上，因此所有面上的数字之和为 $3(1+2+\cdots+8)=3\cdot\frac{8\cdot 9}{2}=108.$ 立方体有六个面，所以每个面的公共和必须是 $108/6=18$。图中给出了一种可能的编号方式。

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