AMC8 2020

AMC8 2020 · Q9

AMC8 2020 · Q9. It mainly tests 3D geometry (surface area).

Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?

Akash 的生日蛋糕是一个 $4 \times 4 \times 4$ 英寸的立方体。蛋糕顶部和四个侧面有糖霜，底部没有糖霜。假设蛋糕被切成 64 个 $1 \times 1 \times 1$ 英寸的小立方体，如下图所示。有多少个小块正好有两个面有糖霜？

(A) 12 12

(B) 16 16

(D) 20 20

(E) 24 24

Answer

Correct choice: (D)

正确答案：（D）

Solution

Notice that, for a small cube which does not form part of the bottom face, it will have exactly $2$ faces with icing on them only if it is one of the $2$ center cubes of an edge of the larger cube. There are $12-4 = 8$ such edges (as we exclude the $4$ edges of the bottom face), so this case yields $2 \cdot 8 = 16$ small cubes. As for the bottom face, we can see that only the $4$ corner cubes have exactly $2$ faces with icing, so the total is $16+4 = \boxed{\textbf{(D) }20}$.

注意，对于不属于底面的小立方体，只有当它是较大立方体边缘的 $2$ 个中心立方体之一时，才正好有两个面有糖霜。有 $12-4 = 8$ 条这样的边（排除底面的 $4$ 条边），所以此情况产生 $2 \cdot 8 = 16$ 个小立方体。至于底面，只有 $4$ 个角立方体正好有两个面有糖霜，因此总数为 $16+4 = \boxed{\textbf{(D) }20}$。

Topics

GE.014 3D geometry (surface area)