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 $1 imes 1 imes 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?

A. 12

B. 16

C. 18

D. 20

E. 24

1. 2020 AMC 8 官方解析

The cuts divide the cake into four $4 \times 4 imes 1$ inch horizontal layers. In the top layer there are $8$ pieces having exactly two sides with icing, namely the non-corner edge pieces that are part of both the top face and a side face. In each of the next $3$ layers, the $4$ corner pieces are the only ones having two sides with icing. This gives a total of $8 + 3 \cdot 4 = 20$ small pieces with icing on exactly two sides.
切口将蛋糕分成了四个$4 \times 4 \times 1$的水平层. 在顶层中，有$8$块恰好在两侧带有糖霜，即非角边块，它们既是顶面又是侧面的一部分. 在接下来的每个$3$层中，只有$4$个角边块具有两个带有糖霜的侧面. 由此可得总共有$8 + 3 \cdot 4 = 20$ 个恰好两面有糖霜的小方块.
First consider the $8$ edges of the cake that are not along the bottom face. Each of those edges has $2$ middle pieces with icing on exactly two sides. In addition to these pieces, the $4$ corner pieces of the bottommost layer also have two sides with icing. This gives a total of $8 \cdot 2 + 4 = 20$ small pieces with icing on exactly two sides.