AMC10 2002 A

AMC10 2002 A · Q18

AMC10 2002 A · Q18. It mainly tests 3D geometry (volume), 3D geometry (surface area).

A $3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $3 \times 3 \times 3$ cube is

一个 $3 \times 3 \times 3$ 立方体由粘合在一起的 27 个标准骰子构成。（标准骰子上，任意一对相对面的数字和为 7。）$3 \times 3 \times 3$ 立方体表面上所有数字的最小可能总和是

(A) 60 60

(B) 72 72

(D) 90 90

(E) 96 96

Answer

Correct choice: (D)

正确答案：（D）

Solution

(D) There are six dice that have a single face on the surface, and these dice can be oriented so that the face with the 1 is showing. They will contribute $6(1)=6$ to the sum. There are twelve dice that have just two faces on the surface because they are along an edge but not at a vertex of the large cube. These dice can be oriented so that the 1 and 2 are showing, and they will contribute $12(1+2)=36$ to the sum. There are eight dice that have three faces on the surface because they are at the vertices of the large cube, and these dice can be oriented so that the 1, 2, and 3 are showing. They will contribute $8(1+2+3)=48$ to the sum. Consequently, the minimum sum of all the numbers showing on the large cube is $6+36+48=90$.

（D）有六个小骰子只有一个面在外表面上，这些骰子可以调整朝向，使得点数为 1 的面朝外显示。它们对总和的贡献为 $6(1)=6$。有十二个小骰子有两个面在外表面上，因为它们位于大立方体的棱上但不在顶点处。这些骰子可以调整朝向，使点数 1 和 2 的面朝外显示，它们对总和的贡献为 $12(1+2)=36$。有八个小骰子有三个面在外表面上，因为它们位于大立方体的顶点处，这些骰子可以调整朝向，使点数 1、2、3 的面朝外显示。它们对总和的贡献为 $8(1+2+3)=48$。因此，大立方体外表面所有可见点数之和的最小值为 $6+36+48=90$。

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